Rotating Star with Spots

A spotted, limb-darkened photosphere rotating under starry (Luger+ 2019) — the surface is rendered as a Planck-blackbody disk per pixel and the synchronized light curve is the integrated flux that a distant observer sees.

Luger, Agol, Foreman-Mackey, Fleming, Lustig-Yaeger, Deitrick · 2019 · AJ 157, 64 · arXiv:1810.06559
MODEL CLASS  FORWARD-MODELLED PHOTOSPHERE  a limb-darkened Planck disk; the light curve is its integrated flux
5772
Teff · K
2.0
v sin i · km/s
90
inclination · °
25.4
Prot · d
1
active spots
0.8
ΔF · %
Photosphere · Limb-Darkened · Single-Hemisphere View

Rotating star with cool spots · live photometry

The 3D sphere is a unit photosphere with quadratic limb darkening I(μ) = 1 − u₁(1−μ) − u₂(1−μ)². Each spot is a cooler patch (T_spot = c·T_eff with contrast c < 1) at fixed lat/lon; the disk color is the Planck-blackbody color at the local temperature. The light-curve panel at lower-left shows the integrated flux a distant observer sees as the star rotates.

View

Surface state · live

rotation phase0.00
t / Prot0.00
F / F★1.0000
visible spots1
v sin i2.0 km/s
Limb darkening: I(μ) = 1 − u₁(1−μ) − u₂(1−μ)². Photosphere color via Planck blackbody at Teff; spots cooler by factor c. Light curve integrated on a 96×96 visible-disk grid per rotation phase.

Light curve · synchronized

Prot = 25.4 d
0rotation phase φ2π
φ = 0.00 · 0.0 d
Entry I · Entry II

A spotted star is one of the few astrophysical objects whose entire detectable signature can be computed from first principles in closed form. Given a surface intensity map (photosphere plus cool spots), a limb-darkening law, an inclination, and a rotation rate, the observed flux as a function of time is an analytic integral over the visible hemisphere — that's the contribution of Luger+ 2019's starry package, which casts the problem in a spherical-harmonic basis and gives the flux integral in closed form. The animation here renders the disk directly through a fragment shader (one pixel = one surface point) and computes the photometric light curve numerically on a 96×96 disk grid; both views are exact in the spherical-rigid-photosphere limit. Below are the boundary conditions where that limit fails.

Source parameters

Stellar parameters

5772 K
M (3000 K) → F (7000 K) → A (10000 K) · sets photosphere color
25.4 d
log scale · 0.1 d (rapid) to 160 d (slow) · controls v sin i
90°
0° = pole-on (no LC modulation) · 90° = equator-on (max)
1.00 R☉
Sets v sin i for the chosen Prot (2πR sin i / P)
0.40
Quadratic (Mandel & Agol 2002) · Sun V-band ≈ 0.40
0.27
Quadratic coefficient · Sun V-band ≈ 0.27
0.15
Ω(θ) = Ωeq(1 − kDR sin²(colat)) · Sun-like ≈ 0.15

Active regions · click the disk above to place a spot · shift-click to remove · or use sliders

Boundary of validity

Where this model is the right oneread this

The starry analytic framework is exact for rigid-body rotation of a spherical limb-darkened photosphere with a finite number of bounded surface features. Within that domain the photometric prediction is correct to numerical precision; the boundaries below are where the spherical rigid-photosphere assumption breaks down or where a real spotted star has structure the model doesn't carry.

Rigid sphere · no oblateness

The star is modelled as a perfect sphere with a single Teff. Rapid rotators (e.g. Achernar at v sin i ≈ 250 km/s) are visibly oblate and show von Zeipel gravity darkening — equator cooler than the pole by > 1000 K. Not in this model.

v sin i = 2 km/s · spherical OK
No magnetic structure

Real starspots have umbra/penumbra concentric structure, are surrounded by brighter faculae, and have Zeeman-broadened photospheric lines. Here a spot is a single uniform-T cool patch.

uniform-T patches
Static spot list within a cycle

Spots are fixed in (lat, lon, radius, contrast) for the duration of the animation. Real spots emerge over days, decay over weeks, and migrate in latitude over a magnetic activity cycle (Spörer's law on the Sun).

no spot evolution
No flares

Magnetic reconnection events overlaid on active regions produce rapid optical/UV brightening (the > 10²⁴ J Carrington-class events on M dwarfs). Not modelled.

no flare physics
Single broadband · LD coefficients fixed

The light curve is shown in a single broadband. Real photometry in different filters (TESS / Kepler / ZTF g, i) gives different LC amplitudes because the limb-darkening coefficients depend on wavelength. A v2 multi-band view would expose the chromatic spot signature.

single LC band
Quadratic LD · valid in mid-disk

The two-parameter LD law fits well for μ > 0.2 (most of the disk) but underestimates intensity near the limb. Stars hotter than ~10000 K or cooler than ~3000 K really want a 4-parameter Claret law.

quadratic LD only

References

  • Luger, R., Agol, E., Foreman-Mackey, D., Fleming, D. P., Lustig-Yaeger, J., Deitrick, R. 2019, AJ 157, 64 — "starry: Analytic Occultation Light Curves" — closed-form spherical-harmonic flux integrals for spotted, limb-darkened bodies — arXiv:1810.06559
  • Mandel, K., Agol, E. 2002, ApJL 580, L171 — Analytic light curves for planetary transits with quadratic limb darkening — arXiv:astro-ph/0210099
  • Claret, A., Bloemen, S. 2011, A&A 529, A75 — Limb-darkening coefficients for Kepler, CoRoT, Spitzer, uvby, UBVRIJHK, and Sloan photometric systems — doi:10.1051/0004-6361/201116451
  • Vogt, S. S., Penrod, G. D., Hatzes, A. P. 1987, ApJ 321, 496 — Doppler imaging of spotted stars (the foundational paper, motivating the v2 line-profile view) — doi:10.1086/165647
  • Strassmeier, K. G. 1999, A&A 347, 225 — Doppler image of HD 12545 with the record 22% spot coverage
  • Berdyugina, S. V. 2005, Living Rev. Solar Phys. 2, 8 — Starspots review: morphology, magnetic structure, lifetime, latitude migration — doi:10.12942/lrsp-2005-8
  • Strassmeier, K. G. 2009, A&A Rev. 17, 251 — Comprehensive review of starspot observations and physics
  • Schrijver, C. J., Zwaan, C. 2000 — Solar and Stellar Magnetic Activity, Cambridge University Press — reference for solar-like activity cycles and spot physics
  • Reinhold, T., Reiners, A., Basri, G. 2013, A&A 560, A4 — Kepler differential rotation measurements: k_DR distribution across the FGK main sequence — arXiv:1308.1508
  • McQuillan, A., Mazeh, T., Aigrain, S. 2014, ApJS 211, 24 — Rotation periods of 34,030 Kepler main-sequence stars from spot-modulation autocorrelation — arXiv:1402.5694
  • Désert, J.-M. et al. 2011, ApJS 197, 14 — Kepler-17 system properties used as preset — arXiv:1107.5750
  • Skelly, M. B. et al. 2010, MNRAS 403, 159 — V410 Tau Doppler-imaging map showing the polar cap
  • Berdyugina, S. V. 1998, A&A 338, 97 — II Peg long-term Doppler imaging program
  • Donati, J.-F. et al. 1999, MNRAS 302, 437 — AB Dor Zeeman-Doppler-imaging baseline
  • Tanner-Helland blackbody color approximation — analytic RGB approximation to Planck spectrum used for the photosphere color shader