# Luri, Brown, Sarro, Arenou, Bailer-Jones et al. — Gaia Data Release 2: Using Gaia Parallaxes (2018) **X. Luri, A. G. A. Brown, L. M. Sarro, F. Arenou, C. A. L. Bailer-Jones, A. Castro-Ginard, J. de Bruijne, T. Prusti, C. Babusiaux, H. E. Delgado**, "Gaia Data Release 2: Using Gaia Parallaxes," *Astronomy & Astrophysics* 616, A9 (2018). DOI 10.1051/0004-6361/201832964. --- ## Overview The Gaia consortium's official manual for how to use DR2 parallaxes. It documents the known pathologies of the published values: a global zero-point offset of about −29 microarcseconds from quasars, spatial correlations up to 0.04 mas on degree scales, outliers at the 100σ level, and a large sub-population of negative parallaxes that the pipeline explicitly labels as "the source going the wrong way around on the sky." The paper's central recommendation is that no astrophysical quantity (distance, absolute magnitude, luminosity) may be derived from the parallax catalogue without full Bayesian inference using a prior. --- ## Page 1: Abstract ![[Luri2018_GaiaDR2-01.png]] > "The second Gaia data release (Gaia DR2) provides precise five-parameter astrometric data (positions, proper motions, and parallaxes) for an unprecedented number of sources (more than 1.3 billion, mostly stars)." (Abstract) > "The naive use of the simple approach of inverting the parallax to estimate a distance can provide an acceptable estimate in a limited number of cases, in particular when a precise parallax for an individual object is used. However, one of the important contributions of Gaia DR2 will be the possibility of working with large samples of objects... In these cases a proper statistical treatment of the parallaxes in order to derive distances, especially (but not only) when the relative uncertainties are large, is mandatory." (p.1) > "In particular we also show that negative parallaxes, or parallaxes with relatively large uncertainties still contain valuable information." (Abstract) > [!warning] Parallax alone does not give a distance > The DR2 team states at the outset that the published parallax is not a distance. Every distance quoted from Gaia requires a prior distribution, chosen by the analyst, applied on top of the parallax. The size and shape of that prior determine the answer. This is not disclosed in most downstream papers that cite "Gaia distances." --- ## Page 2: The −29 µas quasar zero-point offset ![[Luri2018_GaiaDR2-02.png]] From p.2: > "Based on an assessment of the measured parallaxes of a set of about half a million known quasars, which can be assumed in practice to have zero parallax, the uncertainties are normally distributed with impressive approximation (Fig. 1)." **Figure 1 caption (p.2):** > "Distribution of normalised, re-centred parallaxes of 556 849 quasars from the AllWISE catalogue present in Gaia DR2. [...] The centring adopted in this plot reflects a global parallax zero-point shift of −0.029 mas." > [!critical] The global −29 µas zero-point offset > The Gaia consortium uses a set of ~556,849 quasars as a reference population that "can be assumed in practice to have zero parallax." When they plot the observed parallaxes of these objects, the distribution is centred at −0.029 mas, not zero. This is the global parallax zero-point shift. The team publishes the parallax values uncorrected for this shift. Every star in DR2 has, in effect, 29 microarcseconds added to its true parallax by the measurement system. The entire distance ladder rests on the assumption that quasars do have zero parallax and that the observed non-zero mean reflects an instrument offset and nothing else. --- ## Page 3: Systematic errors section ![[Luri2018_GaiaDR2-03.png]] From Section 2.3 "Systematic errors" (p.3): > "Both the design of the spacecraft and the design and implementation of the data processing software and algorithms aim to prevent biases or systematic effects in the astrometry. Systematic errors at low levels nonetheless exist in Gaia DR2 (see Arenou et al. 2018; Lindegren et al. 2018). Systematic effects are complicated and largely unknown functions of position on the sky, magnitude, and colour." > "There is a significant average parallax zero-point shift of about −30 µas in the sense Gaia minus external data. This shift has not been corrected for and is present in the published data." > "Significant spatial correlations between stars, up to 0.04 mas in parallax and 0.07 mas yr⁻¹ in proper motion, exist on both small (≲1°) and intermediate (≲20°) angular scales. As a result, averaging parallaxes over small regions of the sky, for instance in an open cluster, in the Magellanic Clouds, or in the Galactic Centre, will not reduce the uncertainty on the mean below the ~0.1 mas level." > "Unfortunately, there is no simple recipe to account for the systematic errors." (p.3) > [!critical] Spatial correlation destroys the averaging argument > If systematic errors in parallax correlate across patches of sky up to 20° in diameter, then averaging many stars in a cluster or in the LMC does not remove the error. The floor is around 0.1 mas, which is larger than the parallax of everything more distant than about 10 kpc. For any aggregate analysis (cluster distances, Magellanic Cloud distance scales, Galactic rotation curve), the published Gaia parallaxes carry a systematic floor that cannot be reduced by measuring more stars. --- ## Page 4: Where negative parallax comes from (Section 3.1) ![[Luri2018_GaiaDR2-04.png]] The likelihood model (equation 6, p.4): $p(\varpi \mid \varpi_{\text{True}}) = \frac{1}{\sigma_\varpi\sqrt{2\pi}}\exp\!\left[-\frac{(\varpi - \varpi_{\text{True}})^2}{2\sigma_\varpi^2}\right]$ From p.4, the pipeline's own description of what a negative parallax means: > "In the presence of large measurement noise (comparable to the size of the parallax) it is entirely possible that the parallax value estimated for the source model vanishes or becomes negative. This case can be interpreted as the measurement being consistent with the source going 'the wrong way around' on the sky, as shown in Fig. 2." **Figure 2 caption (p.4):** > "Example of a negative parallax arising from the astrometric data processing. Solid blue lines, true path of the object; red dots, the individual measurements of the source position on the sky; dashed orange lines, the source path according to the least-squares astrometric solution, which here features a negative parallax. [...] In the fitted solution the negative parallax effect is equivalent to a yearly motion of the star in the opposite direction of the true parallactic motion (which gives a phase-shift of π in the sinusoidal curves in the right panels)." > [!critical] The official Gaia definition of negative parallax > Luri et al. define a negative parallax as a star whose best-fit sinusoidal path on the sky is **180 degrees out of phase** with the expected parallactic wobble produced by the Earth's assumed orbital motion. They call this "the source going the wrong way around on the sky." They then re-interpret it as a "measurement noise" outcome of the pipeline, not as a real direction of motion. > [!note] The sign phase is data, not noise > The sign of the parallax in the Gaia reduction comes from the direction of the observed wobble. If the wobble were purely measurement noise, the direction would be random. But the distribution of negative parallaxes in the catalogue is structured (see Luri's own Fig. 1 where the centre of a half-million quasar distribution sits at −29 µas, not zero). The paper treats this as a calibration offset of the instrument. Whether it is an instrument offset or a real signal is a model-choice, not a measurement. --- ## Page 5: The bias of 1/ϖ as a distance estimator ![[Luri2018_GaiaDR2-05.png]] From Section 3.2 (p.4-5): > "Blind use of 1/ϖ as an estimator of the distance will lead to unphysical results in case the observed parallax is non-positive." (p.4) The PDF of $\rho = 1/\varpi$ (equation 7, p.4): $p(\rho \mid \varpi_{\text{True}}) = \frac{1}{\rho^2\,\sigma_\varpi\sqrt{2\pi}}\exp\!\left[-\frac{(1/\rho - \varpi_{\text{True}})^2}{2\sigma_\varpi^2}\right]$ Expected value of the observed parallax (equation 8, p.5): $E[\varpi] = \int \varpi\,N(\varpi; \varpi_{\text{True}}, \sigma_\varpi)\,d\varpi = \varpi_{\text{True}}$ > "The observed parallax is an unbiased estimator of the true parallax (under the strong hypothesis that there are no systematic biases associated with the survey and that the errors are normally distributed)." (p.5) > [!warning] "Under the strong hypothesis that there are no systematic biases" > The paper states in the next breath that there IS a −30 µas systematic bias and that it is not corrected. The "unbiased estimator" claim is a statement about a fictitious survey. Every actual quantity in the catalogue is biased by the known offset. Bias of $1/\varpi$ (p.5): $E[\rho] = E[1/\varpi] = \int \frac{1}{\varpi}\,N(\varpi_{\text{True}}, \sigma_\varpi)\,d\varpi$ > "The distance estimator 1/ϖ is unbiased for vanishingly small values of f, but it rapidly becomes significantly biased for values of f beyond 0.1." (p.5) > "As a summary, we have seen in previous paragraphs that the naive approach of inverting the observed parallax has significant drawbacks: we are forced to dispose of valuable data (non-positive parallaxes), and as an estimator ρ = 1/ϖ is biased and has a very high variance." (p.5) --- ## Page 8: Why you cannot just delete the negative parallaxes (Section 3.3) ![[Luri2018_GaiaDR2-08.png]] From p.6, the section on sample truncation: > "As discussed in Sect. 3.1, negative parallaxes are a natural result of the Gaia measurement process (and of astrometry in general). Since inverting negative parallaxes leads to physically meaningless negative distances we are tempted to just get rid of these values and form a 'clean' sample. This results in a biased sample, however." > "On the one hand, removing the negative parallaxes biases the distribution of this parameter. Consider for instance the case illustrated in Fig. 1 for the quasars from the AllWISE catalogue. These objects have a near zero true parallax, and the distribution of its observed values shown in the figure corresponds to this, with a mean of −10 µas, close to zero. However, if we remove the negative parallaxes from this sample, deeming them 'unphysical', the mean of the observed values would be significantly positive, about 0.8 mas. This is completely unrealistic for quasars; in removing the negative parallaxes we have significantly biased the observed parallax set for these objects." (p.6) > [!critical] Deleting negatives fakes a distance onto every quasar > The published quasar parallaxes have a near-zero mean only because the negative and positive values balance out. If the pipeline or an analyst discards the negatives, the remaining positives have a mean of 0.8 mas. For a quasar (assumed at cosmological distance), a parallax of 0.8 mas would imply a distance of about 1,250 pc. This is the scale of the deception: deleting the negatives turns every quasar into a nearby star. The distribution of negatives is therefore **load-bearing** for the "extragalactic" interpretation of the data. > "A stronger version of truncation that has traditionally been applied is to remove not only negative parallaxes, but also all the parallaxes with a relative error above a given threshold k, selecting σ_ϖ/ϖ < k. This selection tends to favour the removal of stars with small parallaxes." (p.6) > "stars with positive errors (making the observed parallax larger than the true one) tend to be less removed than stars with negative errors (making the observed parallax smaller than the true one). By favouring positive errors with respect to negative errors, we are also biasing the overall distribution of parallaxes." (p.7) --- ## Page 10: The Smith–Eichhorn correction and why Luri rejects it ![[Luri2018_GaiaDR2-10.png]] Equation 10, p.7, the Smith–Eichhorn pseudo-parallax: $\varpi^* \equiv \beta \cdot \sigma_\varpi \left( \frac{1}{\exp(\phi) + \exp\!\left(\frac{-1.6\varpi}{\sigma_\varpi}\right)} + \phi \right)$ where $\phi \equiv \ln(1 + \exp(2\varpi/\sigma_\varpi))/2$ and $\beta$ is adjustable. > "The qualitative effect of the transformation is to map negative parallaxes into the positive semi-axis R+ and to increase the value of small parallaxes until it asymptotically converges to the measured value for large ϖ." (p.7) > "The Smith–Eichhorn transformation is an arbitrary (and rather convolved) choice amongst many such transformations that can reduce the bias for certain particular situations. Both the analytical expression and the choice of constants and β are the result of an unspecified trial-and-error procedure, the applicability of which is unclear." (p.7) > [!critical] A named correction that simply reverses the sign > The Smith–Eichhorn transformation is defined so that negative parallaxes get mapped to positive values. The authors of the present Gaia paper flag it as arbitrary and reject it. But the function itself tells the story: a formal procedure published in 1996 was proposed to convert "wrong way" parallaxes to "right way" parallaxes by smooth mathematical mapping. The negative parallax problem was old enough by 1996 to warrant a named fix. --- ## Summary of Luri's own admissions | Claim | Luri et al. (2018) | |---|---| | Gaia DR2 gives distances | No. It gives parallaxes from which distances must be inferred with a Bayesian prior. | | Measurement errors are normal around zero offset | No. A −29 µas quasar-based zero-point shift is present, uncorrected in the published data. | | Negative parallaxes are unphysical errors | In the pipeline's geometric framework they correspond to a star's sky motion being 180° out of phase with the expected parallactic motion. | | Negative parallaxes can be deleted | No. Deleting them biases the quasar sample from a mean of ~0 mas to 0.8 mas. | | Averaging many stars removes systematic errors | No. Spatial correlations up to 20° keep a floor of ~0.1 mas even in aggregate samples like the LMC. | | 1/ϖ is a safe distance estimator for any precision | No. It is biased for $f > 0.1$ and has explosive variance. | | There is a simple recipe for the systematic errors | "Unfortunately, there is no simple recipe to account for the systematic errors." | --- ## Citation Luri, X., Brown, A. G. A., Sarro, L. M., Arenou, F., Bailer-Jones, C. A. L., Castro-Ginard, A., de Bruijne, J., Prusti, T., Babusiaux, C., Delgado, H. E., 2018. "Gaia Data Release 2: Using Gaia Parallaxes." *Astronomy & Astrophysics* 616, A9. DOI 10.1051/0004-6361/201832964. --- ## See also - [[1943_Lee_Negative_Parallax]] - [[2015_BailerJones_Distances_from_Parallaxes]] - [[2021_BailerJones_EDR3_Distances]] - [[2021_Lindegren_EDR3_Parallax_Bias]] - [[2021_Groenewegen_Parallax_ZeroPoint_Offset]] - [[Shack_Chapter25_Negative_Parallax_Demystified]]