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Photon-weighted barycentric correction and its importance for precise radial velocities

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Pith's one-line read The standard barycentric correction for long exposures leaves a systematic radial-velocity error, sometimes over 1 m/s, because the correction curves with Earth's rotation rather than changing linearly.

desk verdict A clean, useful derivation of the second-order barycentric correction error that deserves referee time; the central claim holds up. read the letter →

arxiv 1908.00991 v1 pith:U5SHOZVW submitted 2019-08-02 astro-ph.EP astro-ph.IM

René Tronsgaard , Lars A. Buchhave , Jason T. Wright , Jason D. Eastman , Ryan T. Blackman This is my paper · ORCID
classification astro-ph.EPastro-ph.IM
keywords barycentriccorrectionphoton-weightedmidpointtimeexposuremeterradialvelocityprecisionsecond-ordererrorEarthrotationexoplanetvelocities
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper is trying to establish that the standard way of applying the barycentric correction to a long spectroscopic exposure—evaluating the correction once at the photon-weighted midpoint time—leaves a systematic second-order error because the correction curves over the exposure rather than changing linearly. That error is typically about 10 cm/s for 30–60 minute exposures but can pass 1 m/s for realistic worst-case targets observed low in the east or west near the celestial equator. The correct procedure, the paper argues, is to photon-weight the correction itself, using the exposure meter flux curve $f(t)$ as the weight in $\langle v\rangle = \int v(t)f(t)\,dt/\int f(t)\,dt$. The paper therefore recommends that precision radial velocity instruments record and archive the full exposure meter flux curve, and shows that archival data without the curve can be partly repaired with a constant-flux correction term.

What carries the argument

The central object is the photon-weighted average in Equation (3), with the exposure meter flux curve $f(t)$ acting as the smoothing kernel applied to the instantaneous barycentric correction. Expanding $v(t)$ around $\langle t\rangle$ to second order, the missed term is proportional to the time variance of the photon arrival distribution, $\langle t^2\rangle-\langle t\rangle^2$, multiplied by the local curvature $2\pi^2 V_0\sin\psi/(24\,\mathrm{h})^2$; for a uniform exposure the variance becomes $\Delta t^2/12$, producing the compact scaling $\propto \cos(\mathrm{lat})\cos(\delta)\sin\psi\,(\Delta t)^2$ in Equation (10). This machinery turns a correction that is normally a single number into a filter over the exposure, and it predicts that flux curves concentrated at one end shrink the error while curves with a mid-exposure dip grow it.

What would settle it

Take a 60-minute exposure at a low-latitude site on a target with declination near zero observed through the east, record the exposure-meter flux curve, and compare $\langle v\rangle$ computed from the full flux curve with $v(\langle t\rangle)$ computed at the photon-weighted midpoint; the predicted difference is about 1 m/s (scaled by the actual flux shape). A null result at the 10 cm/s level would show the diurnal-curvature model is incomplete.

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Extended reading notes

Core claim

At the center of the paper is the difference between two averages of the barycentric-corrected velocity over an exposure: the value at the photon-weighted midpoint time, $v(\langle t\rangle)$, and the true photon-weighted average $\langle v\rangle$ from Equation (3). Since the diurnal component of $v_B(t)$ is sinusoidal, evaluating at $\langle t\rangle$ always falls inside the curve's curvature, so the second-order error has a definite sign and scales with $\cos(\mathrm{lat})\cos(\delta)\sin\psi\,(\Delta t)^2$; for a uniform exposure it is about $1.32\,\mathrm{m\,s^{-1}}\cos(\mathrm{lat})\cos(\delta)\sin\psi\,(\Delta t/\mathrm{1\,h})^2$. At a low-latitude observatory the worst case allowed by a 30-degree altitude limit gives 0.25 m/s for a 30-minute exposure and 1.0 m/s for 60 minutes, with a V-shaped flux dip amplifying the effect by about 1.5. Full numerical simulations of the barycentric correction reproduce the analytic scaling, and real exposure-meter data for exposures up to 20 minutes shows the constant-flux approximation recovers the correction to within about 1 cm/s. The paper's conclusion is that instruments aiming at 10 cm/s precision must store the flux curve and apply photon weights to the correction itself.

Load-bearing premise

The paper's error estimates rest on modeling the barycentric velocity over an exposure as a pure diurnal sinusoid with a constant hour-angle rate; everything else in the correction is assumed to curve negligibly over tens of minutes, and the real-data check only reaches 20-minute exposures.

Editorial extensions

If this is right

  • Storing the exposure meter flux curve in the raw data makes the barycentric correction reproducible at the 1 cm/s level even for hour-long exposures, independent of weather and guiding changes.
  • Without photon weighting, a 30-minute low-latitude exposure near the celestial equator can be wrong by 0.25 m/s, and a 60-minute exposure by 1.0 m/s, in the worst observing geometry.
  • For archived data that only has the photon-weighted midpoint time, adding the constant-flux correction term of Equation (10) removes most of the second-order error, and real data out to 20 minutes supports this approximation to about 1 cm/s.
  • Observers can reduce the error by scheduling near the meridian, shortening exposures, or choosing high-declination targets; the error is largest for rising or setting targets observed through the east or west.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper's own tests, the same curvature argument should apply to any time-varying correction over an exposure, including the chromatic dependence of the barycentric correction, so instruments that archive only midpoint times may have hidden systematics in other correction terms as well.
  • The paper does not quantify survey-level impact, but because the error depends on hour angle, nightly-offset fits will partially absorb it; a testable prediction is that archival RV residuals should show a sinusoidal pattern in hour angle with amplitude growing as $\Delta t^2$.
  • One could exploit the opposite sign of the error east and west of the meridian: pairs of observations taken at $\pm$ hour angles would show a symmetric curl if this is the dominant systematic, providing a clean null test.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 7 minor

Summary. This manuscript addresses a systematic error in barycentric corrections for radial-velocity (RV) exposures of finite duration. The authors show that the common practice of evaluating the barycentric correction at the photon-weighted midpoint time of an exposure is exact only if the correction varies linearly with time; because the dominant diurnal component is curved, a residual second-order offset remains between the correction evaluated at the photon-weighted mean time and the flux-weighted mean of the instantaneous corrections. They derive analytic estimates (Eqs. 8-10) from a Taylor expansion of a sinusoidal diurnal model, predict errors of 0.25 m/s for a 30 minute and 1.0 m/s for a 60 minute uniform exposure in a worst-case geometry, verify these scalings with barycorrpy simulations for different exposure times, declinations, flux shapes, and observatory latitudes, and use 1315 EXPRES exposures to show that a uniform-flux approximation captures the second-order error to within about 1 cm/s for exposures shorter than 20 minutes. The paper recommends that instruments record and archive exposure-meter flux curves and, for existing data without flux curves, apply a uniform-flux correction term.

Significance. If correct, this is an important and directly actionable result for the precision-RV community. The central derivation is a parameter-free analytic Taylor expansion with no fitted free parameters, and the resulting coefficient in Eq. (10) and the V-shape enhancement factor of 1.5 are concrete, falsifiable predictions. The analytic results are corroborated by simulations with barycorrpy, a widely used and externally benchmarked package, and by a real-data comparison with EXPRES exposure-meter data. The paper closes a gap left open in earlier barycentric-correction work and gives a practical recommendation for both current and future instruments, which is relevant for the 10 cm/s and 1 m/s RV precision regimes.

minor comments (7)
  1. [Section 5.2 / Fig. 7] The text reports 1315 EXPRES observations, while the Fig. 7 caption states 1316 observations; please reconcile the count.
  2. [Section 5.2] The statement that "There is no indication that longer exposure times would change this picture" is stronger than the data support, because Fig. 7 contains no exposure longer than 20 minutes; I suggest softening the wording or adding a simulation for longer exposures.
  3. [Section 3] Equation (10) is obtained from Eq. (9) via the spherical-trig identity cos(δ)sin(ψ) = -cos(alt)sin(az), but the derivation is relegated to a footnote; stating the identity explicitly in the text would make the derivation easier to verify and reproduce.
  4. [Section 3] The sentence "If photons are concentrated towards one end of the exposure, the error second-order error decreases" contains a typo; please also state explicitly the sign convention for azimuth so that the signs in Figs. 3 and 6 can be interpreted unambiguously.
  5. [Section 5.2] The claim that "In our experience, a typical offset is of order 5% of the exposure time" is anecdotal and unreferenced; providing a quantitative justification or a reference would strengthen this part of the mitigation discussion.
  6. [Section 5.2] Footnote 2 describes a step-function mitigation strategy for data without flux curves, but it does not specify exactly how to implement it; a brief formula would make the recommendation reproducible.
  7. [Section 3] The derivation in Eq. (4) assumes that non-diurnal contributions to the barycentric velocity have negligible curvature over an exposure, and although the barycorrpy simulations support this, the paper would be more self-contained with an explicit order-of-magnitude bound on the neglected terms.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found; the central derivation is a parameter-free Taylor expansion against an external physical model and is not fitted to the paper's own outputs.

full rationale

The paper's load-bearing result is Equation (9), obtained by inserting a second-order Taylor expansion of the barycentric velocity, Equation (7), into the photon-weighted average, Equation (3). The first-order term cancels by the definition of the photon-weighted midpoint time in Equation (6), leaving only the curvature term proportional to the variance of the arrival-time distribution. No quantity appearing in the final estimate is fitted from the data it is later used to explain; the coefficient in Equation (10) follows from analytically evaluating the integrals for a uniform flux curve. The barycorrpy simulations are an independent numerical check of the analytic estimate, not an input to it, and although Wright and Eastman are co-authors of barycorrpy, the analytic derivation does not rely on that code. The EXPRES comparison in Figure 7 uses real exposure-meter fluxes to test whether the uniform-flux approximation is adequate; it is a validation exercise rather than a fit, and it does not define the predicted error. Self-citations such as Wright & Eastman (2014) and Blackman et al. (2017) provide context and code references but are not load-bearing for the derivation. No step reduces by construction to its own inputs, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities; the error formulas depend only on observatory latitude, target altitude and azimuth, exposure time, and fixed physical constants. The stated axioms are domain approximations that are explicitly flagged and supported by full simulations and a short-exposure real-data check.

assumptions (5)
  • domain assumption Over timescales of hours, barycentric velocity is dominated by Earth's diurnal rotation, Eq. (4).
    Basis for the Taylor expansion and all analytic error formulas; supported by barycorrpy simulations, but other components are assumed to have negligible curvature over an exposure.
  • domain assumption Hour angle rate is constant at 2π/24h.
    Used to evaluate first and second derivatives of the barycentric velocity; the true sidereal rate differs by about 0.27 percent.
  • domain assumption Exposure meter flux f(t) is proportional to photon flux in the relevant pixel, Eq. (3).
    Needed so the flux curve can serve as photon weights; the paper restricts itself to a single wavelength channel.
  • domain assumption Results for vmeas = 0 generalize to any vmeas much smaller than c.
    Stated without derivation in Section 3; the (1+vmeas/c) factor is argued to leave the second-order error unaffected at the 1 cm/s level.
  • domain assumption Without a flux curve, a uniform flux distribution is the most realistic assumption for archived data.
    Motivates the constant-flux correction term; validated on EXPRES data but only for exposures shorter than 20 minutes.

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Cite this review

Pith. "Pith review of Photon-weighted barycentric correction and its importance for precise radial velocities." pith.science (2026). https://pith.science/paper/U5SHOZVW

@misc{pith2026190800991,
  author       = {Pith},
  title        = {Pith review of: Photon-weighted barycentric correction and its importance for precise radial velocities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U5SHOZVW}},
  note         = {Machine review of arXiv:1908.00991}
}
abstract

When applying the barycentric correction to a precise radial velocity measurement, it is common practice to calculate its value only at the photon-weighted midpoint time of the observation instead of integrating over the entire exposure. However, since the barycentric correction does not change linearly with time, this leads to systematic errors in the derived radial velocities. The typical magnitude of this second-order effect is of order 10 cm s$^{-1}$, but it depends on several parameters, e.g. the latitude of the observatory, the position of the target on the sky, and the exposure time. We show that there are realistic observing scenarios, where the errors can amount to more than 1 ms$^{-1}$. We therefore recommend that instruments operating in this regime always record and store the exposure meter flux curve (or a similar measure) to be used as photon-weights for the barycentric correction. In existing data, if the flux curve is no longer available, we argue that second-order errors in the barycentric correction can be mitigated by adding a correction term assuming constant flux.

Figures

Figures reproduced from arXiv: 1908.00991 by the authors.

Figure 1
Figure 1. Left panel shows the instantaneous barycentric correction (black curve) for a star during one 60-minutes exposure at Mauna Kea. The fictional, rising target is chosen such that the exposure starts when the star is 30◦ above the horizon (air mass = 2.0), and such that we are looking due east at the geometric midpoint time of the exposure. The lower panel shows what the exposure meter flux could look like during the e… view at source ↗
Figure 2
Figure 2. Left panel: Locations of 25 different observatories that are hosting or will soon be hosting one or more PRV instruments (see e.g. Plavchan et al. 2015; Fischer et al. 2016; Wright & Robertson 2017). Most of these sites are located around latitude ±30◦ , yielding cos(lat) ≈ 0.87. Of these instruments, Mauna Kea (Hawaii) at latitude +20◦ is the PRV site closest to equator, with cos(lat) = 0.94. Right panel: Systemati… view at source ↗
Figure 3
Figure 3. Second-order error, v( hti) − hvi, simulated as function of local hour angle for various exposure times. Each point on the curves corresponds to an exposure at that hour angle. The declination and minimum/maximum hour angle is chosen for each exposure time such that the the telescope is pointing straight east/west at the geometric midpoint of the exposure, while the exposure starts/ends at 30◦ altitude. The exposure… view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: Second-order error, v( hti) − hvi, simulated as function of local hour angle for various shapes of the exposure meter flux curve. The shapes are a linear ramp, a uniform exposure, a centred V-shape, and a V-shape offset from the centre. The dotted curves in the backgro…
Figure 6
Figure 6. Figure 6: Second-order error, v( hti) − hvi, simulated as function of sky position at three different observatories. We simulate a 30 minute exposure with uniform flux, observed at local midnight. The colour indicates the size of the error as a function of the target position at…
Figure 7
Figure 7. Figure 7: For nearly one year of EXPRES observations, we com￾pare the actual second-order error of each measurement with the uniform flux approximation. The multi-channel exposure meter data has been binned to one channel. EXPRES mainly observes bright targets, so the exposure t…

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