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Sandbox Physics
Galaxy colors from photons to redshift distributions

L3 · Galaxy photometry

Reproduce redshift distributions for twenty-four real SDSS galaxies

Sandbox Physics ·

Open the frozen SDSS DR17 CSV and its query beside this guide. You will convert five linear flux measurements per galaxy, run the same finite stellar-population inference used by the Lab, then compare its distributions with spectroscopic redshifts that were not supplied to the fit. Two missed intervals and a calibration-sensitive peak show why validation needs more than a scatter plot.

1. Identify the release and reproduce the selection

The input is a 6,162-byte response from the public SDSS DR17 SkyServer SQL service, saved with its exact query and SHA-256 hash. The query joins primary galaxy spectra to their best imaging objects, requires zero spectroscopic warning, clean imaging and a spectroscopic redshift between 0.02 and 0.8, then takes the first twenty-four spectrum identifiers in order. The actual selected range is 0.02122228 to 0.4482199.

This is a compact, bright, spectroscopically selected exercise, not a random sample of all imaging galaxies. It contains neighboring survey observations and cannot measure the redshift distribution of a faint imaging population. Spectroscopic values enter selection and final validation; they do not enter the flux likelihood or train the templates, priors or zero points.

Keep objID and specObjID as strings. These nineteen-digit identifiers exceed the exact integer range of ordinary JavaScript numbers. The query and original bytes let you identify every source without substituting an invented catalogue row.

            mkdir -p galaxy-colors-reproduction
cd galaxy-colors-reproduction
curl -fSLO https://sandboxphysics.com/stories/galaxy-colors/reproduce-sdss.py
curl -fSLO https://sandboxphysics.com/stories/galaxy-colors/sdss-dr17.csv
curl -fSLO https://sandboxphysics.com/stories/galaxy-colors/photometry-grid.json
curl -fSLO https://sandboxphysics.com/stories/galaxy-colors/sdss-results.json
python3 reproduce-sdss.py --reference sdss-results.json

Reading: SDSS DR17 imaging fluxes and flux calibration · SDSS public-release use and acknowledgement policy

2. Read flux, inverse variance and foreground extinction

Each row contains five modelFlux values in nanomaggies and five modelFluxIvar values in inverse-square nanomaggies. The SDSS model fluxes use a common shape selected in r, which supports consistent galaxy colors. The inverse square root of each inverse variance gives its statistical standard deviation. All one hundred twenty inverse variances in this snapshot are positive.

The catalogue extinction fields are band-specific Milky Way foreground estimates. Multiply both flux and statistical standard deviation by the foreground correction. Next apply the frozen approximate AB zero-point shifts: minus 0.04 magnitude in u, plus 0.02 in z, and zero in the other bands. SDSS describes the z shift as tentative and these shifts as uncertain at roughly the hundredth-magnitude level. The tutorial reruns inference without the shifts as a sensitivity check.

Add three percent of the absolute corrected measured flux in quadrature to the statistical standard deviation, then hold that error fixed across models. This is a declared teaching error floor, not an inferred SDSS covariance or a guarantee of calibrated uncertainties. The snapshot supplies no cross-band covariance; the calculation assumes diagonal errors.

qb=100.4AbMW−0.4ΔmbAB,fb′=qbfb,σb′2=qb2ivarb+(0.03∣fb′∣)2q_b=10^{0.4 A_b^{\mathrm{MW}}-0.4\Delta m_b^{\mathrm{AB}}},\quad f_b'=q_b f_b,\quad \sigma_b'^2=\frac{q_b^2}{\mathrm{ivar}_b}+(0.03|f_b'|)^2
The foreground correction brightens the de-reddened source. A negative magnitude offset also brightens the AB flux. The same multiplicative factors act on the statistical error before the fixed fractional floor is added.

Reading: SDSS DR17 imaging fluxes and flux calibration · Hogg et al.

3. Follow a negative flux through the calculation

Row six has objID 1237650795146510903 and specObjID 299491876468189184. Its original u flux is minus 0.1949482 nanomaggies, with inverse variance 6.303689. The foreground extinction is 0.2578563 magnitude. After both corrections its u flux is minus 0.256485 with standard deviation 0.524073 nanomaggies.

The negative estimate is only about half a standard deviation below zero. It is compatible with a small positive physical flux and belongs in the Gaussian likelihood. Deleting it or treating it as an exact upper limit changes the statistical question. A second negative u flux occurs in row twenty-two; both are retained.

For row six the inferred highest peak is 0.415 and the central interval is approximately 0.2566 to 0.4699. Its spectroscopic redshift, read after the flux fit, is 0.4482199. The interval includes that value. The saved output includes all two hundred forty-one density values, so this comparison can be checked without opening the interactive Lab.

Reading: SDSS DR17 imaging fluxes and flux calibration · Brammer

4. Inspect failures before reducing the sample to one score

Row eighteen, objID 1237648721216405657, has spectroscopic redshift 0.08419658. Its highest photo-z peak is 0.130 and its central interval is 0.1028–0.1922: the interval misses the known redshift. Row twenty-three, objID 1237648720142336237, has spectroscopic redshift 0.06518305, highest peak 0.185 and interval 0.0911–0.1962. That interval misses too.

Row twenty-three is also a useful calibration diagnosis. Omitting the small approximate AB shifts moves its highest peak from 0.185 to the lower grid boundary at zero. The two posterior regions were competing; the zero-point changes swap their ranking. The shifts were fixed from the calibration documentation, not fitted to make the spectroscopic answer agree. The correct lesson is to inspect the distribution and calibration dependence before trusting the single peak.

The limited model has one metallicity, single-burst populations and no emission-line contribution. A poor or displaced fit could arise from those model limitations, measurement problems or calibration, and this small catalogue alone does not identify which explanation caused each failure. The residual vectors in the output show which bands disagree; follow-up spectra and richer templates would test the cause.

RowSpectroscopic redshiftHighest peakCentral 68% interval
660.44821990.44821990.4150.415[0.2566, 0.4699][0.2566,\,0.4699]
18180.084196580.084196580.1300.130[0.1028, 0.1922][0.1028,\,0.1922]
23230.065183050.065183050.1850.185[0.0911, 0.1962][0.0911,\,0.1962]

Reading: SDSS DR17 imaging fluxes and flux calibration · Bruzual & Charlot

5. Read the summary with its sample size attached

Twenty-two of twenty-four central 68 percent intervals contain the spectroscopic value. That fraction is about 92 percent for this particular sample. It is not a calibration certificate: the sample is small and selected, several distributions are broad, and the error floor is a teaching assumption. Neither a high inclusion fraction nor a small median offset proves that every posterior is informative.

Define normalized redshift error as below. The saved median error is 0.004414 and the centered normalized median absolute deviation is 0.05443. No object crosses the chosen catastrophic-error threshold of absolute normalized error greater than 0.15. The two missed intervals remain failures even though neither meets that coarser threshold.

For the next catalogue, keep the release, selection and flux/error units first; state how foreground and zero points were treated; keep the spectroscopic answers out of inference; inspect the full distributions, residuals and boundary mass; then report interval inclusion and a clearly defined outlier metric. To calibrate a survey redshift distribution, the validation sample must also represent that survey’s color, brightness and redshift selection.

δi=zmode,i−zspec,i1+zspec,i,σNMAD=1.48 median⁡i ⁣∣δi−median⁡jδj∣\delta_i=\frac{z_{\mathrm{mode},i}-z_{\mathrm{spec},i}}{1+z_{\mathrm{spec},i}},\qquad \sigma_{\mathrm{NMAD}}=1.48\,\operatorname{median}_i\!\left|\delta_i-\operatorname{median}_j\delta_j\right|
The denominator uses spectroscopic redshift. This exact centered definition is used in the script. A zero catastrophic count in twenty-four selected galaxies says little about rare failures in a much larger or fainter population.

Reading: Brammer · SDSS DR17 imaging fluxes and flux calibration

Download the exact inputs and results

Run the standalone Python script with the CSV, frozen grid and reference results in one directory. These public files also expose the template shapes, filter responses and source manifest.

Try the observation

Recognize a peak swap in a controlled observation

  1. Use the default synthetic observation, then remove u. Compare the two posterior regions before comparing the highest-peak numbers.
  2. Return to the real output and inspect row twenty-three with and without the approximate AB shifts. Distinguish a source changing from an inference changing.

Exercises and explained answers

Why can 22 of 24 intervals contain the truth without demonstrating accurate 68-percent calibration?

Calibration concerns repeated coverage in the target population under the stated procedure. Here the selected sample is tiny, intervals can be broad, and the assumed error floor changes them. The observed fraction is a descriptive check with substantial sampling and selection uncertainty.

Sources and calculation scope

  1. Hogg et al., The K correction, astro-ph/0210394v1 (2002)

    Equations (2)–(12): spectral densities, photon response, redshift factors and the AB K correction. We use luminosity distance and define wavelength densities per Angstrom.

  2. Bruzual & Charlot, Stellar population synthesis at the resolution of 2003, astro-ph/0309134v1

    Sections 2–3 describe the stellar-population model and its spectral evolution. The frozen solar-metallicity Salpeter SSP file is distributed by EzGal; this course fixes its model choices rather than inferring a general galaxy age.

  3. SDSS DR17 imaging fluxes and flux calibration

    The AB-offset and physical-flux sections state the approximate u shift, tentative z shift and their uncertainties. The tutorial freezes TOP 24 clean primary GALAXY spectra with zWarning=0 ordered by specObjID, keeping linear modelFlux values and inverse variances.

  4. SDSS public-release use and acknowledgement policy

    The released catalogue is public data. Credit the Sloan Digital Sky Survey and its collaboration; the tutorial uses catalogue numbers, not survey images.

  5. Brammer, van Dokkum & Coppi, EAZY: A Fast, Public Photometric Redshift Code, arXiv:0807.1533v1 (2008)

    Sections 2.1–2.5: template combinations, template error, priors and output redshifts. Section 3.3 and section 4, especially Figures 8–11, connect the error model and priors to real validation. Our finite SSP posterior is a separate teaching calculation.

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