---
title: "Recompute transmission from one fixed SDSS sightline"
description: "Follow a fixed SDSS quasar coadd through masks, a continuum fit and conditional flux statistics, then measure how continuum assumptions move the answer."
canonical_url: "https://sandboxphysics.com/stories/lyman-alpha-forest/recompute-a-fixed-sdss-sightline/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/lyman-alpha-forest/recompute-a-fixed-sdss-sightline/"
last_modified: "2026-11-05"
---

[The Lyman-alpha forest](https://sandboxphysics.com/stories/lyman-alpha-forest/)

The Lyman-alpha forest · L3

# Recompute transmission from one fixed SDSS sightline

Sandbox Physics · 2026-11-05

This fixed SDSS forest has mean transmission about 0.746 under the continuum estimate used here. The unabsorbed background must be inferred, so its estimate affects the answer. Fixing the file, wavelengths and selection rules lets us reproduce the arithmetic before interpreting the gas. Negative ratios and ratios above one will remain in the calculation.

## Before you begin

Read the conceptual introduction to quasar backlighting and hydrogen absorption. This guide uses arrays, a weighted fit and averages; no cosmological parameter fitting is assumed.

## Choose the object before examining the answer

The target is SDSS J030335.42−002001.1, identified from the published quasar table of Chen and colleagues. Its published identity fixed the selection before the forest statistics were calculated. The fixed spectrum identifiers are plate 2068, MJD 53386 and fibre 316, with reduction label run2d 26. The materials record its exact URL, length and SHA-256 so that a replacement reduction cannot silently change the input.

This FITS file contains exposures combined after instrumental reduction and calibration. It does not contain raw CCD counts. It contains 3,812 coadded pixels. The pipeline classifies the object as QSO, gives redshift about 2.6945615 and reports ZWARNING zero. The paper table gives an approximate 2.72; wavelength conversions below consistently use the value stored in this file.

Read column definitions before calculating. SDSS loglam is the base-ten logarithm of vacuum, heliocentric wavelength in angstroms. Flux is in units of $10^{-17}$ erg per square centimetre per second per angstrom. The inverse variance, ivar, uses the reciprocal squared flux unit. This file does not supply every required unit through column TUNIT metadata, so the official format description is part of the input interpretation.

Reading: Chen et al. (2017), published quasar target table: SDSS J030335.42−002001.1 · SDSS DR17: Spectroscopic Data Basics · Fixed SDSS continuum and flux-statistics exercise

## Give wavelength and masks distinct jobs

Divide observed wavelength by one plus the quasar redshift to define the quasar rest frame. The fixed forest window is 1040–1180 angstroms in that frame. It avoids the region close to the quasar’s Ly-alpha emission, but a wavelength cut alone does not identify and remove all metal or strong hydrogen absorbers.

Foreground hydrogen has its own redshift. Interpreting a pixel as Ly-alpha absorption assigns its observed wavelength divided by 1215.67 angstroms, minus one. The chosen forest spans approximately absorber redshifts 2.16–2.59. Assigning every absorber the quasar’s redshift would place the gas incorrectly.

The default selection requires finite wavelength, flux and inverse variance, positive inverse variance, and and\_mask equal to zero. The SDSS AND mask combines flags shared across contributing exposures; OR retains flags present in any exposure. Their meanings differ. AND zero is not a promise that every possible contaminant is absent.

The fixed rule retains 548 forest pixels and excludes one. Requiring or\_mask zero instead retains 359. We keep that stricter selection as a sensitivity test. Its average describes the newly selected pixel set.

$$
\begin{aligned}\lambda_{\rm rest}&=\frac{10^{\tt loglam}}{1+z_{\rm QSO}}\\ z_{\rm abs}&=\frac{\lambda_{\rm obs}}{1215.67\,\mathrm{\mathring A}}-1\end{aligned}
$$

The first conversion locates windows relative to the quasar. The second locates a foreground absorber. Keep these redshifts separate.

Reading: SDSS DR17: Spectroscopic Data Basics · Fixed SDSS continuum and flux-statistics exercise

## Estimate the light that would have arrived without absorption

Transmission needs a denominator: the continuum flux the quasar would have supplied without this absorption. The observation does not directly measure that missing background. We fit a simple power law using three fixed quasar-rest-frame windows: 1270–1290, 1315–1325 and 1340–1360 angstroms. These redward windows anchor an extrapolation into the forest.

The windows retain 166 pixels. Inverse-variance weighting gives amplitude about 3.31429 at 1280 angstroms, in the original flux unit, and slope about −0.846066. An independent solver profiles out amplitude analytically at each slope, then performs a one-dimensional search. Its parameters agree with the two-dimensional least-squares solution to relative error below four parts in a billion.

That agreement checks the solution of the stated fitting problem. It does not establish the power law as the true unabsorbed continuum. The weighted residual sum is about 288.862 for nominally 164 degrees of freedom. Pixel correlations, emission structure and model mismatch are not fully included, so it is not converted into a certified goodness-of-fit probability.

In the figure, the continuum crosses the three redward fitting windows and extends into the forest. The assumed denominator is now visible alongside the measured flux.

$$
\begin{aligned}C(\lambda)&=A\left(\frac{\lambda_{\rm rest}}{1280\,\mathrm{\mathring A}}\right)^p\\ F_i&=\frac{f_i}{C_i}\end{aligned}
$$

$f_i$ is coadded pixel flux, $C_i$ the fitted and extrapolated continuum, and $F_i$ their ratio.

![Fixed SDSS observed coadd, extrapolated continuum and unclipped forest ratios](https://sandboxphysics.com/_astro/observed.DF8V17ff.svg)

SDSS J030335.42−002001.1: plate2068, MJD53386, fibre316. Blue shading marks the forest; green marks continuum windows. The continuum is our weighted power-law estimate. Lower-panel ratios retain noise below zero and above one. This is observed-derived data, distinct from the synthetic gas screen. [Open the full figure](https://sandboxphysics.com/_astro/observed.DF8V17ff.svg)

Reading: Fixed SDSS continuum and flux-statistics exercise

## Keep the noisy ratios before averaging

The arithmetic mean over 548 retained pixels is 0.7461836. Sample variance is about 0.0996164. Temporarily treating the continuum as fixed and neglecting pixel correlations, the average diagonal noise variance divided by continuum squared is about 0.0355552. Subtraction gives about 0.0640612. This is a conditional noise correction, not a direct measurement of intrinsic cosmic gas variance.

There are 13 ratios below zero and 109 above one. Physical transmission lies between zero and one, but noisy flux, background subtraction and continuum estimation need not respect that interval. Clipping the observed ratios would change their mean, variance and distribution. The distribution uses 20 bins between −0.5 and 1.5, with one underflow and three overflow pixels reported separately.

Effective optical depth takes the negative logarithm after averaging, giving about 0.292784. Averaging individual negative logarithms is a different operation and is undefined over the real numbers for negative pixels.

The OR-zero mask gives mean 0.767696 and effective optical depth 0.264362. This difference records sensitivity to selection. Statistics belong to the particular pixels retained.

$$
\begin{aligned}\overline F&=\frac1N\sum_{i=1}^{N}F_i\\\tau_{\rm eff}&=-\ln\overline F\end{aligned}
$$

Average retained pixel ratios first. This effective optical depth requires a positive overall mean.

Reading: Fixed SDSS continuum and flux-statistics exercise

## Move an assumption deliberately

Multiply the estimated continuum by 0.95 without changing observed flux. Mean transmission rises to about 0.785456 and effective optical depth falls to 0.241490. Multiplying by 1.05 gives mean 0.710651 and effective optical depth 0.341574. Raising the expected background makes the same received light appear more strongly absorbed.

Five percent is a chosen sensitivity perturbation, not a data-derived one-sigma uncertainty. Subtracting or adding 0.3 to the fitted slope gives means about 0.714569 and 0.779293. Amplitude and slope errors affect the extrapolated blueward continuum differently.

High signal-to-noise cannot replace checks of the object and continuum. If the mean ratio exceeds one, inspect emission structure, redshift and the continuum windows. A background estimate or target-identification problem may be exposed. Clipping the mean at one would hide that diagnostic.

Reading: Fixed SDSS continuum and flux-statistics exercise

## Hand a reproducible result to the next physical question

Check file hash, units, redshift, windows, masks, continuum parameters and retained count before comparing the final statistics. The download supplies our calculation, input manifest and expected results. Running the script generates the derived pixel table. Fetch the original FITS separately from the named SDSS source. Reproduce the fixed example before substituting another object.

This result describes one sightline, one window and one continuum model. It does not include a complete instrument line-spread function, pixel covariance, metal rejection, damping-wing/BAL treatment or continuum covariance. Those omissions matter for gas temperatures and dark-matter inference.

The prescribed gas screen in the accompanying calculation sits at redshift three; these observed absorbers are at lower redshift. Similar mean fluxes would not constitute a fit. A physical inference must align simulated redshift, instrumental treatment, selection and statistical windows with the observations being explained.

Reading: SDSS DR17: Spectroscopic Data Basics · Fixed SDSS continuum and flux-statistics exercise

## Exercises and explained answers

1. With observed flux fixed, what happens if the entire continuum rises by five percent?

   Read the explanation

   Every ratio is divided by 1.05, moving the mean from about 0.746184 to 0.710651. Effective optical depth increases by about 0.04879. This comes from the continuum assumption, not a newly observed change in gas density.

2. Why are forest absorber redshifts below the quasar redshift of about 2.69456?

   Read the explanation

   The light leaves the quasar and then crosses foreground gas. Quasar redshift locates rest-frame windows; each absorber is located by the observed wavelength of its pixel.

## Try the prediction

### [Keep observed statistics separate from the forward model](https://sandboxphysics.com/labs/lyman-alpha-sightline/)

1. Hold the numerical sightline fixed, record its mean transmission, and change only ionizing strength.
2. Explain how a specified incident continuum differs from the extrapolated continuum in an observation.
3. List what must be matched before fitting a redshift-three gas screen to this lower-redshift forest.

## Independent calculations

[Download the scientific ZIP](https://sandboxphysics.com/stories/lyman-alpha-forest/materials/lyman-alpha-science-v1.zip) · [Read setup and scope](https://sandboxphysics.com/stories/lyman-alpha-forest/materials/README.md)

Python with NumPy and SciPy recalculates three synthetic spectra offline. The optional observed exercise also needs Astropy and the separately fetched, hash-pinned SDSS coadd. Instructions retain all pixels without clipping noisy ratios.

Scripts, file sizes and SHA-256

- [README.md](https://sandboxphysics.com/stories/lyman-alpha-forest/materials/README.md) (5881 B)
  `0f1259a35614944edbc800caab26c9ddfda8efcbdcadb76a1c407b6ee41c21db`
- [calculate.py](https://sandboxphysics.com/stories/lyman-alpha-forest/materials/calculate.py) (1960 B)
  `789206ae0a4100c1ad25437d59c64f4b55d8482d9d4a03dfde040e900b727b2b`
- [expected-observed.json](https://sandboxphysics.com/stories/lyman-alpha-forest/materials/expected-observed.json) (10116 B)
  `469a2191d51bf9050e1800df9ebb5e5b9aa3083da81db313f3cea8b5f2f28c12`
- [expected-synthetic.json](https://sandboxphysics.com/stories/lyman-alpha-forest/materials/expected-synthetic.json) (51605 B)
  `1409ebb5261bcef0ab0871a40e48bdbce4a32404c08e9b6abe7a024ab7c56ade`
- [fetch\_sdss.py](https://sandboxphysics.com/stories/lyman-alpha-forest/materials/fetch_sdss.py) (1694 B)
  `23d8553b76c604c4e071411d845fe576f32471770313ae7e2e823c756c7ef32c`
- [field-input.json](https://sandboxphysics.com/stories/lyman-alpha-forest/materials/field-input.json) (934 B)
  `1206529ed37acab5d72b1e83458c1db78f7a8cd579964ce01db4c02470c0dc43`
- [input-manifest.json](https://sandboxphysics.com/stories/lyman-alpha-forest/materials/input-manifest.json) (651 B)
  `692fce6cd598d3bc14288384381afd24f7432d4e8f054fb94609e1a469e8dcdb`
- [lyman-alpha-science-v1.zip](https://sandboxphysics.com/stories/lyman-alpha-forest/materials/lyman-alpha-science-v1.zip) (29960 B)
  `0a169f89a41770dd33b6bbefa7cda8ccd1d8579d0c77013ea77e0ccaa6112ea3`
- [model.py](https://sandboxphysics.com/stories/lyman-alpha-forest/materials/model.py) (3656 B)
  `6a05678c4e50a5e4180aa8017633be68dcce3e7f16f55150df90edca9d465a22`
- [sdss.py](https://sandboxphysics.com/stories/lyman-alpha-forest/materials/sdss.py) (6728 B)
  `4c9e99d73bfee1f07c9069caf1442f77cdee2cfba51fd079c06732759b6055be`

[Previous: When different places share one spectral velocity](https://sandboxphysics.com/stories/lyman-alpha-forest/when-several-places-share-one-velocity/) [Next: Read a thermal history in small-scale velocity gradients](https://sandboxphysics.com/stories/lyman-alpha-forest/read-pressure-history-in-small-scale-velocities/)

## Sources and calculation scope

1. [Chen et al. (2017), published quasar target table: SDSS J030335.42−002001.1](https://academic.oup.com/view-large/94840027)

   Target-table entry used only to fix the published object name, position and approximate redshift 2.72 before flux analysis. The exact coadd instead supplies pipeline redshift 2.69456148147583. This was a targeted table reading, not a full-paper review.

2. [SDSS DR17: Spectroscopic Data Basics](https://www.sdss4.org/dr17/spectro/spectro_basics/)

   Official interpretation of calibrated coadds, vacuum heliocentric wavelengths, flux units, inverse variance, logarithmic sampling and AND/OR masks. The fixed file has 3,812 coadded pixels; it is not raw exposure data or a measured unabsorbed continuum.

3. [Fixed SDSS continuum and flux-statistics exercise](https://sandboxphysics.com/stories/lyman-alpha-forest/materials/README.md)

   Plate 2068/MJD 53386/fibre 316, run2d 26, SHA-256 83b81bfc16c91750030758d00646366ff2cada06b7f05a80ac97ad152fc387fa. Original FITS is fetched separately; derived ratios use stated continuum windows and masks. Noise correction is conditional on fixed continuum and diagonal noise, not a gas-temperature or cosmological inference.