Limb Darkening

Look at the Sun and the edge is already telling you something

Project a safe white-light image of the Sun onto a screen and you see something that ought to be surprising: the disk is not uniformly bright. It is luminous in the middle and visibly dimmer around the rim, fading smoothly to an edge that looks soft and slightly grey rather than razor-sharp. The center-to-edge brightness drop is large — in ordinary visible light the very edge of the solar disk emits only about 40% of the intensity at the center. Astronomers call this limb darkening, and it is one of those rare phenomena where a single glance at a star directly reveals the temperature structure of its atmosphere.

The fade is not caused by anything happening at the edge of the Sun specifically. The Sun is a sphere of gas with no solid surface; the edge we see is just the place where our line of sight grazes the limb tangentially. What changes from center to edge is not the star but the geometry of our view into it. At the center we look straight down into the deep, hot gas. At the limb we look in at a shallow, grazing angle and can only see the high, cool gas before the atmosphere becomes opaque. Cooler gas radiates less. The disk therefore fades from a bright center to a dim limb, and the rate of the fade encodes how fast temperature drops with height in the stellar atmosphere.

How it works: optical depth and the slanted sightline

A stellar photosphere is not a surface. It is a layer of gas, several hundred kilometres thick in the Sun's case, that grades from transparent at the top to opaque below. The relevant quantity is optical depth τ, the cumulative opacity integrated along a sightline. You see light emitted from around the depth where τ ≈ 1 — shallower than that the gas is too thin to have emitted much; deeper than that the gas is too opaque for the light to escape. The Eddington-Barbier relation makes this precise: the intensity you receive in a given direction is approximately the source function evaluated at τ ≈ 1 along that direction.

Now compare the two extreme sightlines across a stellar disk:

• Disk center (μ = 1). Your line of sight is radial — straight down toward the star's center, parallel to the local outward normal. It reaches τ ≈ 1 deep in the photosphere, where the gas is hot. You see the bright, deep, hot layer.
• Limb (μ = 0). Your line of sight skims the surface at a grazing angle. Because it travels a long slanted path through the atmosphere, it accumulates τ ≈ 1 while still high up, in the cool, tenuous outer gas. You see only the dim, shallow, cool layer.

The variable that quantifies the viewing angle is μ = cos θ, where θ is the angle between the line of sight and the local surface normal. At disk center θ = 0 so μ = 1; at the limb θ = 90° so μ = 0. Geometrically, μ also equals √(1 − (r/R)²) for a point at projected radius r on a disk of radius R, so μ runs smoothly from 1 at center to 0 at the edge. Because temperature falls outward through the photosphere, and you sample progressively higher (cooler) layers as μ → 0, the emergent intensity falls monotonically from center to limb. That monotone fade is limb darkening.

The linear limb-darkening law

The simplest useful description of the center-to-limb profile is the linear limb-darkening law:

I(μ) / I(0) = 1 − u (1 − μ)

Here I(0) is the intensity at disk center (μ = 1; the "(0)" labels the central direction, θ = 0), and u is the single dimensionless limb-darkening coefficient. Read the law at the two endpoints:

μ = 1 (center): I/I(0) = 1 − u(1 − 1) = 1
μ = 0 (limb):   I/I(0) = 1 − u(1 − 0) = 1 − u

So u is exactly the fractional brightness lost between center and limb. For the Sun in the visible, u ≈ 0.6, which makes the limb intensity 1 − 0.6 = 0.4 of the central value — the Sun is about 40% as bright at its visible edge as at its middle, i.e. roughly 40% fainter relative to a uniform disk depending on how you frame the comparison. The law is a first-order (linear in μ) Taylor approximation; real profiles curve slightly, which is why more flexible forms exist.

Worked example: reading the solar coefficient and a transit floor

Start with the Sun. The gray-atmosphere, Eddington-approximation source function in radiative equilibrium is

S(τ) ≈ (3/4) F (τ + 2/3)

where F is the (constant) flux. The Eddington-Barbier relation says the emergent intensity at viewing angle μ is the source function at τ = μ:

I(μ) ∝ S(τ = μ) ∝ (μ + 2/3)

I(μ)/I(1) = (μ + 2/3) / (1 + 2/3)
          = (μ + 0.667) / 1.667
          = 0.4 + 0.6 μ
          = 1 − 0.6 (1 − μ)

This is the linear law with u = 0.6 dropping straight out of first-principles radiative transfer — and it matches the observed solar value in green-yellow light remarkably well. At the limb (μ = 0) it gives I/I(1) = 0.4, the canonical "40% as bright" result.

Now carry that into an exoplanet transit. A planet of radius ratio k = R_p/R_⋆ crossing a uniform disk blocks a fixed fraction k² of the light, giving a flat-bottomed box of depth k². But the star is limb-darkened. Consider a planet with k = 0.10 (a hot Jupiter, transit depth on a uniform disk = 0.01, i.e. 10,000 ppm), crossing a Sun-like star with u = 0.6, on a central chord (impact parameter b = 0):

• Near ingress and egress the planet sits over the limb, where local intensity is only ~0.4 of central. It blocks dim light, so the dip is shallower than the box value.
• At mid-transit the planet covers the bright disk center, blocking the most intense light. The floor reaches its deepest point — for a central transit with u = 0.6 the mid-transit depth is roughly 30–40% deeper than the average depth across the chord.
• The result is a curved, "U-shaped with rounded shoulders" floor instead of a flat box. Fitting that curve with a uniform-disk model would underestimate k by several percent — a direct error in the inferred planet radius.

This is why every serious transit pipeline (Kepler, TESS, ground-based follow-up) either fits limb-darkening coefficients as free parameters or fixes them from precomputed stellar-atmosphere grids such as Claret & Bloemen's tables, indexed by effective temperature, surface gravity, metallicity, and bandpass.

Variants: from linear to quadratic to physically motivated laws

The linear law is convenient but imperfect; the true center-to-limb profile has curvature the single coefficient cannot capture. A hierarchy of laws trades simplicity for fidelity:

Law	Form (I(μ)/I(0))	Free params	Notes
Uniform	1	0	No darkening; gives a flat-bottomed transit box.
Linear	1 − u(1 − μ)	1	Leading-order; u ≈ 0.6 for the Sun (visible).
Quadratic	1 − a(1 − μ) − b(1 − μ)²	2	The workhorse for transit fits; captures curvature.
Square-root	1 − c(1 − μ) − d(1 − √μ)	2	Better for hot stars where the profile steepens.
Logarithmic	1 − e(1 − μ) − f·μ·ln μ	2	Good fit near the limb where μ → 0.
Power-2	1 − g(1 − μ^h)	2	Compact, accurate across stellar types (Hestroffer; Morello+ 2017).
Non-linear (4-coeff)	1 − Σ cₙ(1 − μ^(n/2))	4	Claret 2000; matches model atmospheres to <0.1%.

For transit work the quadratic law is the de facto standard because two coefficients capture nearly all the curvature while keeping the fit well-conditioned. Kipping (2013) introduced a clever reparameterization (q₁, q₂) that maps the physically allowed quadratic-coefficient region onto the unit square, letting fitters sample it uniformly without wandering into unphysical (limb-brightened or negative-intensity) territory.

Wavelength, temperature, and why blue limbs are darker

Limb darkening is strongly chromatic. The center-to-limb contrast comes from a temperature difference between deep (center) and shallow (limb) layers; how much brightness that temperature difference produces depends on where you are on the Planck curve.

• Blue / ultraviolet (Wien side): intensity rises near-exponentially with temperature, so even a modest ΔT toward the limb causes a large brightness drop. Limb darkening is strong; u can approach 0.8–0.9.
• Red / near-infrared (toward Rayleigh-Jeans): intensity scales roughly linearly with temperature, so the same ΔT produces a gentle fade. u drops toward 0.3–0.4.
• Mid- and far-infrared, millimeter: darkening is weak, and where the emission forms above the temperature minimum (in the chromosphere) it can flip to limb brightening.

This chromaticity is a feature, not a nuisance. Multi-band transit photometry uses the known wavelength dependence of limb darkening to help break the geometric degeneracy between the planet radius ratio and the impact parameter, and it is one reason JWST transmission spectra are fit with wavelength-dependent limb-darkening coefficients rather than a single set.

Where limb darkening shows up — and gets used

• Exoplanet transits. The rounded floor of a transit is the limb-darkening fingerprint. Correctly modeling it is required for unbiased planet radii, and the Rossiter-McLaughlin effect (the radial-velocity anomaly during transit) likewise depends on the stellar surface brightness profile.
• Eclipsing binaries. Both stars are limb-darkened, shaping the curvature of primary and secondary minima; light-curve codes such as PHOEBE and the Wilson-Devinney program include limb-darkening laws as standard ingredients for deriving stellar radii and temperature ratios.
• Optical / infrared interferometry. Resolving a star's disk directly (e.g. with CHARA, the VLTI, or NPOI) measures the center-to-limb profile head-on and tests model atmospheres; the difference between the "limb-darkened" and "uniform-disk" angular diameter is a routine correction of a few percent.
• Gravitational microlensing. When a point-mass lens transits a source star, the finite-source magnification pattern scans the source's surface-brightness profile, and the light curve directly constrains limb-darkening coefficients of distant stars otherwise impossible to resolve.
• Solar physics. The classic center-to-limb variation of the Sun is a standard test of photospheric models and a tool for calibrating instruments and measuring how line strengths change with viewing angle.
• Stellar pulsation and starspots. Limb darkening modulates how a spot or a non-radial pulsation mode contributes to the disk-integrated light depending on where it sits on the visible hemisphere.

Gravity darkening, limb brightening, and other relatives

Limb darkening is geometric — it depends on viewing angle at fixed location on the star. Several related effects modulate the surface brightness for different reasons:

• Gravity darkening (von Zeipel effect). A rapidly rotating star bulges at the equator, lowering the local surface gravity and effective temperature there, so the equator is genuinely cooler and dimmer than the poles. This is a real surface-temperature variation, distinct from the line-of-sight effect of limb darkening, though both shape disk-integrated light and interferometric images.
• Limb brightening. Reverse the temperature gradient — a chromosphere or corona where temperature rises outward, or optically thin emission from an extended envelope — and the slanted limb sightline samples hotter or more abundant emitting gas, making the edge brighter. The Sun shows this in the far-UV and at millimeter wavelengths.
• Center-to-limb line variation. Spectral lines change shape and strength from center to limb because they form at different heights; this is used to probe atmospheric velocity fields and granulation.

Derivation: from radiative transfer to I(μ) = 1 − u(1 − μ)

The formal emergent intensity from a plane-parallel atmosphere, looking out at angle μ, is the integral of the source function weighted by the attenuation along the slanted path:

I(0, μ) = ∫₀^∞ S(τ) e^(−τ/μ) dτ/μ

Assume the source function is linear in optical depth, S(τ) = a + bτ (the gray, radiative-equilibrium case gives a = (3/4)F·(2/3) and b = (3/4)F). Substituting and integrating term by term:

∫₀^∞ (a + bτ) e^(−τ/μ) dτ/μ = a + bμ

⇒ I(0, μ) = a + bμ = S(τ = μ)

The middle step recovers the Eddington-Barbier relation: the emergent intensity in direction μ equals the source function at optical depth τ = μ. Normalizing by the central intensity I(0, μ = 1) = a + b:

I(μ)/I(1) = (a + bμ)/(a + b)
          = (a/(a+b)) + (b/(a+b)) μ
          = 1 − [b/(a+b)] (1 − μ)

which is precisely the linear law I(μ)/I(0) = 1 − u(1 − μ) with u = b/(a + b). For the gray Eddington atmosphere a/b = 2/3, so u = b/(a+b) = 1/(2/3 + 1) = 0.6 — the standard solar value. The whole effect collapses to a single statement: because the source function rises with depth and the limb sightline samples shallower depth, the limb is dimmer, and the steepness of S(τ) sets u.

Common pitfalls and misconceptions

• "The edge of the Sun is cooler than the center of the Sun." No — the gas at the geometric limb is not intrinsically different from gas elsewhere. What differs is the height in the atmosphere your sightline can see through before it goes opaque. Limb darkening is a viewing-angle effect, not a horizontal temperature map.
• Confusing u as a fraction blocked. u is the fractional intensity drop from center to limb in the linear model, not the total light lost from the disk. Integrating the profile over the disk shows limb darkening reduces the total emergent flux by a smaller amount than u suggests, because most of the disk area is at intermediate μ.
• Fitting a transit with a uniform disk. Ignoring limb darkening biases the recovered planet radius ratio (typically by a few percent) and the impact parameter, and can mimic or hide transit-timing and spot-crossing features.
• Treating limb-darkening coefficients as wavelength-independent. u changes substantially across a bandpass; using a single value for a wide filter or a spectroscopic transit introduces systematic error.
• Assuming every star limb-darkens the same way. Coefficients depend on T_eff, surface gravity, metallicity, and wavelength. A hot A star, a Sun-like G dwarf, and a cool M dwarf have very different profiles; using the wrong grid value propagates straight into the planet parameters.
• Mixing up limb darkening with gravity darkening. Limb darkening is geometric (line-of-sight); gravity darkening is a real equator-to-pole temperature variation on a fast rotator. Both shape light curves but for entirely different reasons.