SCIENCE · INVESTIGATE AND EXPLAIN

Distinguish an absorption line’s depth from its integrated deficit

Spectral lines arise when matter exchanges photons at particular transition energies. A hot dense source can supply a continuum, while excited low-density material can add emission lines and intervening material can remove light at selected wavelengths. The observed result depends on the source and viewing geometry, not merely on the presence of an element somewhere along the path.

Learning outcome: Vary central optical depth and width separately. Retain depth, integrated area and the thin-limit comparison, then identify one omitted physical or instrumental factor that would affect a real abundance inference.

Starting recap

Keep photon energy/count, density convention, propagation, collection/resolution, calibration and source-quality assumptions distinct.

All lessons are open directly. Your saved experiments remain when you leave and return.

Try

At τ₀=1 and σ=0.2 nm, central transmission≈0.367879 and depth≈0.632121. Integrated equivalent width≈0.363494 nm, below the thin approximation≈0.501326 nm. Doubling σ at fixed τ₀ would double area without changing depth.

Open the experiment

Predict

Can two absorption lines have the same central depth but different equivalent widths?

Your prediction and reason

Experiment

Vary central optical depth and width separately. Retain depth, integrated area and the thin-limit comparison, then identify one omitted physical or instrumental factor that would affect a real abundance inference.

Separate absorption depth, width and integrated equivalent width

Worked teaching model · synthetic values · Original pure absorbing screen with unit continuum and τ(Δλ)=τ0 exp[−Δλ²/(2σ²)]; transmitted exp(−τ), no emission/scattering/blending/instrument convolution. Equivalent width integrated over ±8σ by midpoint quadrature. Not a direct abundance determination.

Central optical depth: 1 dimensionless Gaussian optical-depth width: 0.2 nm

Reset keeps your prediction, captures and practice history.

Saturation makes line depth and equivalent width nonlinear in optical depth.

Changing width can alter equivalent width without changing the central depth. A real abundance inference also needs transition strength, level populations, covering, broadening and continuum/instrument evidence; the original screen supplies none of those as measured facts.

Original calculated points, not measured observations
Wavelength offset / nm	Transmitted / continuum
-0.8000000	0.9996646
-0.7840000	0.9995396
-0.7680000	0.9993721
-0.7520000	0.9991491
-0.7360000	0.9988543
-0.7200000	0.9984674
-0.7040000	0.9979629
-0.6880000	0.9973098
-0.6720000	0.9964704
-0.6560000	0.9953991
-0.6400000	0.9940418
-0.6240000	0.9923346
-0.6080000	0.9902034
-0.5920000	0.9875626
-0.5760000	0.9843157
-0.5600000	0.9803544
-0.5440000	0.9755603
-0.5280000	0.9698054
-0.5120000	0.9629547
-0.4960000	0.9548697
-0.4800000	0.9454117
-0.4640000	0.9344478
-0.4480000	0.9218564
-0.4320000	0.9075345
-0.4160000	0.8914048
-0.4000000	0.8734230
-0.3840000	0.8535851
-0.3680000	0.8319328
-0.3520000	0.8085580
-0.3360000	0.7836048
-0.3200000	0.7572686
-0.3040000	0.7297929
-0.2880000	0.7014627
-0.2720000	0.6725958
-0.2560000	0.6435316
-0.2400000	0.6146193
-0.2240000	0.5862053
-0.2080000	0.5586220
-0.1920000	0.5321772
-0.1760000	0.5071465
-0.1600000	0.4837684
-0.1440000	0.4622411
-0.1280000	0.4427233
-0.1120000	0.4253363
-0.09600000	0.4101682
-0.08000000	0.3972791
-0.06400000	0.3867067
-0.04800000	0.3784729
-0.03200000	0.3725882
-0.01600000	0.3690567
0	0.3678794
0.01600000	0.3690567
0.03200000	0.3725882
0.04800000	0.3784729
0.06400000	0.3867067
0.08000000	0.3972791
0.09600000	0.4101682
0.1120000	0.4253363
0.1280000	0.4427233
0.1440000	0.4622411
0.1600000	0.4837684
0.1760000	0.5071465
0.1920000	0.5321772
0.2080000	0.5586220
0.2240000	0.5862053
0.2400000	0.6146193
0.2560000	0.6435316
0.2720000	0.6725958
0.2880000	0.7014627
0.3040000	0.7297929
0.3200000	0.7572686
0.3360000	0.7836048
0.3520000	0.8085580
0.3680000	0.8319328
0.3840000	0.8535851
0.4000000	0.8734230
0.4160000	0.8914048
0.4320000	0.9075345
0.4480000	0.9218564
0.4640000	0.9344478
0.4800000	0.9454117
0.4960000	0.9548697
0.5120000	0.9629547
0.5280000	0.9698054
0.5440000	0.9755603
0.5600000	0.9803544
0.5760000	0.9843157
0.5920000	0.9875626
0.6080000	0.9902034
0.6240000	0.9923346
0.6400000	0.9940418
0.6560000	0.9953991
0.6720000	0.9964704
0.6880000	0.9973098
0.7040000	0.9979629
0.7200000	0.9984674
0.7360000	0.9988543
0.7520000	0.9991491
0.7680000	0.9993721
0.7840000	0.9995396
0.8000000	0.9996646

Explicit physical or source units
Quantity	Value	Unit
Line-center transmitted continuum fraction	0.3678794	fraction
Line-center absorption depth	0.6321206	fraction
Integrated equivalent width	0.3634938	nm
Optically thin equivalent-width approximation	0.5013257	nm
Exact/thin equivalent width	0.7250652	ratio

Actual SDSS records are explicitly labeled; all other packets are original models. No direct solar viewing, radiation experiment or instrument operation is requested.

Saturation makes line depth and equivalent width nonlinear in optical depth.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

1. Define the light and measurement

2. Connect physical and instrumental mechanisms

The original pure absorbing screen has optical depth τ(Δλ)=τ₀ exp[−Δλ²/(2σ²)] and transmitted continuum fraction exp(−τ). It omits emission, scattering into the beam, partial covering and instrumental convolution. Central transmitted fraction is exp(−τ₀), so absorption depth is 1−exp(−τ₀), independent of the supplied width σ.

Equivalent width integrates the missing continuum-normalized area: W=∫[1−Fλ/Fcontinuum]dλ. It has wavelength units, not flux units. When optical depth is very small, W≈τ₀σ√(2π). Saturation makes the exact result smaller than that linear extrapolation because the central intensity cannot fall below zero. Changing width can increase W without changing central depth.

An abundance inference also requires transition strength, level populations, excitation/ionization, broadening, continuum placement and instrument evidence. The original numerical integral extends to eight widths on each side and is not a fitted atomic line or a column-density measurement. A deep line alone is not a universal abundance scale.

3. Work the evidence

4. Predict and investigate

Can two absorption lines have the same central depth but different equivalent widths?

Apply the evidence to a changed case

At τ₀=2 and σ=0.2 nm, compare central transmission and W with the τ₀=1 case.

Retain the evidence frame, calculations and assumptions before opening public feedback.

Compare a worked response

Transmission falls to≈0.135335 and W grows to≈0.557961 nm, not twice≈0.363494. The thin extrapolation would be≈1.00265 nm and overstates the integrated deficit because absorption saturates.

Source and boundaries

Primary source. Checked 30 October 2026. Original teaching and fictional numerical models are separate from the explicitly labeled actual SDSS archive; no direct observing, radiation experiment or instrument operation is requested.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

At τ₀=2 and σ=0.2 nm, compare central transmission and W with the τ₀=1 case.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Test a wavelength calibration with an unused spectral reference Next: Separate a spectral shift from an additive calibration error

Save a science backup or import one

Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.

A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.

Choose a science backup (maximum 36 MB)

Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.

Complete file sets and course backups

A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.

Export scope

The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.

Choose the manifest and all parts together

0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.

Science storage: loading. No account or remote computation is required.