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Sandbox Physics
Stellar Activity and Radial Velocity

AC18 · L2

The speed hidden in a line

Sandbox Physics ·

A wavelength shift is a measurement. A planet mass is a conclusion reached through several more steps. Keeping those steps visible makes a tiny velocity much easier to interpret.

From a wavelength to a velocity

Choose the line-of-sight coordinate to increase away from us. In the low-speed limit a receding source stretches wavelengths. Compare a line to its rest wavelength, form the fractional change and multiply by the speed of light. Our new surface model uses metres, seconds and nominal solar and terrestrial gravitational parameters; the original binary mode retains its documented astronomical-unit teaching convention.

vr≃cλ−λ0λ0v_r\simeq c\frac{\lambda-\lambda_0}{\lambda_0}
This is the first-order Doppler relation, after accounting for the observing frame and reference wavelength. Stellar convection, gravitational shifts and instrument offsets can contribute to a reported absolute velocity.

For a line at 600 nanometres, a 1 metre-per-second shift is about 0.000002 nanometres. Precision spectroscopy does not read that displacement from a single screen pixel. It combines calibrated information across many line profiles. The Lab’s absorption notch is an intentionally simple local-profile model; it is not a simulated HARPS reduction.

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

Let the companion supply the acceleration

Write the relative orbital semimajor axis as aa and the stellar orbit as a⋆a_\star. Newton’s force law and the two-body center of mass give the first two equalities below. The orbital angular frequency then turns the stellar orbital radius into a speed. Projection supplies the sine of the inclination.

n2a3=G(M⋆+m),a⋆=mM⋆+man^2a^3=G(M_\star+m),\qquad a_\star=\frac{m}{M_\star+m}a
Here n=2π/Pn=2\pi/P and inclination zero means the orbit is face-on. The radius refers to the relative orbit before the mass fraction is applied.
K=(2πGP)1/3msin⁡i(M⋆+m)2/31−e2K=\left(\frac{2\pi G}{P}\right)^{1/3}\frac{m\sin i}{(M_\star+m)^{2/3}\sqrt{1-e^2}}
For an eccentric Kepler ellipse, the speed scale gains the factor involving eccentricity. This semi-amplitude is conditional on an orbital model; an activity centroid is not automatically an orbital velocity.

A nominal Earth gravitational mass around a nominal Sun in a circular one-Julian-year orbit, viewed edge-on, produces about 0.089 metres per second. An inclination of 30 degrees halves that amplitude. The same observed amplitude would then require about twice the companion mass in the small-companion limit. This is why a single-lined velocity often yields a minimum mass, rather than the mass itself.

Reading: IAU 2015 Resolution B3 — nominal conversion constants · Haywood et al. — CoRoT-7, arXiv:1407.1044v1

Mind whose periapse you named

The Lab’s angle names the companion’s periapse. The stellar position and velocity point oppositely around the common center. Thus its formula has a minus sign. The more familiar plus-sign expression instead names the star’s periapse, shifted by half a turn. Both conventions describe the same orbit if used consistently.

v⋆=−K[cos⁡(ν+ωcompanion)+ecos⁡ωcompanion]+γv_\star=-K\left[\cos(\nu+\omega_{\rm companion})+e\cos\omega_{\rm companion}\right]+\gamma
True anomaly is measured along the companion’s relative orbit; the constant offset is the chosen velocity reference. The displayed synthetic record sets that reference to zero.

For a circular orbit the sine and cosine coefficients absorb the phase, and periapse no longer has an independently meaningful direction. The real-data tutorial therefore uses two circular components with free phases, rather than claiming to measure eccentricity from one short campaign. The known short period is supplied externally, not selected by searching every possible period.

Reading: IAU 2015 Resolution B3 — nominal conversion constants · Haywood et al. — CoRoT-7, arXiv:1407.1044v1

What would finish a mass measurement?

You still need a stellar mass and an inclination. A transit can constrain inclination under a transit geometry model. Astrometry can constrain a sky orbit when its amplitude and orientation are resolved. Neither exists as an observed datum in this Lab’s synthetic surface scene. The face-on preset lets you see the degeneracy: substantial motion across the sky can accompany zero orbital radial velocity.

The original double-lined binary mode offers a complementary limit. Two velocity amplitudes constrain the mass ratio; adding a resolved orbit or an adopted inclination and scale supplies the remaining mass information. Its conditional fit holds the documented orbit shape and timing fixed. Do not carry its small amplitude error bars into a claim that all orbital and calibration uncertainties vanished.

Reading: Haywood et al. — CoRoT-7, arXiv:1407.1044v1 · Suárez Mascareño et al. — GJ 1002, arXiv:2212.07332v2

Exercises and explanations

  1. In a circular small-companion system, the period doubles while stellar mass and edge-on companion mass stay fixed. By what factor does the amplitude change?

    Read the explanation

    By 2−1/3≃0.7942^{-1/3}\simeq0.794. Kepler’s law makes the orbit larger, but the longer traversal time wins. This scaling assumes the orbit remains Newtonian and the companion mass stays small compared with the star.

  2. A line centroid gives 3 metres per second, but the flux-only spot model predicts 2 metres per second with the same sign. Can you subtract them and announce the mass corresponding to the remainder?

    Read the explanation

    Only within a justified joint model with uncertainty in both contributions. Surface contrast, latitude and convective shifts may be unknown, and their uncertainties are correlated with the orbital amplitude. A deterministic subtraction would hide that uncertainty.

Try the prediction

Hide an orbit along the sky

  1. Select the quiet-star preset. Reduce orbital inclination in the advanced controls while keeping companion mass fixed.
  2. Compare the radial-velocity reference with the unamplified angular track below the scene.
  3. Return to a tilted orbit and change the period. Predict the amplitude trend before looking at the data.

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. Haywood et al. — CoRoT-7, arXiv:1407.1044v1

    3 July 2014 version. Sections 2–3 and 4.2, Equations 1–12 and Table A1. All 71 HARPS2012 numerical rows are retained; the tutorial does not reproduce the full contemporaneous-photometry analysis.

  3. Suárez Mascareño et al. — GJ 1002, arXiv:2212.07332v2

    25 January 2023 arXiv version; manuscript dated 26 January. Sections 2.1–2.2 and 4.1–4.4, Table 2, Table B.1, Figures 9–10. Multiple instrument states and minimum-mass interpretation are reading targets; no original figures are republished.

  4. 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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