---
title: "What can one wind line determine?"
description: "Infer from saved integer photons, derive the mass-flow and ion-opacity degeneracy, and test a proper two-parameter prior with independent repetitions."
canonical_url: "https://sandboxphysics.com/stories/stellar-winds/what-a-wind-line-can-determine/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/stellar-winds/what-a-wind-line-can-determine/"
last_modified: "2026-11-03"
---

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

Stellar winds · L3

# What can one wind line determine?

Sandbox Physics · 2026-11-03

One spectrum can tightly constrain a combination of parameters while barely separating its ingredients. Starting with saved synthetic photons, we infer a mass-flow and ion-opacity product, then examine why mass flow remains broad. Every probability result is conditional on explicit geometry, detector calibration and priors.

## Before you begin

Read the resonance-surface chapter. Use integrals and probability density; Poisson counts, prior transformations and marginalization are explained as they arise.

## Start from saved integer photons

The exercise saves integer counts in 256 velocity bins, their width, continuum-photon calibration and instrument response. This is a synthetic photon detector. IUE ABS\_CAL is processed energy flux and cannot become a Poisson count record merely by renaming a column.

First calculate the wind photon spectrum, convolve it with the fixed instrument Gaussian and integrate each bin. Generate counts only afterward. This permits the instrument to distribute photons among adjacent bins. Adding noise before arbitrary smoothing would produce a different noise correlation.

The conditional example fixes beta exponent one, asymptotic speed 200 kilometres per second, aperture eight stellar radii, instrument standard deviation eight kilometres per second and continuum calibration 1600 photons per kilometre per second. Instrument width is separate from thermal wind width.

$$
\begin{aligned}\lambda_i(\theta)&=C_N\int_{\mathrm{bin}\,i}[G*f_\theta](u)\,du\\N_i&\sim\mathrm{Poisson}(\lambda_i)\end{aligned}
$$

$f_\theta$ is the wind photon spectrum, $G$ has unit integral and $C_N$ is continuum photons per velocity unit. The bins are conditionally independent under the declared Poisson-arrival and independent-assignment model.

Reading: Standalone stellar-wind scientific reproduction

## Write the probability of those counts

A Poisson distribution describes the count obtained when the mean arrival number is known. For mean $\lambda_i$, the probability of observing $N_i$ photons follows the expression below. Multiply the conditional probabilities across bins to obtain the likelihood of the whole saved spectrum.

Taking logarithms changes products into sums. The factorial term is constant across parameters for one fixed dataset and may be omitted from relative parameter comparisons. It remains part of the original probability distribution. Square-root-count error bars are a large-count approximation; inference here retains the Poisson expression.

infer\_saved\_counts receives saved counts and the means of a declared model family. Generating truths are saved separately and read afterward for coverage validation. Editing a truth annotation cannot change the inferred result.

$$
\mathcal L(\theta)=\prod_i\frac{e^{-\lambda_i(\theta)}\lambda_i(\theta)^{N_i}}{N_i!}
$$

$N_i!$ is an integer factorial. Unknown continuum, response or geometry would require additional parameters or calibration uncertainty; they are fixed in this exercise.

Reading: Standalone stellar-wind scientific reproduction

## Two physically different flows can have the same line

Define mass-flow multiplier $m=\dot M/\dot M_{\rm ref}$ and relative line-opacity multiplier $q$. The latter can represent relative ionization, abundance or level-population changes; it is not an absolute ion fraction. At fixed velocity and geometry, optical depth depends on their product $\theta=mq$.

Mass multiplier one with opacity multiplier one therefore gives the same photon profile as half the mass flow with twice the opacity. With the same noise realization, even the saved integer bins coincide. Density differs, but this line has not measured it separately.

The force diagnostic cannot casually be promoted to an extra observation that breaks the degeneracy. The flow is prescribed, the one-line force is insufficient and other forces are absent. Adding a dynamical constraint requires the relevant physics and evidence.

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

## Transform the proper two-parameter prior

Assign independent LogUniform priors from 0.5 to 2 to both $m$ and $q$. Equal intervals in their logarithms have equal prior probability. With $x=\ln m$, $y=\ln q$ and $a=\ln2$, both log coordinates are uniform on $[-a,a]$.

Returning to linear coordinates requires a Jacobian. The resulting joint density contains the two parameters in its denominator. Dropping that factor would select a different prior.

Now set $s=x+y=\ln\theta$. The number of allowed parameter pairs at a given sum is proportional to a diagonal segment through the log-coordinate square. Segments are longest centrally and vanish at the ends, producing a triangular log-product prior. Linear product density requires an additional division by $\theta$.

Multiply likelihood by this prior and normalize to form the posterior. A mass-flow marginal integrates over all allowed opacity multipliers. Fixing opacity to one instead asks a different conditional question.

$$
\begin{aligned}p(m,q)&=\frac{1}{4a^2mq}\\0.5&\le m,q\le2\end{aligned}
$$

This is density per $dm\,dq$ for two independent log-uniform parameters, normalized to one.

$$
\begin{aligned}p_s(s)&=\frac{2a-|s|}{4a^2},\quad|s|\le2a\\p_\theta(\theta)&=\frac{p_s(\ln\theta)}{\theta}\end{aligned}
$$

Full product support is 0.25 to 4. Prior volume determines product weights and the distribution along the degenerate ridge.

$$
p(x\mid N)\propto\int_{x-a}^{x+a}\mathcal L(e^s)\,ds
$$

This integrates over uniform $y$ at fixed $x$. A direct two-dimensional posterior integration independently checks the marginal calculation.

Reading: Standalone stellar-wind scientific reproduction

## Work through the saved observation

The default record contains 872258 photons. Its calibrated absorption-positive velocity equivalent width is about 14.84 kilometres per second. Counting noise alone gives an estimated standard deviation about 0.584 kilometres per second, conditional on the known continuum. Continuum and physical-model uncertainty are additional questions.

Standalone Python recomputes transfer, convolution and binning. Across the full product support, the total mean discrepancy from finer calculations is about 0.009 Poisson-noise units. Product grids with 1201 and 2401 points change interval endpoints by less than 0.6 percent of the interval width.

The product posterior has median about 0.991 and central 95 percent interval about 0.974 to 1.009. The mass-flow multiplier interval remains about 0.518 to 1.914, nearly spanning its original 0.5 to 2 range. The opacity marginal is identical because the likelihood and priors are symmetric under exchanging the two parameters.

The narrow ridge in the figure is a collection of answers with almost the same product. Its wide mass-flow range and its boundaries depend strongly on the prior. This is not a separate precise mass-loss measurement.

![Saved photons, a two-parameter product ridge and correctly transformed posterior marginals](https://sandboxphysics.com/stories/stellar-winds/materials/conditional-count-inference.png)

Top panels show saved counts with the independent mean and the posterior in log coordinates. Bottom panels give product and mass densities in linear coordinates, including the transformation factors. The cyan cross is a separately declared generating truth, excluded from inference.

```
python3 conditional_count_inference.py --measurement measurement/synthetic-photon-measurement.json --out-dir rerun
```

Reading: Standalone stellar-wind scientific reproduction

## Test prior-averaged and fixed-truth coverage separately

A central 95 percent credible interval is a posterior statement given data and prior: 2.5 percent probability lies in each excluded tail. It does not automatically promise 95 percent repeated-experiment coverage at every fixed true parameter.

The first check draws true mass and opacity independently from the stated priors, then generates 240 independent Poisson observations. Product intervals cover 232 times; mass-flow intervals cover 227 times. Binomial uncertainty accompanies those counts. This tests coverage averaged over the declared prior.

The second check fixes three parameter pairs: $(m,q)=(1,1)$, $(0.51,0.51)$ and $(1.99,0.51)$. Each has 100 independently generated records. Product coverage is 94, 95 and 95; mass-flow coverage is 100, 100 and zero. Original counts and these failures are retained.

The final pair has product about 1.015, close to the central case, but mass near its prior boundary. The spectrum constrains the product; prior volume and marginalization allocate mass along the ridge. A central interval can repeatedly exclude the fixed mass value 1.99 after trimming its boundary tails.

Zero coverage in that case is not an unlucky precise mass-loss measurement. It exposes non-identifiability and failure of a universal fixed-parameter coverage interpretation. Both the prior-averaged and fixed-truth results belong beside the posterior plot.

Reading: Standalone stellar-wind scientific reproduction

## Identify the additional conditions needed for real data

Real continuum, response, ion populations, velocity law and interstellar absorption may be unknown, and lines can overlap. They change the predicted mean, its errors and parameter relationships. Reducing synthetic counting noise cannot by itself remove a product degeneracy.

IUE energy flux and its FN noise are at different calibration stages. They do not provide this exercise with independent Poisson counts and a complete physical-error vector. The next tutorial retains native orders, quality gaps and continuum sensitivity, then compares shapes under explicit isolated-line conditions. Real mass-loss work needs multiple diagnostics and a corresponding atmosphere model.

Reading: IUE NEWSIPS manual, high-dispersion MXHI format · IUE data-analysis FAQ · Brands, Backs, de Koter et al.2025, X-Shooting ULLYSES XII

## Exercises and explanations

1. Why does multiplying two independent log-uniform parameters not give a flat product prior?

   Read the explanation

   Different products admit different numbers of parameter pairs. Diagonal segments through the log-prior square are longest centrally and shortest at the ends, giving triangular log-product density. Transforming back to linear product adds division by that product.

2. Does zero fixed-mass coverage in 100 trials prove that numerical photon means are inaccurate?

   Read the explanation

   No. The mean differences from refined references are far below Poisson noise here. The spectrum identifies only the product, and the fixed mass lies near a prior boundary. Central credible intervals do not guarantee pointwise 95 percent coverage. Retain this outcome beside prior-averaged coverage.

3. Fixing opacity to one would give a narrow mass-flow interval. Does that contradict the broad marginal?

   Read the explanation

   No. Fixing opacity adds information, making the product directly equal to the mass multiplier. Integrating uncertain opacity retains many pairs that explain the same spectrum. The intervals answer different conditional questions.

## Try the prediction

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

1. Save the default observation. Halve mass flow and double opacity, checking the unchanged photon profile at the same noise realization despite different density.
2. Increase mass flow around the saturated preset. Find the spectral regions whose response to added material weakens.
3. Run the independent count exercise. Compare the ridge and both marginals before reading prior-averaged and fixed-truth coverage tables separately.

[Previous: Resonance surfaces in an accelerating wind](https://sandboxphysics.com/stories/stellar-winds/resonance-surfaces-in-an-accelerating-wind/) [Next: Replaying the IUE ultraviolet spectra of P Cygni](https://sandboxphysics.com/stories/stellar-winds/replay-p-cygni-iue-spectra/)

## Sources and calculation scope

1. [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.

2. [IUE NEWSIPS manual, high-dispersion MXHI format](https://archive.stsci.edu/iue/manual/newsips/node155.html)

   Native17-field binary rows, padded768-element vectors, one-based STARTPIX, vacuum heliocentric wavelengths, ABS\_CAL physical units and FN noise.

3. [IUE data-analysis FAQ](https://archive.stsci.edu/iue/faq.html)

   Questions13–15: reject nonzero quality flags for quantitative use; high-dispersion cosmic rays are not automatically removed; exposure alone does not define a noise weight.

4. [Brands, Backs, de Koter et al.2025, X-Shooting ULLYSES XII](https://arxiv.org/html/2503.14687v1)

   Pinned v1; sections3/5.3–5.5, Figures1/4/7 and Tables7/8. Twenty-five LMC O-type targets yield24parameter sets. Their model-conditional population/fitness results are external, not P Cygni measurements.

5. [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.