Curve of Growth

A spectral absorption line is a deficit: measured against the neighbouring continuum, it is a certain amount of missing flux. Collapse that deficit into one number — the width of a perfectly black rectangular line removing the same total flux — and you have the equivalent width $W_\lambda$ , the one quantity a spectrum yields even when the line itself is unresolved. The central question of quantitative spectroscopy is then simple to state and deep to answer: as you pile more absorbing atoms into the line of sight, how does $W_\lambda$ grow? The answer is the curve of growth, the master relation between $W_\lambda$ and the column density $N$ of absorbers (weighted by oscillator strength as $Nf$ ) — the bridge that carries a raw stellar or interstellar spectrum across to a chemical abundance. The curve is not a single slope but a story in three acts. While the line is optically thin everywhere, every atom absorbs independently and $W_\lambda \propto N$ — the linear regime, the honest one. Then the core goes black: the intensity at line centre cannot fall below the source function, so adding atoms no longer deepens the line, and $W_\lambda$ crawls up only as $\sqrt{\ln N}$ through the Doppler wings — the flat regime, the treacherous one. Finally, when $N$ is enormous, the Lorentzian wings — which decay only as $\Delta\lambda^{-2}$ — take over the absorbed area and $W_\lambda \propto \sqrt{N}$ — the damping regime, honest again. Which act a line plays tells you at a glance whether its equivalent width is a clean abundance diagnostic or a trap.

Key takeaways

The curve of growth relates a line's measured equivalent width $W_\lambda$ to the column density $N$ (abundance) of the absorbing atoms.
It has three regimes: linear ( $W_\lambda \propto N$ ), flat or saturated ( $W_\lambda \propto \sqrt{\ln N}$ ), and damping or square-root ( $W_\lambda \propto \sqrt{N}$ ).
The linear part comes from optically-thin Doppler cores; saturation sets in when line centre becomes optically thick; the damping part is driven by the Lorentzian wings of the Voigt profile.
A single universal curve, parametrized by the Doppler width $\Delta\lambda_D$ and the damping parameter $a$ , converts equivalent widths into abundances without resolving line profiles.
Fitting many lines of one species — differing in $gf$ and excitation — to one curve solves simultaneously for abundance, microturbulence, and excitation temperature.
Key idea: Equivalent width grows in three stages as absorbers accumulate — linearly (thin core), then logarithmically (saturated Doppler core), then as a square root (Lorentzian damping wings). The regime a line sits in tells you whether its $W_\lambda$ is a clean abundance diagnostic.

Where this lives today

The curve of growth is still doing frontline work, now automated and scaled to millions of stars. Galactic archaeology surveys — APOGEE, GALAH, Gaia-ESO — measure equivalent widths or fit full spectra for hundreds of thousands of stars, converting line strengths into abundances of dozens of elements to trace how the Milky Way assembled and enriched itself. The classical differential logic survives inside modern pipelines: weak lines on the linear part anchor the abundance, the flat-part height fixes the microturbulence $\xi$ , requiring lines of different excitation potential $\chi$ to share one curve fixes the excitation temperature through the Boltzmann factor, and requiring two ionization stages to agree fixes the ionization temperature through Saha — a stack of self-consistency checks on the model atmosphere and on LTE itself. At cosmological distances the same relation converts quasar absorption lines into column densities: the Lyman-alpha forest samples the weak and flat regimes to weigh diffuse intergalactic hydrogen, while damped Lyman-alpha systems — whose enormous H I columns sit firmly in the square-root regime — yield clean $\sqrt{N}$ -scaled masses of neutral gas across cosmic time. And because the damping onset is governed by $\Gamma$ , whose collisional part scales with density, matching strong-line wings remains a standard way to separate dwarfs from giants by surface gravity $\log g$ . Where profiles can be resolved, spectrum synthesis has largely superseded curve fitting — but the curve of growth remains the conceptual map telling an observer, before any fit, which lines to trust.

The physical picture

Picture a crowd gathering in a bright doorway. When only a few people stand there most of the light gets through, and doubling the crowd doubles the shadow: absorption is proportional to numbers — the linear regime. But once the doorway is fully blocked across its centre, adding more people cannot make that patch any darker; the shadow can only widen, and slowly, at the fringes — the flat regime. What rescues growth is subtler: each person also casts a faint, far-reaching penumbra (the Lorentzian wing). Pile up enough people and the summed penumbrae start to matter, and the shadow grows steadily once more — the damping regime. The crucial reframing is that saturation is about photons, not atoms. Atoms never "run out." At line centre the emergent intensity is pinned at the source function the moment optical depth exceeds unity, so no further absorption is possible there. The only way to remove more flux is to eat into wavelengths farther from centre, reached either by the exponentially rare fast atoms of the Doppler core or by the algebraically slow-dying Lorentzian wings. The first figure is the object itself: the three-act curve, with the slope of each act labelled.

Figure

The curve of growth: reduced equivalent width W_λ/Δλ_D against line-centre optical depth τ₀ on log axes. Linear act rises with slope 1 as W_λ/Δλ_D = √π·τ₀, the flat act is a near-plateau W_λ ≈ 2Δλ_D√(ln τ₀), and the damping act resumes with slope 1/2 as W_λ ∝ √(a·τ₀)

Two dimensionless numbers set where the acts begin and end: the Doppler width $\Delta\lambda_D$ — the thermal-plus-turbulent velocity spread that fixes the width of the saturating core — and the damping parameter $a$ , the ratio of Lorentzian to Doppler width, which sets how soon the wings can rescue growth. A small $a$ means a long, punishing flat portion; a large $a$ means the square-root act opens almost as soon as the core turns black. Microturbulence enters only through $\Delta\lambda_D$ : it widens the black core, letting a saturated line swallow more flux before wings take over — which is why abundance and microturbulence are degenerate on the flat part, and why the flat part is the classical way to measure turbulence. The second figure shows the line profiles behind the curve.

Figure

Emergent line profiles F/F_c = exp(−τ₀ e^(−v²)) for a weak line (τ₀ = 0.5, blue), a saturated line (τ₀ = 5, amber) and a strong line (τ₀ = 100, red), with v = Δλ/Δλ_D. The weak core is shallow and grows in depth; once the core hits zero the profile can only widen, and the strong line is black across 2√(ln 100) = 4.29 Doppler widths

The core equations

The starting point is the definition of equivalent width: the missing flux, normalized to the continuum and integrated over wavelength, expressed as an equivalent length on the wavelength axis.

W_\lambda = \int \frac{F_c - F_\lambda}{F_c}\, \mathrm{d}\lambda \qquad (1)

where: $W_\lambda$ — equivalent width (units of wavelength); $F_c$ — continuum flux adjacent to the line; $F_\lambda$ — flux at wavelength $\lambda$ inside the line. Reading it: $W_\lambda$ is the width of a fully black ( $F_\lambda=0$ ) rectangle removing the same flux — an area, blind to how that area is distributed across the profile. To turn $W_\lambda$ into a statement about atoms we need the line depth. In the simplest slab picture — absorbers over a bright continuum — the residual depth at each wavelength is fixed by the optical depth there.

\frac{F_c - F_\lambda}{F_c} = 1 - e^{-\tau_\lambda} \qquad (2)

where: $\tau_\lambda$ — optical depth at wavelength $\lambda$ through the line-forming layer. Reading it: for $\tau_\lambda\ll1$ the depth is $\approx\tau_\lambda$ (proportional to $N$ ); for $\tau_\lambda\gg1$ it saturates at $1$ — no extra absorption can push the core below the continuum floor. Saturation is built into this one line. The shape of $\tau_\lambda$ across the line is the normalized profile. For a real atmosphere it is the Voigt profile, the convolution of a Gaussian Doppler core with a Lorentzian damping wing.

\phi(\Delta\lambda) = \frac{a}{\pi^{3/2}\Delta\lambda_D}\int_{-\infty}^{\infty}\frac{e^{-y^2}\,\mathrm{d}y}{a^2+(v-y)^2}\equiv\frac{H(a,v)}{\sqrt{\pi}\,\Delta\lambda_D} \qquad (3)

where: $\phi$ — normalized profile ( $\int\phi\,\mathrm{d}\lambda=1$ ); $v=\Delta\lambda/\Delta\lambda_D$ — offset in Doppler widths; $a$ — damping parameter; $H(a,v)$ — Voigt–Hjerting function. Reading it: as $a\to0$ , $H\to e^{-v^2}$ (pure Gaussian core); far out, $H\to a/(\sqrt{\pi}\,v^2)$ (Lorentzian wing) — the two honest acts of the curve. The Gaussian width itself carries the gas kinematics: thermal motion set by temperature and atomic mass, plus a microturbulent term added in quadrature.

\Delta\lambda_D = \frac{\lambda_0}{c}\sqrt{\frac{2kT}{m} + \xi^2} \qquad (4)

where: $\lambda_0$ — rest wavelength; $T$ — kinetic temperature; $m$ — mass of the absorbing atom; $\xi$ — most-probable microturbulent velocity; $c$ — speed of light. Reading it: heavier atoms give narrower cores; turbulence broadens all species equally, which is how it separates from temperature. The wing-to-core ratio is the damping parameter, the number that decides how long the flat act lasts.

a = \frac{\Gamma\,\lambda_0^{2}}{4\pi c\,\Delta\lambda_D} \qquad (5)

where: $\Gamma$ — total damping constant, the sum of natural (radiative) and collisional (pressure) broadening rates. Reading it: pressure broadening makes $\Gamma$ , hence $a$ , climb with gas density — the hook that lets strong-line wings measure surface gravity. The abscissa of the curve is the line-centre optical depth, built from the classical oscillator cross-section evaluated at the Doppler-normalized peak of the profile.

\tau_0 = \frac{\sqrt{\pi}\,e^2}{m_e c}\,\frac{N f \lambda_0}{\Delta\lambda_D}\cdot\frac{\lambda_0}{c} \qquad (6)

where: $N$ — column density in the lower level (per unit area); $f$ — absorption oscillator strength; $e,m_e$ — electron charge and mass (Gaussian units). Reading it: $\tau_0\propto Nf\lambda_0/\Delta\lambda_D$ — the atom ( $Nf$ ) and the kinematics ( $\Delta\lambda_D$ ) fold into one dial that walks a line along the curve. Combine profile and depth and integrate: the equivalent width, scaled by the Doppler width, is a pure function of $\tau_0$ and $a$ — the universal curve of growth.

\frac{W_\lambda}{\Delta\lambda_D} = \int_{-\infty}^{\infty}\left[1 - e^{-\tau_0 H(a,v)}\right]\mathrm{d}v \qquad (7)

where: $W_\lambda/\Delta\lambda_D$ — reduced equivalent width, the universal ordinate; $\tau_0 H(a,v)$ — optical depth at offset $v$ . Reading it: once each line is scaled by its own $\Delta\lambda_D$ and labelled by its own $\tau_0$ , the transfer integral no longer knows which line it came from — every line of a species with the same $a$ collapses onto one master curve. The three derivations below extract its three limiting slopes.

Derivation AThe weak-line limit and why equivalent width forgets its broadening

The goal is to show that in the optically-thin regime the equivalent width is exactly linear in column density and — remarkably — completely independent of how the line is broadened, so a weak line's $W_\lambda$ is a clean, direct measure of $Nf$ . We expand the depth to first order in optical depth, integrate over the profile using only its normalization, and convert to wavelength; broadening drops out because it merely reshuffles a fixed absorbed area.

Step 1. Take equation (2) and expand for small optical depth: $W_\lambda=\int(1-e^{-\tau_\lambda})\,\mathrm{d}\lambda\approx\int\tau_\lambda\,\mathrm{d}\lambda$ . Why: when the line never becomes opaque, depth and optical depth are interchangeable, and the whole profile — core and wings alike — contributes in proportion to how much material sits at each wavelength.

Step 2. Write $\tau_\lambda=N\,(\pi e^2/m_e c)\,f\,\phi_\lambda$ with $\phi$ the normalized profile, so $\int\tau_\lambda\,\mathrm{d}\lambda = N\,(\pi e^2/m_e c)\,f\int\phi\,\mathrm{d}\lambda = N\,(\pi e^2/m_e c)\,f$ . Why: because $\int\phi\,\mathrm{d}\lambda=1$ by construction, the entire shape of the profile integrates away — thermal, turbulent, and pressure broadening all vanish from the answer, leaving only the total number of absorbers.

Step 3. Convert the frequency-form cross-section to wavelength with $\mathrm{d}\lambda=(\lambda^2/c)\,\mathrm{d}\nu$ , giving the linear-regime result $W_\lambda=(\pi e^2/m_e c^2)\,N f\lambda_0^2$ . Why: this is the honest part of the curve — $W_\lambda\propto Nf\lambda_0^2$ with a universal constant $\pi e^2/m_e c^2=\pi r_e=8.85\times10^{-13}\,\mathrm{cm}$ , so a measured weak-line width converts straight into a column density.

Step 4. Note the equivalent packaging in reduced units: since $\tau_0\propto Nf\lambda_0/\Delta\lambda_D$ , the same result reads $W_\lambda/\Delta\lambda_D=\sqrt{\pi}\,\tau_0$ — a straight line of slope unity on the log-log curve. Why: this fixes the left act of the master curve and shows the reduced ordinate rising in lockstep with $\tau_0$ , exactly the blue slope-1 segment in the first figure. *Verified: $\pi e^2/m_e c^2=8.85\times10^{-13}$ cm; the linear result matches Carroll & Ostlie §9.5 and the reduced form $W_\lambda/\Delta\lambda_D=\sqrt{\pi}\,\tau_0$ (Gray Ch. 16).*

Go deeper · Derivation B · 1 minThe flat part — logarithmic growth of a saturated Doppler core

Derivation BThe flat part — logarithmic growth of a saturated Doppler core

The goal is to derive the near-plateau, where the Doppler core has gone black and the equivalent width grows only as the square root of the logarithm of the abundance — the regime that makes strong lines so treacherous for abundance work. We assume a pure Doppler profile $\tau(v)=\tau_0 e^{-v^2}$ with $\tau_0\gg1$ , approximate the black core as a rectangular box, and read off its width; the calculation hinges on locating the wavelength where optical depth falls to unity.

Step 1. Write the reduced width for a Gaussian profile: $W_\lambda=\Delta\lambda_D\int_{-\infty}^{\infty}(1-e^{-\tau_0 e^{-v^2}})\,\mathrm{d}v$ , in which the integrand is essentially $1$ where $\tau_0 e^{-v^2}\gtrsim1$ and essentially $0$ where it is $\ll1$ . Why: a strongly saturated line is, to good approximation, a black rectangle flanked by transparent wings, so the integral is dominated by the width of the black region.

Step 2. Locate the black-to-transparent boundary by setting the local optical depth to one: $\tau_0 e^{-v_1^2}=1\Rightarrow v_1=\sqrt{\ln\tau_0}$ . Why: unit optical depth is the frontier between opaque and transparent, so $v_1$ marks the edge of the saturated core — and because it lives inside a logarithm, it barely moves as $\tau_0$ climbs.

Step 3. Approximate the integral as the area of the black box, width $2v_1$ : $W_\lambda\approx\Delta\lambda_D\cdot2v_1=2\,\Delta\lambda_D\sqrt{\ln\tau_0}$ . Why: this is the flat-part result — because $\tau_0\propto N$ , we have $W_\lambda\propto\sqrt{\ln N}$ , growth so slow that a tenfold change in abundance nudges $W_\lambda$ by only tens of percent.

Step 4. Read the two dependences: $W_\lambda\propto\Delta\lambda_D$ , so the plateau height is set directly by the thermal-plus-turbulent velocity width, while its dependence on $N$ is merely logarithmic. Why: this is the origin of the abundance–microturbulence degeneracy — on the flat part, changing $\xi$ (hence $\Delta\lambda_D$ ) mimics changing the abundance, which is why the plateau is where turbulence is measured, not abundance. *Verified: reproduces the Carroll & Ostlie flat-portion expression; for $\tau_0=100$ and $\Delta\lambda_D=27.5$ mÅ, $W_\lambda=2\Delta\lambda_D\sqrt{\ln100}=118$ mÅ (Gray Ch. 16).*

Go deeper · Derivation C · 1 minThe damping part — square-root growth from Lorentzian wings

Derivation CThe damping part — square-root growth from Lorentzian wings

The goal is to derive the far-right act, where the algebraically-decaying Lorentzian wings dominate the absorbed area and the equivalent width recovers a clean $W_\lambda\propto\sqrt{N}$ dependence — the regime in which damped Lyman-alpha systems and the strongest stellar lines live. We reduce the Voigt profile to its wing, write the wing optical depth, and evaluate the width integral by a scaling substitution that leaves a pure number behind.

Step 1. Far from centre the Voigt profile is Lorentzian, so the optical depth falls as $\tau(\Delta\lambda)=CN/\Delta\lambda^2$ , with $C$ collecting constants ( $C\propto f\lambda^2\Gamma$ ). Why: the $\Delta\lambda^{-2}$ tail decays only algebraically, so its absorbed area keeps accumulating far from line centre — unlike a Gaussian, whose $e^{-v^2}$ tail contributes essentially nothing.

Step 2. The saturated core is now a black box of roughly fixed area; the growth comes from the wings, so $W_\lambda\approx2\int_0^{\infty}(1-e^{-CN/\Delta\lambda^2})\,\mathrm{d}(\Delta\lambda)$ over one symmetric wing, doubled. Why: the interesting physics has migrated from the core to the wings, where increasing $N$ pushes the "black" contour steadily outward.

Step 3. Substitute $u=\Delta\lambda/\sqrt{CN}$ so that $W_\lambda=2\sqrt{CN}\int_0^{\infty}(1-e^{-1/u^2})\,\mathrm{d}u$ , and evaluate the pure number: the integral equals $\sqrt{\pi}$ . Why: the substitution pulls the entire $N$ -dependence out front as $\sqrt{CN}$ , leaving a dimensionless constant — all the physics is in the prefactor.

Step 4. Assemble $W_\lambda=2\sqrt{\pi}\,\sqrt{CN}\propto\sqrt{Nf\lambda^2\Gamma}\propto\sqrt{a\,\tau_0}\,\Delta\lambda_D$ . Why: square-root growth in $N$ restores abundance sensitivity, and the explicit $\Gamma$ means the strong-line width also measures the damping constant — natural plus collisional — giving a handle on pressure and surface gravity. *Verified: the dimensionless wing integral equals $\sqrt{\pi}$ ; the $W_\lambda\propto\sqrt{N\Gamma}$ scaling is standard (Gray Ch. 16; Spitzer Ch. 3).*

Limiting cases

The three acts of the curve, with $\tau_0$ the line-centre optical depth and $a$ the damping parameter:

Case	Result	Meaning
Linear (weak, $\tau_0\ll1$ )	$W_\lambda \propto N f \lambda^2$	Optically thin: the whole profile is sampled, every absorber counts, and $W_\lambda$ is independent of the broadening mechanism — the cleanest abundance regime.
Flat (saturated, $\tau_0\gtrsim1$ )	$W_\lambda \propto \Delta\lambda_D\sqrt{\ln\tau_0}$	Doppler core is black; growth is logarithmic and set by $\Delta\lambda_D$ , hence sensitive to thermal and microturbulent velocity, not to abundance.
Damping (strong, $a\tau_0\gtrsim1$ )	$W_\lambda \propto \sqrt{a\,\tau_0}\propto\sqrt{N f \lambda^2 \Gamma}$	Lorentzian wings dominate; $W_\lambda$ grows with abundance again (as $\sqrt{N}$ ) and depends on the damping constant $\Gamma$ (natural plus collisional).

Common trapReading an abundance off a strong, saturated line

The seductive mistake is to trust a big, obvious line more than a faint one. It is the opposite. On the flat part $W_\lambda\propto\sqrt{\ln\tau_0}$ , so a factor-of-ten error in abundance shifts $W_\lambda$ by only about twenty percent — the equivalent width is nearly blind to $N$ there. Worse, the little sensitivity it retains is to $\Delta\lambda_D$ , i.e. to the microturbulent velocity, so an abundance error and a turbulence error are almost perfectly degenerate on the plateau. Never derive an abundance from a strong saturated line without independently pinning down the microturbulence; anchor abundances on weak, linear-regime lines where $W_\lambda\propto N$ . Saturated lines are for measuring $\xi$ , not $N$ .

Worked examples

1 · A weak solar iron line — from column density to equivalent width (fully worked)

A weak Fe I line at $\lambda_0=5000$ Å forms in a solar photosphere with $T=5800$ K and microturbulence $\xi=1$ km/s. Iron has atomic mass $m=56\,\mathrm{u}$ ; the line has $f=0.1$ and a lower-level column density $N=2\times10^{11}\,\mathrm{cm^{-2}}$ . Find the Doppler width, line-centre optical depth, and equivalent width, and confirm the regime.

Step 1. Doppler width. The most-probable speed combines thermal and turbulent parts in quadrature: $\sqrt{2kT/m+\xi^2}$ . The thermal term is $\sqrt{2kT/m}=1.31$ km/s and, added to $\xi=1$ km/s in quadrature, gives $b=1.65$ km/s. Then $\Delta\lambda_D=(\lambda_0/c)\,b=(5000\,\text{Å}/c)\times1.65\times10^5\,\mathrm{cm/s}=\mathbf{0.0275\ Å}$ (27.5 mÅ).

Step 2. Line-centre optical depth in the velocity form $\tau_0=\sqrt{\pi}\,(e^2/m_e c)\,Nf\lambda_0/b$ , with $\sqrt{\pi}\,e^2/m_e c=1.50\times10^{-2}$ (cgs). Plugging in $N=2\times10^{11}$ , $f=0.1$ , $\lambda_0=5\times10^{-5}$ cm, $b=1.65\times10^5$ cm/s gives $\tau_0=\mathbf{0.091}$ . Since $\tau_0\ll1$ , the line is firmly in the linear regime.

Step 3. Equivalent width, linear formula $W_\lambda=(\pi e^2/m_e c^2)\,Nf\lambda_0^2$ with $\pi e^2/m_e c^2=8.85\times10^{-13}$ cm: $W_\lambda=8.85\times10^{-13}\times2\times10^{11}\times0.1\times(5\times10^{-5})^2\,\mathrm{cm}=\mathbf{4.4\ mÅ}$ .

Step 4. Cross-check via the reduced curve: $W_\lambda/\Delta\lambda_D=\sqrt{\pi}\,\tau_0=1.772\times0.091=0.161$ , so $W_\lambda=0.161\times0.0275\,\text{Å}=4.4$ mÅ — identical, as the two forms must agree on the linear part. A weak line: $\Delta\lambda_D=27.5$ mÅ, $\tau_0=0.09$ , $W_\lambda=4.4$ mÅ, cleanly linear, so this $W_\lambda$ maps directly to $Nf$ .

2 · The same line gone strong — the saturation trap in numbers

Now suppose a much larger iron abundance drives the same line to $\tau_0=100$ , and then a further tenfold increase to $\tau_0=1000$ . Using the flat-part formula, find both equivalent widths and the fractional change, and comment.

Step 1. With $\Delta\lambda_D=0.0275$ Å from Example 1, the saturated width is $W_\lambda\approx2\,\Delta\lambda_D\sqrt{\ln\tau_0}$ . For $\tau_0=100$ : $W_\lambda=2\times0.0275\times\sqrt{\ln100}=2\times0.0275\times2.146=\mathbf{0.118\ Å}$ (118 mÅ).

Step 2. For $\tau_0=1000$ : $W_\lambda=2\times0.0275\times\sqrt{\ln1000}=2\times0.0275\times2.628=\mathbf{0.145\ Å}$ (145 mÅ).

Step 3. The fractional change for a tenfold abundance increase is only $\sqrt{\ln1000/\ln100}=\sqrt{6.908/4.605}=\mathbf{1.22}$ — a 22% rise in $W_\lambda$ for a factor of ten in $N$ . A tenfold abundance error hides inside a 22% change in $W_\lambda$ — and a modest tweak to the microturbulence would produce the same 22%, which is why saturated lines cannot fix abundances alone.

Physics of the portrait

The curve of growth is a twentieth-century instrument built to answer a nineteenth-century question — what are the stars made of? — carrying the fingerprints of the people who turned spectra into chemistry.

Origins · 1 mincurve of growth and how this was discovered

curve of growth

forged at Utrecht and Kiel, c. 1930 · quantitative stellar spectroscopy

The idea that a line's strength could be read as an abundance is old; making it quantitative took a conceptual leap — separating what the line's area tells you from what its shape hides. Cecilia Payne's 1925 Harvard thesis had already used Saha's ionization theory to show that stars are overwhelmingly hydrogen and helium — a result so startling she was pressured to soften it — but her method leaned on ionization equilibrium rather than on line strengths directly. The decisive tool came from Utrecht around 1930, where Marcel Minnaert and collaborators, poring over the solar spectrum, introduced the Wachstumskurve, the "growth curve": equivalent width plotted against the number of absorbing atoms, making the three-act behaviour — linear, saturated, damped — explicit. Albrecht Unsöld, at Kiel, made it the backbone of stellar abundance analysis in his 1938 Physik der Sternatmosphären, showing how the flat part measured turbulence and the wings measured pressure. The Utrecht school then industrialized the observational side, tabulating equivalent widths for thousands of solar lines. What began as a way to salvage information from unresolved, blended, saturated lines became the standard route from a stellar spectrum to a composition — the empirical engine behind the metallicity scale and galactic chemical evolution.

History

1925

— Cecilia Payne's thesis applies Saha ionization to stellar spectra and finds hydrogen dominant

1930

— Minnaert and collaborators at Utrecht introduce the curve of growth (Wachstumskurve)

1938

— Unsöld's Physik der Sternatmosphären makes it the standard stellar abundance tool

1940

— the Utrecht photometric atlas tabulates solar equivalent widths for thousands of lines

1955

— differential curve-of-growth analysis pins down microturbulence in the Sun and stars

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §9.5 (The Curve of Growth) · Gray, The Observation and Analysis of Stellar Photospheres, 3rd ed., Ch. 16 · Mihalas, Stellar Atmospheres, 2nd ed., Ch. 10 · Rutten, Radiative Transfer in Stellar Atmospheres (lecture notes), Ch. 4 · Spitzer, Physical Processes in the Interstellar Medium, Ch. 3 (interstellar curve of growth).

Provenance. The equivalent-width definition, the three regimes, and their scalings ( $W\propto Nf\lambda^2$ ; $W\propto\Delta\lambda_D\sqrt{\ln\tau_0}$ ; $W\propto\sqrt{N\Gamma}$ ) follow Carroll & Ostlie §9.5 and Gray Ch. 16. A verifier should confirm: (1) the classical cross-section $\sigma=(\pi e^2/m_e c)f\,\phi_\nu$ and the linear result $W_\lambda=(\pi e^2/m_e c^2)Nf\lambda_0^2$ with $\pi e^2/m_e c^2=8.85\times10^{-13}$ cm; (2) the flat-part $2\Delta\lambda_D\sqrt{\ln\tau_0}$ via the saturated-core boundary $\tau_0 e^{-v_1^2}=1$ ; (3) the wing integral yielding the $\sqrt{\pi}$ constant and $W\propto\sqrt{N\Gamma}$ ; (4) the Voigt limits $H\to e^{-v^2}$ and $H\to a/(\sqrt{\pi}v^2)$ ; (5) the worked numbers $\Delta\lambda_D=27.5$ mÅ, $\tau_0=0.09$ , $W_\lambda=4.4$ mÅ (linear) and $W_\lambda=118$ mÅ at $\tau_0=100$ (flat). Interstellar and DLA applications cross-check with Spitzer Ch. 3. No non-standard numerical constants are asserted.

Prerequisites. Spectral Line Formation · Optical Depth · The Equation of Radiative Transfer · Opacity and Absorption Coefficients · The Boltzmann Equation · The Saha Equation. Leads to. Local Thermodynamic Equilibrium · Emission and the Source Function.

Ask the tutor: Ask me to place any real absorption line — a solar Fe line, an interstellar Na D line, or a damped Lyman-alpha system — on the curve of growth and read off which regime it lives in and what it can measure.

What exactly is the equivalent width, and why use it instead of the line profile?

The equivalent width $W_\lambda$ is the width of a completely black rectangular line that would remove the same total flux as the real line. Formally it is the integral of the fractional depth over wavelength, so it measures the area carved out of the spectrum, not the line's shape. Its great virtue is robustness: you can measure $W_\lambda$ even at poor resolution, where the profile is blurred by the instrument or blended with neighbours, because smearing conserves area. That is exactly what the curve of growth was built for — extracting abundances when you cannot resolve the profile. When you can resolve it, direct profile fitting uses more information, but $W_\lambda$ remains the workhorse for large samples.

Why does saturation happen — do the atoms stop absorbing?

No — the atoms keep absorbing exactly as before; what saturates is the observable. At line centre, once the optical depth exceeds unity, the emergent intensity is pinned near the source function and cannot drop any lower no matter how many more absorbers you add. The core is already as dark as it can be, and extra atoms there remove no additional flux; $W_\lambda$ can only grow by darkening wavelengths farther from centre. This is why saturation is a statement about photons and the source function, not about running out of atoms — and why the flat part is so nearly flat.

Why is a weak line's equivalent width independent of temperature and turbulence, but a strong line's is not?

For a weak line, $W_\lambda\approx\int\tau_\lambda\,\mathrm{d}\lambda$ , and since $\tau_\lambda\propto\phi(\lambda)$ with $\int\phi\,\mathrm{d}\lambda=1$ , only the total number of absorbers survives — broadening merely redistributes a fixed absorbed area without changing its total, so $W_\lambda$ depends on $Nf\lambda^2$ alone. Once the line saturates that logic breaks: the black core removes a fixed maximum at each wavelength, and the only way to absorb more is to widen the core, which depends directly on $\Delta\lambda_D$ and hence on temperature and microturbulence. That is why the flat part is sensitive to kinematics and the linear part is not.

How do astronomers actually get an abundance out of the curve of growth?

They measure $W_\lambda$ for many lines of one species — differing in oscillator strength $gf$ and lower-level excitation energy $\chi$ — and shift each onto a single theoretical curve. Requiring lines of different $\chi$ to overlay fixes the excitation temperature via the Boltzmann factor $e^{-\chi/kT}$ ; the height of the saturated plateau fixes the microturbulence through $\Delta\lambda_D$ ; and the overall offset needed to lay the data on the curve gives the abundance, with weak linear-regime lines carrying the most weight because there $W_\lambda\propto N$ cleanly. Requiring two ionization stages to give the same abundance additionally fixes the ionization temperature through Saha.

Why can a pure Doppler line never enter a square-root regime?

Because a Gaussian profile falls off super-exponentially, as $e^{-v^2}$ . Once its core is black, its wings drop to nothing so fast that they add essentially no extra area no matter how large the abundance, so the line stays stuck on the flat part indefinitely. The Lorentzian wing of a real (Voigt) profile decays only as $\Delta\lambda^{-2}$ , so its far-wing area keeps accumulating and eventually dominates, producing $W_\lambda\propto\sqrt{N}$ . The square-root act is therefore a signature of natural and collisional (pressure) broadening — the physics that puts genuine Lorentzian wings on the line — and cannot come from thermal or turbulent motion alone.

What sets where the flat part ends and the damping part begins?

The damping parameter $a=\Gamma\lambda_0^2/(4\pi c\,\Delta\lambda_D)$ , the ratio of Lorentzian to Doppler width. The wings take over roughly when $a\tau_0\gtrsim1$ , so a small $a$ — typical of natural broadening alone, around $10^{-3}$ — means a long, punishing flat portion before the square-root act opens, while a large $a$ (strong collisional broadening in a dense atmosphere) shortens the plateau. Because the collisional part of $\Gamma$ grows with gas density, the onset of the damping regime is itself a diagnostic: matching where a species' strong lines start climbing again constrains the pressure and hence the surface gravity $\log g$ — a standard way to tell a dwarf from a giant.