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Sandbox Physics
Stellar Activity and Radial Velocity

AC18 · L2

Follow a spot’s missing light

Sandbox Physics ·

The star’s right half has not slowed down. It has become less visible. A weighted average turns that distinction into an apparent velocity, and the same weights also move the star’s light center.

Start with the light that reaches us

Use projected coordinates across a unit stellar disk. The observer lies on negative zz, the visible surface has z<0z<0 and μ=−z\mu=-z is the local line-of-sight cosine. The quadratic quiet intensity describes the darker limb. The chosen coefficients are illustrative, fixed for this solar-radius teaching star, not a fit to CoRoT-7.

Iq(μ)=1−0.45(1−μ)−0.20(1−μ)2,W=Iq(μ)[1−csg]I_q(\mu)=1-0.45(1-\mu)-0.20(1-\mu)^2,\qquad W=I_q(\mu)[1-c_s g]
The contrast is positive for a dark region and negative for a bright one. Projected area belongs in every integral. On a surface-area grid that same projection would require an extra factor of the line-of-sight cosine.

The region is a smooth spherical Gaussian, g=exp⁡[−(1−n⋅ns)/σs2]g=\exp[-(1-\mathbf n\cdot\mathbf n_s)/\sigma_s^2], with angular width in radians. This avoids a hard cell-membership edge while keeping a localized region that rotates. It is not a uniform circular spot with a sharp radius. Surface latitude and longitude specify its center; direct dragging changes those physical coordinates.

Reading: Dumusque, Boisse & Santos — SOAP 2.0, arXiv:1409.3594v1 · IAU 2015 Resolution B3 — nominal conversion constants

Add the absorption, then take its moment

Assume every patch has the same symmetric Gaussian absorption profile, width and equivalent width. Its center combines the whole-star reflex velocity, local rotation and a prescribed active convective redshift. A patch’s continuum weight also multiplies its absorption deficit. Summing first, then integrating the line, gives a particularly useful simplification: each symmetric local profile has first moment equal to its center.

vcell=v⋆+ΩR⋆sin⁡i⋆ x+Δvcg,v‾=∫Wvcell dA∫W dAv_{\rm cell}=v_\star+\Omega R_\star\sin i_\star\,x+\Delta v_c g,\qquad \overline v=\frac{\int Wv_{\rm cell}\,\mathrm dA}{\int W\,\mathrm dA}
Positive velocity recedes. The displayed line is synthesized from these same weighted centers. A real cross-correlation velocity or bisector can respond differently when local line shapes and equivalent widths vary.

With no active perturbation, symmetry cancels the odd rotational moment exactly. The centroid becomes the Kepler reflex velocity. In the implementation the quiet odd moment is removed analytically before summation, which avoids cancellation between large positive and negative contributions. The plotted difference from a quiet spectrum reveals a small distortion that would otherwise be difficult to see beside the full line.

Reading: Dumusque, Boisse & Santos — SOAP 2.0, arXiv:1409.3594v1 · Rajpaul et al. — joint activity processes, arXiv:1506.07304v1

A one-percent patch, worked all the way through

Imagine an otherwise symmetric unit-flux disk. Remove a fraction 0.01 of its original light from a small patch moving away at 1000 metres per second, at horizontal position half a stellar radius. Ignore convection and companions. The remaining flux is 0.99. The original first moment was zero; after removal it is minus 10 in flux-weighted velocity units. Divide by the remaining flux to obtain about minus 10.1 metres per second.

v‾=−0.01(1000)0.99 m s−1,XphotR⋆=−0.01(0.5)0.99\overline v=\frac{-0.01(1000)}{0.99}\ \mathrm{m\,s^{-1}},\qquad \frac{X_{\rm phot}}{R_\star}=\frac{-0.01(0.5)}{0.99}
The denominator matters because the star is now fainter. The horizontal photocenter is about minus 0.00505 stellar radii. At 20 parsecs, a nominal solar radius subtends about 233 microarcseconds, giving about minus 1.18 microarcseconds.

In a rigidly rotating, flux-only model, the rotational velocity is linear in horizontal position. The integrals therefore share an exact proportionality after subtracting the stellar orbit. This is a useful independent check, not a universal activity correction.

Δvrot=Ωsin⁡i⋆ Xphot,surface\Delta v_{\rm rot}=\Omega\sin i_\star\,X_{\rm phot,surface}
The position here is a physical length along the rotational velocity gradient. Adding an active convective shift breaks this relation even if the continuum brightness is unchanged.

Reading: Dumusque, Boisse & Santos — SOAP 2.0, arXiv:1409.3594v1 · IAU 2015 Resolution B3 — nominal conversion constants

What the numerical surface does and leaves out

The browser integrates a smooth visible-hemisphere coordinate chart using Gauss–Legendre nodes. Ordinary presets use order 48 in each coordinate; a two-degree region raises that order to 129. The independent reference uses polar area quadrature and an independently solved eccentric anomaly. Across 35 preset and extreme samples, the adaptive grid differs by at most 0.0000161 metres per second in centroid and about 0.000000501 microarcseconds in photocenter. These are finite reference-comparison errors, not physical accuracy claims.

A fixed order-48 grid misses the narrow-region extreme by 0.175 metres per second. That failed resolution test is retained. The sampled spectrum uses 80-metre-per-second velocity bins with linear deposition, which preserves the first moment; this bin width is not an instrument’s achievable precision. The continuous centroid and the sampled profile agree to the stored numerical tolerance.

The model omits differential rotation, evolving multiple regions, granulation, limb-dependent convection, spectral-type changes and occultations. Its active-area proxy has no calcium calibration; its Gaussian-equivalent line width is not the published HARPS fitted FWHM. It predicts a controlled mechanism, not the true surface of the observed star.

Actual synthetic records generated by check-surface.ts. Nominal solar radius and GM, distance 20 pc; sample phase 0.9 radian of rotation. Full settings and 121 acquired times per case are in default-results.json. These are not HARPS observations.
CaseContrastConvective shift · m/sNominal Earth massesCentroid · m/sFlux / quiet
Spot and planet0.70.71001002525−18.3-18.30.9820.982
Quiet orbit0000909015.715.711
Only a spot0.70.70000−24.3-24.30.9820.982
Bright magnetic patch−0.08-0.084004000034.434.41.011.01
Face-on0.70.71001001001001.371.370.9850.985

Reading: Dumusque, Boisse & Santos — SOAP 2.0, arXiv:1409.3594v1 · Rajpaul et al. — joint activity processes, arXiv:1506.07304v1

Exercises and explanations

  1. Double the removed-light fraction in the worked example. Does the centroid exactly double?

    Read the explanation

    It becomes −20/0.98≃−20.4-20/0.98\simeq-20.4 metres per second, slightly more than twice the original magnitude. The missing first moment doubles, but the remaining-flux denominator also decreases.

  2. A photocenter stays fixed while the line shifts. Which assumption in the proportionality could have failed?

    Read the explanation

    A changed local convective shift is one possibility: it changes the line center without needing a matching brightness displacement. Unequal local equivalent widths or line-shape changes also break the shared weighting assumption. It would be premature to identify the residual as a planet.

Try the prediction

Test the shared weighting

  1. Choose “A spot without a planet” and move the region across the equator. Compare the sign of horizontal photocenter and rotational velocity.
  2. Add convective suppression in the advanced controls without changing contrast. The photometric weights stay fixed while the spectral center changes.
  3. Export a record. Its current observation is evaluated directly at the displayed time; the 121 campaign samples are a separate fixed sampling grid.

Sources and calculation scope

  1. Dumusque, Boisse & Santos — SOAP 2.0, arXiv:1409.3594v1

    11 September 2014 version. Sections 2.1–2.3, Equations 1–7 and Figure 1: surface-profile integration, flux contrast and convective blueshift. Our identical Gaussian local profiles are a simpler independent mechanism model.

  2. Rajpaul et al. — joint activity processes, arXiv:1506.07304v1

    24 June 2015 version. Sections 3.1–3.4 and 4.1–4.2, Figures 3–4, Appendix A.2. Derivative covariance signs follow a lag equal to first time minus second time; kernel-width normalizations must be kept with their equations.

  3. IAU 2015 Resolution B3 — nominal conversion constants

    Official adopted table: nominal solar and terrestrial gravitational parameters and nominal solar radius are exact conversion constants, not uncertainty-free measurements of the real Sun and Earth.

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