---
title: "Resonance surfaces in an accelerating wind"
description: "Derive the projected gradient, Sobolev optical depth and scattering source, then calculate one stellar-disk ray and its contribution to a wind spectrum."
canonical_url: "https://sandboxphysics.com/stories/stellar-winds/resonance-surfaces-in-an-accelerating-wind/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/stellar-winds/resonance-surfaces-in-an-accelerating-wind/"
last_modified: "2026-11-03"
---

[Reading a stellar wind](https://sandboxphysics.com/stories/stellar-winds/)

Stellar winds · L2

# Resonance surfaces in an accelerating wind

Sandbox Physics · 2026-11-03

A received frequency usually resonates with only certain positions in a wind. Those positions share a projected velocity and form a resonance surface. Combining that surface with the stellar disk, occultation and aperture takes us from a prescribed flow to a P Cygni spectrum.

## Before you begin

Read the photon-momentum chapter first. Use derivatives, integrals and a geometric picture of spherical coordinates; the viewing projection and local approximation are developed below.

## Define the viewing direction and boundaries

Place the observer on positive $z$. Gas flows radially outward, at distance $r$ from the centre. A ray has projected impact parameter $p$, with $\mu=z/r$. Display received Doppler velocity $u$ with red positive, so approaching near-side material has negative $u$.

The line-forming calculation extends from the photosphere to eight stellar radii. A prescribed beta velocity law sets the speed, with exponent $\beta_{\rm w}$ controlling the acceleration shape. Changing the exponent also changes $b$ so the base speed remains five percent of the asymptotic speed.

For $\beta_{\rm w}=1$ and asymptotic speed 200 kilometres per second, the outer boundary reaches only 176.25 kilometres per second. A finite-volume edge and an infinite-radius asymptote are different quantities. Intrinsic width, ion populations and observational processing also affect a spectral blue edge.

$$
\begin{aligned}u&=-\mu v(r)\\v(r)&=v_\infty\left(1-\frac{bR_\star}{r}\right)^{\beta_{\rm w}}\\b&=1-0.05^{1/\beta_{\rm w}}\end{aligned}
$$

The velocity law is an input. $R_\star$ is photospheric radius, $v_\infty$ is asymptotic speed, and $\beta_{\rm w}$ is a shape exponent, separate from escape probabilities.

Reading: Lamers, Cerruti-Sola & Perinotto1987, the SEI line-profile method · Standalone stellar-wind scientific reproduction

## Carry the same mass through every shell

Steady spherical flow carries the same mass through every shell each second. Shell area multiplied by density and radial speed gives that mass-flow rate. Solving for density shows how expansion and acceleration both dilute the wind.

Double radius and double speed at fixed mass flow, and density falls by a factor of eight. Residence time tells the same story: slower material leaves more mass in a shell. The weighted tracer distribution can therefore follow the same continuity law as the transfer calculation.

Density must update when the terminal-speed scale changes. Stretching a previously drawn spectrum while leaving the matter unchanged would disconnect the spectrum from its physical flow.

$$
\dot M=4\pi r^2\rho(r)v(r),\qquad\rho(r)=\frac{\dot M}{4\pi r^2v(r)}
$$

In SI, $\dot M$ is kilograms per second, $r$ is metres and $v$ is metres per second. Density is kilograms per cubic metre.

Reading: Owocki2014, Theory of Winds from Hot, Luminous Massive Stars

## The gradient along a ray includes geometry

Moving along a fixed-impact ray changes both speed and the angle between the radial velocity and the sightline. Differentiating $v\mu$ must retain both changes. Geometry gives $dr/dz=\mu$ and $d\mu/dz=(1-\mu^2)/r$.

The chain rule then gives the directional gradient below. A radial ray mainly samples radial acceleration. A nearly tangential ray instead depends strongly on the changing projection of the spherical flow. A slowly changing speed magnitude does not imply a constant projected velocity.

The declared monotonic outflow has positive gradient, so a given frequency has at most one local resonance along a ray. A nonmonotonic wind can connect several resonance positions by radiation and requires a different transfer treatment.

$$
D(\mu,r)=\frac{d(v\mu)}{dz}=\mu^2\frac{dv}{dr}+(1-\mu^2)\frac vr
$$

$D$ has inverse-second units. Since $du/dz=-D$, optical depth uses its positive magnitude.

Reading: Lamers, Cerruti-Sola & Perinotto1987, the SEI line-profile method

## Integrate the narrow resonance layer

Describe the intrinsic line with $\phi(u)$, normalized to unit integral over velocity. Let $\kappa_V$ be opacity integrated over that velocity coordinate, with SI units $\mathrm{m^3\,kg^{-1}\,s^{-1}}$. The local extinction coefficient $\kappa_V\rho\phi$ then has inverse-length units.

If resonance occupies a sufficiently thin layer, density, directional gradient and source can be treated as constant across it. Change the transfer integral from position to projected velocity. The unit line-profile integral leaves opacity times density divided by the directional gradient: the Sobolev optical depth.

Normalizing radius and velocity by the stellar radius and asymptotic speed gives the scale $T$. Mass flow and line opacity appear in its numerator; speed enters squared in the denominator. This is the origin of the coupled width and depth response when the flow is accelerated.

$$
\int\phi(u)\,du=1,\qquad\tau(\mu,r)=\frac{\kappa_V\rho}{D(\mu,r)}
$$

$\kappa_V\rho$ and $D$ both have inverse-second units, so $\tau$ is dimensionless. Thermal or microscopic turbulent width is separate from instrument width.

$$
T=\frac{\kappa_V\dot M}{4\pi R_\star v_\infty^2}
$$

Reference opacity can be multiplied by a relative ion/opacity factor. An absolute ion-number fraction is bounded by one; this factor instead multiplies a reference opacity.

Reading: Lamers, Cerruti-Sola & Perinotto1987, the SEI line-profile method · Owocki2014, Theory of Winds from Hot, Luminous Massive Stars

## Calculate the scattering source

The star occupies a finite solid angle as seen from the gas. Its limb defines direction cosine $\mu_\star$. Radiation arriving from that cone supplies photons; local reabsorption can scatter them repeatedly before they escape the resonance layer.

For an isotropic conservative isolated line, define mean escape probability $B_{\rm esc}$ and the stellar illumination term $B_\star$. Local scattering balance is $S=(1-B_{\rm esc})S+B_\star I_c$. Rearranging gives the source relative to the uniform stellar continuum intensity.

In the thin limit, escape tends to unity and the source reduces to geometric dilution. Thick-line escape depends on the directional gradients. At the photospheric boundary the star fills a hemisphere; even angular opacity then makes the source exactly half the stellar intensity. That supplies a useful analytic check.

$$
\begin{aligned}\mu_\star&=\sqrt{1-R_\star^2/r^2}\\B_{\rm esc}&=\frac12\int_{-1}^{1}\frac{1-e^{-\tau(\mu)}}{\tau(\mu)}\,d\mu\end{aligned}
$$

$I_c$ is photon continuum intensity of the uniform photosphere. Escape uses the full directional optical depth.

$$
\begin{aligned}B_\star&=\frac12\int_{\mu_\star}^{1}\frac{1-e^{-\tau(\mu)}}{\tau(\mu)}\,d\mu\\S/I_c&=B_\star/B_{\rm esc}\end{aligned}
$$

This is the specified local Sobolev scattering balance. Complete non-LTE populations and finite-width comoving transfer are outside this calculation.

Reading: Lamers, Cerruti-Sola & Perinotto1987, the SEI line-profile method · Standalone stellar-wind scientific reproduction

## Work through a complete stellar-disk ray

Choose $\beta_{\rm w}=1$ and $T=0.2$. Examine a near-side point at twice the photospheric radius with $\mu=0.9$. Speed is 0.525 times the asymptotic value, and impact parameter is about 0.872 stellar radii. The ray therefore really crosses the visible stellar disk. At asymptotic speed 200 kilometres per second, its red-positive velocity is minus 94.5 kilometres per second.

In stellar-radius and asymptotic-speed units, the radial derivative is 0.2375 and speed divided by radius is 0.2625. Their directional combination is 0.24225. Density and opacity then give optical depth about 0.39314. Independent angular integration gives source ratio 0.06634.

Normalize incoming disk intensity to one. Direct transmission is about 0.67493; adding local scattering gives outgoing intensity about 0.69650. Scattering fills part of the absorption. This is one ideal ray before the full wind integration and instrumental response.

Place its counterpart on the rear side and the same impact parameter lies behind the opaque disk. That path is blocked. At greater impact parameters, rear-side light can pass beside the star and contribute to red emission.

$$
I_{\rm out}=I_{\rm in}e^{-\tau}+S(1-e^{-\tau})
$$

A foreground disk ray has $I_{\rm in}=I_c$; an off-disk ray has zero incident background. Occultation additionally tests whether the observer path hits the photosphere.

Reading: Standalone stellar-wind scientific reproduction · Lamers, Cerruti-Sola & Perinotto1987, the SEI line-profile method

## Add the rays into a spectrum

At each frequency, integrate outgoing intensity over projected annuli. Their area supplies the impact-parameter weight, and the uniform stellar disk supplies the continuum normalization. An aperture containing the whole disk has continuum one outside the line.

Scattered photons can be captured, returned to the opaque star or missed with the halo. With one continuum normalization and dimensionless velocity integral, these three areas sum to the area removed from direct starlight. Absorption-positive equivalent width equals the returned and missed areas multiplied by the velocity scale.

A full halo aperture removes the missed term but retains stellar interception. This relation concerns photon number under a flat continuum and the stated Doppler convention. Arbitrary wavelength equivalent widths and exact energy flux require their own conversion and work terms.

$$
f(u)=2\int_0^{p_{\rm ap}/R_\star}\frac{I(u,\hat p)}{I_c}\hat p\,d\hat p,\qquad\hat p=p/R_\star
$$

The uniform disk has $2\int_0^1\hat p\,d\hat p=1$. Every ray uses the same flow and scattering source.

$$
W_v=\int[1-f(u)]\,du=v_\infty(A_{\rm back}+A_{\rm miss})
$$

$A_{\rm back}$ and $A_{\rm miss}$ are returned and missed photon areas normalized over $du/v_\infty$. A saved Poisson observation fluctuates around the model result.

Reading: Lamers, Cerruti-Sola & Perinotto1987, the SEI line-profile method · Standalone stellar-wind scientific reproduction

## Check discretization and physical approximation separately

With intrinsic Doppler width $v_D$, resonance thickness is approximately $v_D/D$. It must be small compared with density and source variation scales and with the distance to a boundary. Comparing only with stellar radius can overlook rapidly changing illumination close to the photosphere.

The independent finite-width check holds the source fixed and solves the transfer differential equation. At width 0.00005 times the asymptotic speed, the largest difference across four rays is about $3.9\times10^{-5}$. Width 0.05 already gives a counterexample near 0.033. This tests formal integration; finite-width source accuracy still requires a fuller calculation.

That intrinsic Doppler width is defined by $e^{-(\Delta u/v_D)^2}$. The fixed eight-kilometre-per-second Gaussian standard deviation belongs to the instrument, applied after wind transfer. Interchanging those widths changes both the physical and measurement models.

The production solver is also checked against independent impact-ray integration over 197 presets, corners, faces and interior configurations. Its maximum detector-profile difference is about 0.072 percent of continuum. Runtime checks compare refined spectra, sources and forces. These tests measure numerical differences within the declared calculation, with finite-width and non-LTE approximations retaining separate tests.

Reading: Lamers, Cerruti-Sola & Perinotto1987, the SEI line-profile method · Standalone stellar-wind scientific reproduction

## Exercises and explanations

1. What is lost by using the radial velocity derivative on every sightline, especially near a tangent?

   Read the explanation

   The changing direction of spherical flow contributes a projection term. Near a tangent the radial-acceleration term has little weight and the geometric term can dominate. Dropping it miscalculates resonance length and optical depth.

2. Why did the worked ray explicitly check that its impact parameter is smaller than the photospheric radius?

   Read the explanation

   That ensures a bright stellar disk lies behind the near-side resonance, permitting incoming intensity one. An off-disk ray instead starts with zero background and has only the scattered contribution.

3. Should closing the aperture change the local source function or the collection integral?

   Read the explanation

   Only the collection integral changes here. Aperture does not alter stellar illumination or local scattering; it excludes projected annuli. The same flow and source therefore produce greater missed-halo area.

## Try the prediction

### [Change the flow and follow its scattered light](https://sandboxphysics.com/labs/stellar-wind-p-cygni/)

1. Compare blue and red resonance surfaces. Predict which path intersects the disk from its impact parameter, then find the marked occultation.
2. Change acceleration shape alone, then speed scale alone. Predict the resonance positions, density and depth response separately.
3. Close the aperture from the complete halo toward the stellar disk. Explain the rising missed term while the local source stays unchanged.

[Previous: How much momentum can light carry?](https://sandboxphysics.com/stories/stellar-winds/photon-momentum-and-wind-energy/) [Next: What can one wind line determine?](https://sandboxphysics.com/stories/stellar-winds/what-a-wind-line-can-determine/)

## Sources and calculation scope

1. [Owocki2014, Theory of Winds from Hot, Luminous Massive Stars](https://arxiv.org/html/1409.2084v1)

   Pinned v1; section2 for CAK/Sobolev steady winds and sections3.1–3.4 for line instability, clumping, porosity and velocity gaps. Linked theory source; original is not redistributed.

2. [Lamers, Cerruti-Sola & Perinotto1987, the SEI line-profile method](https://adsabs.harvard.edu/pdf/1987ApJ...314..726L)

   ApJ314,726, sectionsII–V and equations4–10/14–19. Our local escape singlet and fixed-source finite-width test retain their separate approximations; no complete SEI/non-LTE solver is claimed.

3. [Standalone stellar-wind scientific reproduction](https://sandboxphysics.com/stories/stellar-winds/materials/stellar-wind-science.zip)

   Original small data, independent transfer/photon/force/finite-width references,197producer checks and a proper synthetic Poisson exercise. Fixed truths and prior-averaged coverage remain separate.