M073 · Which part of a star is moving?

Radial Velocities, Active Stars and Binary Masses

Does a repeating Doppler shift prove that a planet is there? A repeating Doppler shift can come from orbital motion or a rotating active surface. A planet shifts the whole stellar spectrum; an active region changes the contribution of different surface velocities. Compare line shape, brightness and activity alongside the velocity, then test their joint explanation.

Drag a rotating star’s active region to see false Doppler shifts. Compare its light and photocenter with orbital motion, or measure double-star masses.

Full-line centroid · metres per second $-18.3$

Continuum / quiet star $0.982$

Projected active fraction $0.0264$

Photocenter offset · microarcseconds $2.75$

Move an active patch

Active-region angular width

8

\sigma_{\rm spot}\;[{}^{\circ}]

Darkening contrast; negative brightens

0.7

c_{\rm spot}

Stellar rotation

23.81

P_{\rm rot}\;[\mathrm d]

Planet mass · nominal Earth masses

25

m_{\rm p}/\mathcal M_\oplus^{\rm N}

Spin axis / line of sight

60

i_{\rm spin}\;[{}^{\circ}]

Drag the visible active region to move it across the stellar surface. The line and photocenter come from the same bright and dark patches.

Orbit, convection and approximation limits

Local convective shift suppression

100

\Delta v_{\rm conv}\;[\mathrm{m\,s^{-1}}]

Planet orbital period

12.7

P_{\rm p}\;[\mathrm d]

Orbit inclination

60

i_{\rm orbit}\;[{}^{\circ}]

Orbital eccentricity

0.1

e

Companion periapse direction

35

\omega_{\rm companion}\;[{}^{\circ}]

Distance

20

d\;[\mathrm{pc}]

Negative contrast makes a bright patch. The active-region proxy measures its limb-weighted projected coverage; it is not a calibrated calcium index. The local convective shift is prescribed. Spin and orbit inclinations can differ.

Read the course and compare real HARPS posterior models

Model assumptions

The single-line mode sums identical symmetric Gaussian local absorption profiles over a spherical star with fixed quadratic limb darkening. One prescribed spherical-Gaussian active region changes continuum brightness and can suppress a fixed local convective shift. The line centroid is the complete absorption-deficit first moment; the projected activity fraction is a teaching proxy, without a calibrated calcium emission model. Photocenter motion combines surface brightness and the same Kepler reflex orbit. Nominal solar radius and gravitational parameters set the physical scale. Surface evolution, differential rotation, realistic line bisectors, oscillations, granulation, relativistic shifts, finite companion occultations and instrumental noise are omitted. The real HARPS tutorial is separate. The original binary mode retains its complete 48-pair synthetic double-line velocity/relative-sky record, true versus adopted inclination/period/distance, known sky orientation and enlarged point-mass star markers; eclipses and collisions are not computed. Positive radial velocity recedes from the observer in both modes.

Synthetic surface integration and the same Kepler orbit. Cyan is the total line centroid; the gray reflex reference excludes surface activity.

The unresolved absorption line at the current time

Line centroid and orbital reflex reference

Continuum change and projected active fraction

Brightness photocenter in the observer sky

Physical units and numerical/display conventions

The stellar brightness and active region use one prescribed surface law. A Gaussian angular width describes a smooth region, not a sharp circular edge. The gray spectrum reference is a quiet rotating star with the same orbital shift. Each local absorption line has the same width and depth; real line asymmetries and calcium radiative transfer are outside this model.

The star has one nominal solar radius and gravitational mass parameter. Planet mass uses nominal terrestrial gravitational units. Photocenter includes brightness structure as well as stellar orbital position; the scene enlarges the reflex displacement, while all numerical sky positions use their unexpanded physical values.

Saved 60-day synthetic observation; surface animation marks the current day.

Light recorded from the rotating surface

Noisy exposure averagesSame-surface noiseless reference

Clock probability, keeping alternatives

One-pattern clockTwo-pattern clock

Age conditional on a fixed empirical scale

Three cluster anchors; the complete clock mixture enters the age calculation.

The same surface also changes an absorption line

$t_{\rm obs}=1.43\,\mathrm d$

Current instantaneous synthetic lineSame rotating surface without the active regionDifference, on its own scale below

This line uses the current surface clock and settings. Each curve is normalized to its own continuum; the local Gaussian width is prescribed. The light curve above contains saved exposure averages.

A separate wind-braking clock

This teaching model keeps stellar inertia fixed. Its prescribed wind torque slows rotation; it is independent of the empirical age density above. The green point is an evolutionary snapshot, not the current observing day.

Initial spin period in daysElapsed evolutionary age in Gyr

$t=1.00\,\mathrm{Gyr},\quad P=9.55\,\mathrm d$

The coefficient and saturation threshold are stated in the course. No contraction, structure evolution or mass/metallicity calibration is included; this curve never supplies the age inference.

Why can the clock fail?

A dark Gaussian patch on a solar-size sphere changes continuum brightness. The fixed limb law, local rotation speed and any prescribed convective shift use the same cells. A latitude-dependent rotation law moves the patch and its meridians together. The Gaussian envelope fades brightness, convection and the active-area proxy together; its width is not a physical birth or death date.

Inference reads saved fluxes, errors, exposure lengths and times. Four stationary harmonics have proper Gaussian amplitude priors. A declared equal prior keeps one-pattern and two-pattern clocks; daily aliases also remain. A Bayes factor against a constant-plus-trend model is not a false-alarm probability. Weak signals, large residuals or an observing window shorter than two rotations withhold age.

The calibration uses observed rotation periods from Praesepe, NGC6811 and Ruprecht147. Its effective scatter includes unknown period errors and physical dispersion. The temperature support is $4245.37\leq T_{\rm eff}/\mathrm K\leq5832.12$ ; fixed ages span $0.67\leq t/\mathrm{Gyr}\leq2.7$ . These are calibration boundaries, not physical age limits. Cool-star stalling and shared extinction/age-scale assumptions remain. No absolute age precision is claimed.

Read the course and reproduce the cluster comparison

Start with the essentials

Focus question: Does a repeating velocity prove an orbit?
One-sentence intuition: The full-line centroid is a brightness-weighted velocity, and its weights can change as the star turns.

Core mathematical model

Same surface, several observables

F=\int W\,\mathrm dA,\quad \overline v=F^{-1}\int Wv\,\mathrm dA,\quad X=F^{-1}\int Wx\,\mathrm dA

The spectrum, light and photocenter use the same projected-area light weights. The activity proxy is a prescribed area response, not observed calcium.

The quiet Kepler limit

K=\left(\frac{2\pi G}{P}\right)^{1/3}\frac{m\sin i}{(M_\star+m)^{2/3}\sqrt{1-e^2}}

A symmetric quiet surface recovers this orbital scale. Single-line velocities constrain minimum mass unless inclination is independently known.

One latent activity process

y_v=m_{\rm Kep}+aG+b\dot G+\epsilon_v,\qquad y_h=AG+\epsilon_h

The public tutorial retains a posterior over the allocations and full covariances. It does not subtract one best activity curve and call the remainder certain.

Common difficulties

The darkest moment

Typical misconceptionThe greatest brightness dip must be the largest velocity excursion.

Better mental modelA central dark region removes little rotational velocity. Off-center missing light can produce a larger centroid shift.

A successful fit

Typical misconceptionA small residual proves that the chosen planet is real.

Better mental modelActivity can mimic or absorb an orbit. Check independent indicators, prior sensitivity, observing-window injections and sampling diagnostics.

Run the experiment

01
Remove the companion
Choose A spot without a planet, then drag the region across the visible disk.
What to observe: The gray orbital reference is zero, but the cyan centroid and light change.
02
Change the mechanism
Choose A bright magnetic patch, then change convective suppression in the advanced controls.
What to observe: The line center changes even when contrast is held fixed. Brightness alone cannot remove that contribution.
03
Recover the quiet orbit
Choose A quiet star wobbles and vary orbital inclination.
What to observe: The centroid follows the companion-only reference; face-on projection hides the radial component.
04
Keep the original binary
Open Binary orbit. Fit the original 48 paired observations under its adopted geometry.
What to observe: The two velocities constrain mass ratio; adopted inclination and distance still matter for absolute masses.

Original binary assumptions

Original binary scope: isolated Newtonian point masses in AU, Julian years and solar masses with the documented teaching gravitational constant. No eclipses, spectra, light-travel time or relativity. Orbit shape and epoch are known; period, inclination and distance are adopted. Amplitude standard errors are conditional residual errors, not full mass uncertainties.

Physics tutorial

Measuring a mass means measuring a geometry

BackgroundAn isolated pair of stars orbits its barycenter under Newtonian gravity. Spectroscopy sees motion along the line of sight; astrometry sees the two projected sky coordinates. Each one hides a different part of the same three-dimensional orbit.

Why it mattersAn orbit viewed face-on can be moving rapidly while its line-of-sight velocity stays constant. Miller's discussion of binary mass functions explains the missing projection. Here the orbital shape and epoch are given, so the regression tests amplitudes under an explicitly adopted geometry.

Start with the essentials

Focus question: Which observations determine mass, and which assumptions still remain?
One-sentence intuition: Fitting both velocity curves together measures the amplitudes and the mass ratio, but the total mass also needs the viewing geometry. Angular motion needs the distance as well. Clean residuals alone cannot confirm either outside assumption.

Core mathematical model

Two bodies, one relative orbit

\mathbf r_1=-\frac{M_2}{M_1+M_2}\mathbf r,\qquad\mathbf r_2=\frac{M_1}{M_1+M_2}\mathbf r

The heavier star moves on the smaller orbit around the barycenter. The relative orbit shown is the separation vector between the stars.

Period and total mass

P^2=\frac{4\pi^2a^3}{G(M_1+M_2)}

In the teaching units of AU, years and solar masses, the gravitational constant equals four pi squared. The semimajor axis is that of the relative orbit.

Projected sky motion

X=x\cos\omega-y\sin\omega,\quad Y=(x\sin\omega+y\cos\omega)\cos i,\quad Z=(x\sin\omega+y\cos\omega)\sin i

Tilting the orbit shrinks one sky coordinate and creates velocity along the line of sight. The angular separation also depends on distance.

Fitting the velocities of both stars

v_1=\gamma-K_1[\cos(\nu+\omega)+e\cos\omega],\quad v_2=\gamma+K_2[\cos(\nu+\omega)+e\cos\omega]

The linear regression fits a shared systemic velocity and two amplitudes. The orbit's shape and phase are taken from independent information, and a wrong adopted period changes the basis functions of the fit.

What spectroscopy determines

\frac{M_2}{M_1}=\frac{K_1}{K_2},\qquad (M_1+M_2)\sin^3i=\frac{P(K_1+K_2)^3(1-e^2)^{3/2}}{2\pi G}

Without the viewing inclination, the projected mass is only a lower bound, not the true mass. Seen nearly face-on, the radial signal fades away; the Lab requires both amplitudes to exceed three conditional standard errors.

Distance enters cubed

a_{\rm fit}=\theta_{a,\rm fit}d_{\rm fit},\quad M_{\rm ast}=\frac{4\pi^2a_{\rm fit}^3}{GP_{\rm fit}^2},\quad M_{\rm RV}=\frac{M_{\rm proj}}{\sin^3i_{\rm fit}}

The fit of the angular semimajor axis assumes the adopted projection geometry. If the distance is off by some factor, the astrometric mass is off by that factor cubed, even when the angular residuals are tiny.

Common difficulties

Projection is not mass

Typical misconceptionA fitted velocity amplitude determines the mass by itself.

Better mental modelRadial velocities give the mass times the cube of the sine of the inclination. Look at the whole family of masses that fits.

A vanishing signal does not reveal the masses

Typical misconceptionFace-on motion means the stars have no mass or do not orbit.

Better mental modelThe orbit is still there on the sky. Both radial signals vanish together, so the mass ratio cannot be determined.

Good residuals do not verify assumptions

Typical misconceptionA tiny residual proves the adopted distance and inclination.

Better mental modelDifferent physical masses and viewing geometries can produce the same velocity record. The distance comes from outside the angular fit.

Standard errors are conditional

Typical misconceptionThe amplitude standard error is a complete error bar on the mass.

Better mental modelIt leaves out the uncertainty in period, orbital shape, phase, distance and inclination. Close to face-on, the inversion is poorly determined.

Run the experiment

01
Recover a clean binary
Compare the loaded velocity record with its mass estimates. Choose Recover the masses for the noiseless limit; complete results include the true and astrometric masses.
What to observe: The amplitudes give the mass ratio, and the correct adopted inclination gives back the total mass.
02
Separate viewing from analysis
Drag the purple star to change the true inclination, then open Advanced controls and set the adopted inclination to the same value.
What to observe: The projected mass changes with viewing angle. The corrected mass matches only when the adopted angle is right.
03
Diagnose a bad period
In Advanced controls, compare Wrong period and Noisy observation record, then open Conditional residuals and press New observation record.
What to observe: A wrong period leaves a pattern in the residuals, while seeded Gaussian errors scatter around the conditional fit.
04
Question a clean-looking result
Select Wrong adopted distance, then Face-on weak signal.
What to observe: A clean angular fit can still give a biased mass. Near face-on, spectroscopy cannot resolve the masses even though the orbit is visible on the sky.

Start with the essentials

Focus question: When does a starspot clock give a useful stellar age?
One-sentence intuition: Infer periods from saved observations, retain alternatives, then combine their likelihood with a temperature-dependent calibration.

Core mathematical model

A latitude-dependent clock

\Omega(\lambda)=\Omega_{\rm eq}(1-\alpha\sin^2\lambda)

The same phase law moves the patch and its surface meridian; local cell velocities follow it too.

An empirical age likelihood

\mathcal L(t)=\sum_m\pi_m\int\!\int Z(m/P)p(P\mid t,T)L_T(T)\,dP\,dT

The original saved-flux likelihood enters once. Every supported period contributes.

A separate teaching wind clock

P^2=P_0^2+8\pi^2\kappa t

The unsaturated constant-inertia branch gives this relation. Evolutionary time is in Gyr; the star animation uses days.

Common difficulties

A pattern can repeat twice per turn

Typical misconceptionThe tallest period peak always equals equatorial spin.

Better mental modelLatitude and multiple active longitudes can change the clock. Saved light retains alternatives instead of reading injected geometry.

The calibration endpoint is conditional

Typical misconceptionThe oldest supported cluster sets a physical upper age bound.

Better mental modelProbability at an edge asks for older calibrators or another age method. A finite support alone establishes no physical limit.

Run the experiment

01
Move the patch to high latitude
Choose the high-latitude preset, pause, and drag the visible dark region.
What to observe: Its local clock changes under differential rotation; the equatorial period remains a separate simulation parameter.
02
Compare a changing pattern
Select the short-lived and pole-on presets in turn.
What to observe: The saved-data checks withhold age for nonstationary or unsupported modulation.
03
Change only the age assumption
Restore the lasting clock and compare the two age priors.
What to observe: The same saved light yields a different conditional age density when the prior changes.

Core mathematical model

Same surface, several observables

The quiet Kepler limit

One latent activity process

Common difficulties

The darkest moment

A successful fit

Run the experiment

Remove the companion

Change the mechanism

Recover the quiet orbit

Keep the original binary

Core mathematical model

Two bodies, one relative orbit

Period and total mass

Projected sky motion

Fitting the velocities of both stars

What spectroscopy determines

Distance enters cubed

Common difficulties

Projection is not mass

A vanishing signal does not reveal the masses

Good residuals do not verify assumptions

Standard errors are conditional

Run the experiment

Recover a clean binary

Separate viewing from analysis

Diagnose a bad period

Question a clean-looking result

Core mathematical model

A latitude-dependent clock

An empirical age likelihood

A separate teaching wind clock

Common difficulties

A pattern can repeat twice per turn

The calibration endpoint is conditional

Run the experiment

Move the patch to high latitude

Compare a changing pattern

Change only the age assumption

Related experiments and articles

From starspot clocks to stellar age

A star can fake a wobble

Gravity leaves a trace

A planet moves its star