Radial velocities for the HIPPARCOSGaia HundredThousandProperMotion project^{⋆}
Research and Scientific Support Department in the Directorate of Science
and Robotic Exploration of the European Space Agency,
Postbus 299,
2200AG
Noordwijk,
The Netherlands
email: jdbruijn@rssd.esa.int
Received:
14
March
2012
Accepted:
20
July
2012
Context. The HundredThousandProperMotion (HTPM) project will determine the proper motions of ~113 500 stars using a ~23year baseline. The proper motions will be based on spacebased measurements exclusively, with the Hipparcos data, with epoch 1991.25, as first epoch and with the first intermediaterelease Gaia astrometry, with epoch ~2014.5, as second epoch. The expected HTPM propermotion standard errors are 30−190 μas yr^{1}, depending on stellar magnitude.
Aims. Depending on the astrometric characteristics of an object, in particular its distance and velocity, its radial velocity can have a significant impact on the determination of its proper motion. The impact of this perspective acceleration is largest for fastmoving, nearby stars. Our goal is to determine, for each star in the Hipparcos catalogue, the radialvelocity standard error that is required to guarantee a negligible contribution of perspective acceleration to the HTPM propermotion precision.
Methods. We employ two evaluation criteria, both based on MonteCarlo simulations, with which we determine which stars need to be spectroscopically (re)measured. Both criteria take the Hipparcos measurement errors into account. The first criterion, the Gaussian criterion, is applicable to nearby stars. For distant stars, this criterion works but returns overly pessimistic results. We therefore use a second criterion, the robust criterion, which is equivalent to the Gaussian criterion for nearby stars but avoids biases for distant stars and/or objects without literature radial velocity. The robust criterion is hence our prefered choice for all stars, regardless of distance.
Results. For each star in the Hipparcos catalogue, we determine the confidence level with which the available radial velocity and its standard error, taken from the XHIP compilation catalogue, are acceptable. We find that for 97 stars, the radial velocities available in the literature are insufficiently precise for a 68.27% confidence level. If requiring this level to be 95.45%, or even 99.73%, the number of stars increases to 247 or 382, respectively. We also identify 109 stars for which radial velocities are currently unknown yet need to be acquired to meet the 68.27% confidence level. For higher confidence levels (95.45% or 99.73%), the number of such stars increases to 1071 or 6180, respectively.
Conclusions. To satisfy the radialvelocity requirements coming from our study will be a daunting task consuming a significant amount of spectroscopic telescope time. The required radialvelocity measurement precisions vary from source to source. Typically, they are modest, below 25 km s^{1}, but they can be as stringent as 0.04 km s^{1} for individual objects like Barnard’s star. Fortunately, the followup spectroscopy is not timecritical since the HTPM proper motions can be corrected a posteriori once (improved) radial velocities become available.
Key words: techniques: radial velocities / astronomical databases: miscellaneous / catalogs / astrometry / parallaxes / proper motions
The results data file is only available in electronic form at the CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsarc.ustrasbg.fr/vizbin/qcat?J/A+A/546/A61
© ESO, 2012
1. Introduction
Gaia (e.g., Perryman et al. 2001; Lindegren et al. 2008) is the upcoming astrometry mission of the European Space Agency (ESA), following up on the success of the Hipparcos mission (ESA 1997a; Perryman et al. 1997; Perryman 2009). Gaia’s science objective is to unravel the kinematical, dynamical, and chemical structure and evolution of our galaxy, the Milky Way (e.g., Gómez et al. 2010). In addition, Gaia’s data will revolutionise many other areas of (astro)physics, e.g., stellar structure and evolution, stellar variability, double and multiple stars, solarsystem bodies, fundamental physics, and exoplanets (e.g., Pourbaix 2008; Tanga et al. 2008; Mignard & Klioner 2010; Eyer et al. 2011; Sozzetti 2011; Mouret 2011). During its fiveyear lifetime, Gaia will survey the full sky and repeatedly observe the brightest 1000 million objects, down to 20th magnitude (e.g., de Bruijne et al. 2010). Gaia’s science data comprises absolute astrometry, broadband photometry, and lowresolution spectrophotometry. Mediumresolution spectroscopic data will be obtained for the brightest 150 million sources, down to 17th magnitude. The final Gaia catalogue, due in ~2021, will contain astrometry (positions, parallaxes, and proper motions) with standard errors less than 10 microarcsecond (μas, μas yr^{1} for proper motions) for stars brighter than 12 mag, 25 μas for stars at 15th magnitude, and 300 μas at magnitude 20 (de Bruijne 2012). Millimagnitudeprecision photometry (Jordi et al. 2010) allows to get a handle on effective temperature, surface gravity, metallicity, and reddening of all stars (BailerJones 2010). The spectroscopic data allows the determination of radial velocities with errors of 1 km s^{1} at the bright end and 15 km s^{1} at magnitude 17 (Wilkinson et al. 2005; Katz et al. 2011) as well as astrophysical diagnostics such as effective temperature and metallicity for the brightest few million objects (Kordopatis et al. 2011). Clearly, these performances will only be reached with a total of five years of collected data and only after careful calibration.
Intermediate releases of the data – obviously with lower quality and/or reduced contents compared to the final catalogue – are planned, the first one around two years after launch, which is currently foreseen for the second half of 2013. The HundredThousandProperMotion (HTPM) project (Mignard 2009), conceived and led by François Mignard at the Observatoire de la Côte d’Azur, is part of the first intermediate release. Its goal is to determine the absolute proper motions of the ~113 500 brightest stars in the sky using Hipparcos astrometry for the first epoch and early Gaia astrometry for the second. Clearly, the HTPM catalogue will have a limited lifetime since it will be superseded by the final Gaia catalogue in ~2021. Nevertheless, the HTPM is a scientifically interesting as well as unique catalogue: the ~23year temporal baseline, with a mean Hipparcos epoch of 1991.25 and a mean Gaia epoch around 2014.5, allows a significant improvement of the Hipparcos proper motions, which have typical precisions at the level of 1 milliarcsec yr^{1} (mas yr^{1}): the expected HTPM propermotion standard errors^{1} are 40−190 μas yr^{1} for the proper motion in right ascension μ_{α ∗ } and 30−150 μas yr^{1} for the proper motion in declination μ_{δ}, primarily depending on magnitude (we use the common Hipparcos notation α^{ ∗ } = α cosδ; ESA 1997a, Sect. 1.2.5). A clear advantage of combining astrometric data from the Hipparcos and Gaia missions is that the associated proper motions will be, by construction and IAU resolution, in the system of the International Celestial Reference System (ICRS), i.e., the proper motions will be absolute rather than relative. In this light, it is important to realise that massive, modernday propermotion catalogues, such as UCAC3 (Zacharias et al. 2010), often contain relative proper motions only and that they can suffer from substantial, regional, systematic distortions in their propermotion systems, up to levels of 10 mas yr^{1} or more (e.g., Röser et al. 2008, 2010; Liu et al. 2011).
It is a wellknown geometrical feature, for instance already described by Seeliger in 1900, that for fastmoving, nearby stars, it is essential to know the radial velocity for a precise measurement and determination of proper motion. In fact, this socalled secular or perspective acceleration on the sky was taken into account in the determination of the Hipparcos proper motions for 21 stars (ESA 1997a, Sect. 1.2.8) and the same will be done for Gaia, albeit for a larger sample of nearby stars. Clearly, the inverse relationship also holds: with a precise proper motion available, a socalled astrometric radial velocity can be determined, independent of the spectroscopically measured quantity (see Lindegren & Dravins 2003, for a precise definition and meaning of [astrometric] radial velocity). With this method, Dravins et al. (1999) determined^{2} the astrometric radial velocities for 17 stars, from Hipparcos proper motions combined with Astrographic Catalogue positions at earlier epochs. Although Dravins et al. (1999) reached relatively modest astrometricradialvelocity precisions, typically a few tens of km s^{1}, their results are interesting since they provide direct and independent constraints on various physical phenomena affecting spectroscopic radial velocities, for instance gravitational redshifts, stellar rotation, convection, and pulsation. In our study, however, we approach (astrometric) radial velocities from the other direction since our interest is to determine accurate HTPM proper motions which are not biased by unmodelled perspective effects. In other words: we aim to establish for which stars in the forthcoming HTPM catalogue the currently available (spectroscopic) radial velocity and associated standard error are sufficient to guarantee, with a certain confidence level, a negligible perspectiveaccelerationinduced error in the HTPM proper motion. For stars without a literature value of the radial velocity, we establish whether – and, if yes, with what standard error – a radial velocity needs to be acquired prior to the construction of the HTPM catalogue. Section 2 describes the available astrometric and spectroscopic data. The propagation model of star positions is outlined in Sect. 3. We investigate the influence of the radial velocity on HTPM proper motions in Sect. 4 and develop two evaluation criteria in Sect. 5. We employ these in Sect. 6. We discuss our results in Sect. 7 and give our final conclusions in Sect. 8.
2. The XHIP catalogue
As source for the Hipparcos astrometry and literature radial velocities, we used the eXtended Hipparcos compilation catalogue (CDS catalogue V/137), also known as XHIP (Anderson & Francis 2012). This catalogue complements the 117 955 entries with astrometry in the Hipparcos catalogue with a set of 116 096 spectral classifications, 46 392 radial velocities, and 18 549 iron abundances from various literature sources.
2.1. Astrometry
The starting point for the XHIP compilation was the new reduction of the Hipparcos data (van Leeuwen 2007, 2008, CDS catalogue I/311), also known as HIP2. Realising that stars with multiple components were solved individually, rather than as systems, by van Leeuwen for the sake of expediency, Anderson & Francis reverted to the original HIP1 astrometry (ESA 1997a,b, CDS catalogue I/239) in those cases where multiplicity is indicated and where the formal parallax standard error in HIP2 exceeds that in HIP1. This applies to 1922 entries. In addition, Anderson & Francis included the Tycho2 catalogue (Høg et al. 2000a,b, CDS catalogue I/259) in their XHIP propermotion data. In the absence of Tycho2 proper motions, HIP2 proper motions were forcibly used. When multiplicity is indicated, Hipparcos proper motions were replaced by Tycho2 values in those cases where the latter are more precise. When multiplicity is not indicated, Tycho2 proper motions replaced Hipparcos values if the associated standard errors exceed the Tycho2 standard errors by a factor three or more. In all other cases, a mean HIP2 – Tycho2 proper motion was constructed and used, weighted by the inverse squared standard errors; this applies to 92 269 entries.
2.2. Radial velocities
The XHIP catalogue contains radial velocities for 46 392 of the 117 955 entries, carefully compiled by Anderson & Francis from 47 literature sources. The vast majority of measurements have formal measurement precisions, i.e., radialvelocity standard errors (1753 measurements lack standard errors; see Sect. 6.2). In addition, all radial velocities have a quality flag:

An “A” rating (35 932 entries) indicates that thestandard errors are generally reliable.

A “B” rating (4239 entries) indicates potential, small, uncorrected, systematic errors.

A “C” rating (3465 entries) indicates larger systematic errors, while not excluding suitability for population analyses.

A “D” rating (2756 entries) indicates serious problems, meaning that these stars may not be suitable for statistical analyses. A “D” rating is assigned whenever:

1.
the radialvelocity standard error is not available;

2.
the star is an un(re)solved binary;

3.
the star is a WolfRayet star or a white dwarf that is not a component of a resolved binary; or

4.
different measurements yield inconsistent results.

1.
The majority of stars in the XHIP catalogue (71 563 entries to be precise) have no measured radial velocity. All we can reasonably assume for these stars is that their radialvelocity distribution is statistically identical to the radialvelocity distribution of the entries with known radial velocities. Figure 1 shows this distribution. It is fairly well represented by a normal distribution with a mean μ = −2.21 km s^{1} and standard deviation σ = 22.44 km s^{1} (the median is −2.00 km s^{1}). The observed distribution has lowamplitude, broad wings as well as a small number of real “outliers”, with heliocentric radial velocities up to plusorminus several hundred km s^{1}. A small fraction of these stars are earlytype runaway^{3} stars (Hoogerwerf et al. 2001) but the majority represent nearby stars in the (nonrotating) halo of our galaxy. The bulk of the stars, those in the main peak, are (thin)disc stars, corotating with the Sun around the galactic centre. In theory, the main peak can be understood, and also be modelled in detail and hence be used to statistically predict the radial velocities for objects without literature values, as a combination of the reflex of the solar motion with respect to the local standard of rest (Delhaye 1965; Schönrich et al. 2010), the effect of differential galactic rotation (Oort 1927), and the random motion of stars (Schwarzschild 1907). In practice, however, such a modelling effort would be massive, touching on a wide variety of (sometimes poorly understood) issues such as the asymmetric drift, the tilt and vertex deviation of the velocity ellipsoid, mixing and heating of stars as function of age, the height of the sun above the galactic plane (Joshi 2007), the dynamical coupling of the local kinematics to the galactic bar and spiral arms (Antoja et al. 2011), largescale deviations of the local velocity field caused by the Gould Belt (Elias et al. 2006), migration of stars in the disc (Schönrich & Binney 2009), etc. To model these effects, and hence be able to predict a more refined radial velocity for any star as function of its galactic coordinates, distance, and age, is clearly beyond the scope of this paper. We will come back to this issue in Sect. 6.2.
Fig. 1 Distribution of all 46 392 radial velocities contained in the XHIP catalogue. The smooth, red curve fits the histogram with a Gaussian normal distribution. The bestfit mean and standard deviation are μ = −2.21 km s^{1} and σ = 22.44 km s^{1}, respectively. 

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3. Propagation model
3.1. The full model
Let us denote the celestial position of a star at time t_{0} (in years) in equatorial coordinates in radians by (α_{0},δ_{0}), its distance in parsec by r = 1000 ϖ^{1}, with the parallax ϖ in mas, and its propermotion components in equatorial coordinates in mas yr^{1} by (μ_{α},μ_{δ}). The threedimensional position of a star in Cartesian equatorial coordinates at time t_{0} is then given by: (1)With v_{r} the star’s radial velocity in km s^{1}, it is customary to define the “radial proper motion” as μ_{r} = v_{r} r^{1}. The linear velocity in pc yr^{1} is then given by: (2)where A_{V} = 4.740,470,446 km yr s^{1} is the astronomical unit and the factor pc s km^{1} yr^{1} changes km s^{1} to pc yr^{1} (ESA 1997a, Table 1.2.2) so that B × A_{V} = 4.848,136,811 × 10^{6} pc AU^{1}. Transforming these equations into Cartesian coordinates leads to: (3)Since the motion of stars, or the barycentre of multiple systems, is to nearperfect approximation rectilinear over time scales of a few decades, the position of a star at time t_{1}, after a time t = t_{1} − t_{0}, now simply follows by applying the propagation model: (4)Transforming this back into equatorial coordinates returns the celestial position (α(t),δ(t)) of the star at time t: (5)So, in summary, it is straightforward to compute the future position of a star on the sky once the initial celestial coordinates, the proper motion, parallax, radial velocity, and time interval are given. This, however, is what nature and Gaia will do for us: the initial celestial coordinates correspond to the Hipparcos epoch (1991.25) and the final coordinates (α(t),δ(t)), with t ≈ 23 yr, will come from the first Gaia astrometry. The HTPM project will determine the propermotion components (μ_{α},μ_{δ}) from known initial and final celestial coordinates for given time interval, parallax, and radial velocity. Unfortunately, it is not possible to express the propermotion components in a closed (analytical) form as function of (α_{0},δ_{0}), (α(t),δ(t)), ϖ, v_{r}, and t since the underlying set of equations is coupled. The derivation of the propermotion components hence requires a numerical solution. We implemented this solution using NewtonRaphson iteration and refer to this solution as the “full model”. This model, however, is relatively slow for practical implementation. We hence decided to also implement a “truncated model” with analytical terms up to and including t^{3} (Sect. 3.2), which is about ten times faster and sufficiently precise for our application (Sect. 3.3).
3.2. The truncated model
Mignard (2009) shows that the full model (Sect. 3.1) can be truncated up to and including thirdorder terms in time t without significant loss in accuracy (Sect. 3.3). Equations (6) and (7) below show the forward propagation for right ascension and declination, respectively. By forward propagation, we mean computing the positional displacements Δα and Δδ of a star for given proper motion, parallax, radial velocity, and time interval t = t_{1} − t_{0}: Equations (8) and (8) below show the backward solution for the proper motion in right ascension and declination, respectively. By backward solution, we mean computing the propermotion components (μ_{α},μ_{δ}) from the initial and final celestial positions (α_{0},δ_{0}) and (α(t),δ(t)), for given parallax, radial velocity, and time interval t = t_{1} − t_{0}: It is straightforward to insert Eqs. (6) and (7) into Eqs. (8) and (8) to demonstrate that only terms of order t^{4} and higher are left.
Propermotion errors accumulated over t = 25 yr as function of declination due to truncation of the full model up to and including first, second, and thirdorder terms in time for an “extreme”, i.e., nearby, fastmoving star: ϖ = 500 mas (r = 2 pc), μ_{α} = μ_{δ} = 2000 mas yr^{1}, and v_{r} = 50 km s^{1}.
3.3. Accuracy of the truncated model
To quantify that the truncation of the full model up to and including thirdorder terms in time is sufficient for the HTPM application, Table 1 shows the errors in derived proper motions over an interval of 25 years induced by the truncation of the model when using the approximated Eqs. (6), (8) for an “extreme” star (i.e., nearby, fastmoving and hence sensitive to perspectiveacceleration effects) as function of declination. Four cases have been considered. First, Eqs. (6), (7) up to and including firstorder terms in time were used for the forward propagation and Eqs. (8), (8) were used for the backward solution. This is indicated by the heading . The difference between the proper motion used as input and the proper motion derived from Eqs. (8), (8) is listed in the table and can reach several mas yr^{1} close to the celestial poles. The second case (“”) is similar to the first case but includes also secondorder terms in time for the forward propagation. The propermotion errors are now much reduced, by about three orders of magnitude, but can still reach 10 μas yr^{1}, which is significant given the predicted HTPM standard errors (30–190 μas yr^{1}, depending on magnitude). The third case (“”) also includes thirdorder terms in time. The propermotion errors are now negligible, reaching only up to 10 nanoarcsec yr^{1}. The fourth case uses the full model for the forward propagation and the NewtonRaphson iteration for the backward solution and recovers the input proper motions with subnanoarcsecond yr^{1} errors.
Fig. 2 Schematic diagram showing how to quantify the sensitivity of the proper motion to a change in (read: measurement error of) radial velocity. A change in the radial velocity Δv_{r} introduced before the backward solution leads to a certain (HTPM) propermotion error Δμ. The linear dependence is commented on in Sect. 4.3. Since the magnitude of the propermotion error does not depend on the sign but only on the magnitude of the radialvelocity variation, the sensitivity curve is symmetric with respect to the true radial velocity. The dashed horizontal line denotes the maximum perspectiveaccelerationinduced propermotion error we are willing to accept in the HTPM proper motion. The distance Σ between the intersection points of the dashed horizontal line and the solid sensitivity curves determines the tolerance on the radialvelocity error. 

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4. The influence of radial velocity
4.1. Principle of the method
To quantify the influence of radial velocity on the HTPM proper motion for a star, we take the Hipparcos astrometric data (with epoch 1991.25) and the literature radial velocity from the XHIP catalogue (Sect. 2 – see Sect. 6.2 for stars without literature radial velocity) and use Eqs. (6), (7) to predict the star’s celestial position in 2014.5^{4}, i.e., the mean epoch of the first intermediaterelease Gaia astrometry. We then use backward solution, i.e., apply Eqs. (8), (8), to recover the proper motion from the given Hipparcos and Gaia positions on the sky, assuming that the parallax and time interval are known. Clearly, if the radial velocity (radial proper motion) used in the backward solution is identical to the radial velocity used in the forward propagation, the derived (HTPM) proper motion is essentially identical to the input (XHIP) proper motion (Table 1). However, by varying the radial velocity used for the backward solution away from the input value, the sensitivity of the HTPM proper motion on radial velocity is readily established. This sensitivity does not depend on the sign but only on the magnitude of the radialvelocity variation. Figure 2 schematically shows this idea. The abscissa shows the change in radial velocity, Δv_{r}, with respect to the input value used for the forward propagation. The ordinate shows Δμ (either Δμ_{α} or Δμ_{δ}), i.e., the difference between the input value of the proper motion (either μ_{α} or μ_{δ}) and the HTPM proper motion derived from the backward solution. The dashed horizontal line represents the maximum perspectiveaccelerationinduced propermotion error that we are willing to accept. If we denote the radialvelocity interval spanned by the intersections between the dashed, horizontal threshold line and the two solid sensitivity curves by Σ (either Σ_{α} or Σ_{δ}), the tolerance on the radialvelocity standard error is easily expressed as . The question now is: what is the probability that the error in radial velocity (i.e., true radial velocity minus catalogue value) is smaller than or larger than ? Naturally, we want this probability to be smaller than a chosen threshold 1 − c, where c denotes the confidence level (for instance c = 0.6827 for a “1σ result”): (10)where we have assumed that the error distribution for ΔV_{r} is a normal distribution with zero mean and variance and where denotes the error function with argument x/: (11)From Eq. (10), one can easily deduce: (12)where denotes the inverse of (e.g., ). For the “special case” c = 68.27%, Eq. (12) hence simplifies to .
Fig. 3 Sensitivity of the HTPM proper motion in right ascension to radial velocity for HIP 70890 (Proxima Centauri). The sensitivity is linear and has a value C_{α} = −74.59 μas yr^{1} per km s^{1} (Sect. 4.3). The dashed horizontal line indicates the maximumtolerable perspectiveaccelerationinduced propermotion error caused by an incorrect radial velocity. Since the expected HTPM standard error in right ascension is 97 μas yr^{1} for this star, we set this threshold to 97/10 = 9.7 μas yr^{1}. This implies, for a confidence level c = 68.27%, that the maximumacceptable radialvelocity standard error σ_{vr} for this object is km s^{1}. 

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4.2. Example for a real star
Figure 3 is similar to the schematic Fig. 2 but shows data for a real star, HIP 70890. This object, also known as Proxima Centauri (or α Cen C), is a nearby (ϖ = 772.33 ± 2.60 mas), fastmoving (μ_{α} = −3775.64 ± 1.52 mas yr^{1} and μ_{δ} = 768.16 ± 1.82 mas yr^{1}) M6Ve flaring emissionline star which is known to have a significant perspective acceleration. HIP 70890 is actually one of the 21 stars in the Hipparcos catalogue for which the perspective acceleration was taken into account (ESA 1997a, Sect. 1.2.8). Figure 3 was constructed by comparing the input proper motion used in the forward propagation with the proper motion resulting from the backward solution while using a progressively differing radial velocity in the backward solution from the (fixed) value used in the forward propagation. The actual radialvelocity variation probed in this figure is small, only ± 0.5 km s^{1}.
HIP 70890 is relatively faint (Hp = 10.7613 mag) and the expected HTPM propermotion standard error is 97 μas yr^{1} (see Sect. 6.1 for details). If using a ten times lower threshold, i.e., 9.7 μas yr^{1}, for the perspectiveaccelerationinduced propermotion error caused by an incorrect radial velocity (see Sect. 6.1 for details), we find that Σ = 0.26 km s^{1}. This implies, for a confidence level c = 68.27%, that the radial velocity should have been measured for this object with a standard error smaller than km s^{1}. The literature radial velocity for this star is v_{r} = −22.40 ± 0.50 km s^{1} (with quality grade “B”; Sect. 2.2), which is not precise enough. New spectroscopic measurements are thus needed for this object to reduce the standard error by a factor ~4.
The discussion above has implicitly focused on the rightascension propermotion component μ_{α}, and the associated Σ_{α}, since the sensitivty of μ_{δ} is a factor ~4 less stringent for this star. It is generally sufficient to consider the most constraining case for a given star, i.e., either Σ_{α} or Σ_{δ}. Therefore, we drop from here on the subscript α and δ on Σ, implicitly meaning that it either refers to Σ_{α} or Σ_{δ}, depending on which one is largest. Typically, this is the largest propermotion component, i.e.,  μ_{α}  >  μ_{δ}  → Σ = Σ_{α} and  μ_{α}  <  μ_{δ}  → Σ = Σ_{δ}.
4.3. Derivation of the sensitivity
Figure 3 shows that the sensitivity of proper motion to radial velocity is linear. This can be understood by substitution of Eqs. (8), (8) in Eqs. (6), (7), after replacing v_{r}, as used in the forward propagation, by v_{r} + Δv_{r} in the backward solution: (13)which immediately shows Δμ_{α} ∝ Δv_{r}. A similar analysis for the sensitivity coefficient C_{δ} yields: (14)The coefficients C_{α} and C_{δ} quantify the propermotionestimation error caused by a biased knowledge of v_{r} given measured displacements Δα^{ ∗ } and Δδ. They can hence more formally be defined as the partial derivatives of μ_{α} and μ_{δ} from Eqs. (8) and (8) with respect to v_{r} with Δα^{ ∗ } and Δδ kept constant: Clearly, our relations confirm the wellknown, classical result (e.g., Dravins et al. 1999, Eq. (4)) that perspectiveaccelerationinduced propermotion errors are proportional to the product of the time interval, the parallax ϖ ∝ r^{1}, the propermotion components μ_{α,δ}, and the radialvelocity error Δv_{r}: Δμ_{α,δ} = C_{α,δ}Δv_{r} = r^{1}μ_{α,δ}tΔv_{r}. Equations (13), (14) show that the perspectiveaccelerationinduced propermotion error caused by a radialvelocity error does not depend on the radial velocity v_{r} itself but only on the error Δv_{r}. This may look counterintuitive at first sight, since the proper motion itself is sensitive to the precise value of the radial velocity. The error in the proper motion, however, is sensitive only to the radialvelocity error. In other words, the slopes of the Vshaped wedge in Fig. 3 do not depend on the absolute but only on the relative labelling of the abscissa.
4.4. Taking measurement errors into account
So far, we ignored the measurement errors of the astrometric parameters α, δ, ϖ, μ_{α}, and μ_{δ}. A natural way to take these errors into account is by MonteCarlo simulations: rather than deriving Σ once, namely based on the astrometric parameters contained in the XHIP catalogue, we calculate Σ a large number of times (typically N = 10 000), where in each run we do not use the catalogue astrometry but randomly distorted values drawn from normal distributions centred on the measured astrometry and with standard deviations equal to the standard errors of the astrometric parameters (denoted N(mean,variance)). We also randomly draw the radial velocity in each run from the normal probability distribution .
Fig. 4 Histogram of the distribution of Σ in the N = 10 000 MonteCarlo simulations for star HIP38 (ϖ = 23.64 ± 0.66 mas, so 3% relative error). The smooth, red curve is a Gaussian fit of the histogram; it provides a good representation. 

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The MonteCarlo simulations yield N = 10 000 values for Σ; the interpretation of this distribution will be addressed in Sect. 5. Two representative examples of the distribution of Σ are shown in Figs. 4 and 5. The first distribution (Fig. 4, representing HIP38) is for a nearby star with a welldetermined parallax: ϖ = 23.64 ± 0.66 mas, i.e., 3% relative parallax error. The smooth, red curve in Fig. 4 is a Gaussian fit of the histogram; it provides a good representation of the data. The second distribution (Fig. 5, representing HIP8) is for a distant star with a less welldetermined parallax: ϖ = 4.98 ± 1.85 mas, i.e., 37% relative parallax error. This results in an asymmetric distribution of Σ with a tail towards large Σ values. This is easily explained since we essentially have , meaning that the distribution of Σ reflects the probability distribution function of ϖ^{1} ∝ r. The latter is wellknown (e.g., Kovalevsky 1998; Arenou & Luri 1999) for its extended tail towards large distances and its (light) contraction for small distances. The smooth, red curve in Fig. 5 is a Gaussian fit of the histogram; it provides an inadequate representation of the data. We will come back to this in Sect. 5.2.
To avoid dealing with (a significant number of) negative parallaxes in the MonteCarlo simulations, we decided to ignore 11 171 entries with insignificant parallax measurements in the XHIP catalogue; these include 3920 entries with ϖ ≤ 0 and 7251 entries with 0 < ϖ/σ_{ϖ} ≤ 1 (recall that negative parallaxes are a natural outcome of the Hipparcos astrometric data reduction, e.g., Arenou et al. 1995). This choice does not influence the main conclusions of this paper: perspectiveaccelerationinduced HTPM propermotion errors are significant only for nearby stars whereas negative and lowsignificance parallax measurements generally indicate large distances.
Fig. 5 Histogram of the distribution of Σ in the N = 10 000 MonteCarlo simulations for star HIP8 (ϖ = 4.98 ± 1.85 mas, so 37% relative error). The smooth, red curve is a Gaussian fit of the histogram; it provides a poor representation and does not account for the tail in the distribution. 

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5. Evaluation criteria
From the MonteCarlo distribution of Σ (Sect. 4.4), we want to extract information to decide whether the radial velocity available in the literature is sufficiently precise or not. For this, we develop two evaluation criteria.
5.1. The Gaussian evaluation criterion
The first criterion, which we refer to as the Gaussian criterion, is based on Gaussian interpretations of probability distributions. It can be applied to all stars but, since we have seen that distant stars do not have a Gaussian Σ distribution, but rather a distribution with tails towards large Σ values (Fig. 5), the Gaussian criterion is unbiased, and hence useful, only for nearby stars. For these stars, the MonteCarlo distribution of Σ is well described by a Gaussian function with mean μ_{Σ} and standard deviation σ_{Σ} (Fig. 4). The equations derived in Sect. 4.1 by ignoring astrometric errors are easily generalised by recognising that both the radialvelocity error and the distribution of Σ have Gaussian distributions, with standard deviations σ_{vr} and σ_{Σ}, respectively: (17)generalises to: (18)Using Eq. (8.259.1) from Gradshteyn & Ryzhik (2007), Eq. (18) simplifies to: (19)which correctly reduces to Eq. (10) for the limiting, “errorfree” case μ_{Σ} → Σ and σ_{Σ} → 0 in which the Gaussian distribution of Σ collapses into a delta function at μ_{Σ} = Σ, i.e., δ(Σ).
For the example of HIP 70890 discussed in Sect. 4.2, we find μ_{Σ} = 0.26 km s^{1} and σ_{Σ} = 0.000,89 km s^{1} so that, with σ_{vr} = 0.50 km s^{1}, Eq. (19) returns c = 20.58%.
It is trivial, after rearranging Eq. (19) to: (20)to compute the required standard error of the radial velocity to comply with a certain confidence level c. For instance, if we require a c = 99.73% confidence level (for a “3σ result”), the radialvelocity standard error of HIP 70890 has to be 0.04 km s^{1}. We finally note that Eq. (20) correctly reduces to Eq. (12) for the limiting, “errorfree” case μ_{Σ} → Σ and σ_{Σ} → 0.
A limitation of Eq. (20) is that the argument of the square root has to be nonnegative. This is physically easy to understand when realising that, in the Gaussian approximation, one has σ_{Σ} ≈ μ_{Σ}(σ_{ϖ}/ϖ) (see Sect. 4.4), so that: (21)So, for instance, if a certain star has ϖ/σ_{ϖ} = 2 (a “2σ parallax”), the Gaussian methodology will only allow to derive the radialvelocity standard error σ_{vr} required to meet a confidence level c = 95.45% or lower.
5.2. The robust criterion
Since the MonteCarlo distribution of Σ values is not Gaussianly distributed for distant stars (Fig. 5), the Gaussian criterion returns incorrect estimates; in fact, the estimates are not just incorrect but also biased since the Gaussian criterion systematically underestimates the mean value of Σ (i.e., μ_{Σ}) and hence systematically provides too conservative (small) estimates for σ_{vr} through Eq. (20). Rather than fitting a Gaussian function, we need a more robust estimator of the location and width of the Σ distribution than the Gaussian mean and standard deviation. This estimator is contained in the data itself and provides, what we call, the robust criterion.
Let us denote the individual values of Σ derived from the N = 10 000 MonteCarlo simulations by Σ_{i}, with i = 1,...,N. Equation (18) is then readily generalised for arbitrary distributions of Σ to: (22)The inverse relation generalising Eq. (20) by expressing σ_{vr} as function of c, required to determine the precision of the radial velocity required to comply with a certain confidence level c, is not analytical; we hence solve for it numerically.
The robust criterion generalises the Gaussian criterion. Both criteria return the same results for nearby stars which have a symmetric Gaussian distribution of Σ values. In general, therefore, the robust criterion is the prefered criterion for all stars, regardless of their distance.
6. Application of the evaluation criteria
6.1. Target propermotionerror threshold
Before we can apply the Gaussian and robust criteria, we have to decide on a target propermotionerror threshold for each star (i.e., the location of the dashed horizontal lines in Figs. 2 and 3). We adopt as a general rule that the maximum perspectiveaccelerationinduced HTPM propermotion error caused by radialvelocity errors shall be an order of magnitude smaller than the predicted standard error of the HTPM proper motion itself (Sect. 7.2 discusses this choice in more detail). The latter quantity has been studied by Mignard (2009) and can be parametrised as: where H = max { 6.5,Hp [mag] } with Hp the Hipparcos broadband magnitude. These relations are shown in Fig. 6. The predicted HTPM standard errors include residual errors caused by the correction for the parallactic effect in the Gaia data (see footnote 1), the expected number and temporal distribution of the Gaia fieldofview transits for the Gaia nominal sky scanning law, and the expected locationestimation precision (“centroiding error”) of Gaia’s CCDlevel data.
6.2. Stars without literature radial velocities
As already discussed in Sect. 2.2, the majority of stars in the XHIP catalogue do not have a literature radial velocity. These 71 563 objects are treated as stars with radial velocity, with three exceptions:

1.
For each of theN = 10 000 MonteCarlo simulations, the radial velocity is randomly taken from the full list of 46 392 radial velocites contained in the XHIP catalogue (Fig. 1). In practice, this choice does not influence our results since we are not sensitive to the absolute value of the radial velocity (Sect. 4.3). But at least in principle this choice means that there is a finite probability to assign a halostarlike, i.e., large radial velocity in one (or more) of the MonteCarlo runs. Regarding the HTPM Project, the best choice for stars without known radial velocity is to use v_{r} = −2.00 km s^{1}, which is the median value of the distribution (recall that the mean equals −2.21 km s^{1}).

2.
For the Gaussian evaluation criterion (Sect. 5.1), we use σ_{vr} = 22.44 km s^{1} from the Gaussian fit to the distribution of all literature radial velocities. We also follow this recipe for the Gaussian criterion for the 1753 entries which do have a radial velocity but which do not have an associated standard error in the XHIP catalogue (this concerns 23 entries with quality grade “C” and 1730 entries with quality grade “D”). Clearly, this approach ignores the broad wings of the distribution visible in Fig. 1 as well as a small but finite number of halo stars and runaway stars with heliocentric radial velocities up to plusorminus several hundred km s^{1} (see Sect. 2.2). As a result, the Gaussian criterion systematically returns an overly optimistic (i.e., too large) confidence level c_{Gauss} for stars without literature radial velocity (Figs. 7 and 8). This bias comes in addition to the bias for distant stars for which the Gaussian criterion returns too conservative (small) estimates for σ_{vr} (Sect. 5.2).

3.
For the robust evaluation criterion (Sect. 5.2), we do not make a priori assumptions except that the overall radialvelocity distribution of stars without known radial velocity is the same as the distribution of stars with literature radial velocity, including broad wings and “outliers”. We thus use Eq. (22) in the form: (25)where the probability P that Δv_{r} is contained in the interval is calculated as the fraction of all stars with literature radial velocities in the XHIP catalogue which has (recall that the median radial velocity equals −2.00 km s^{1}). We thus cater for the broad wings of the observed radialvelocity distribution (Figs. 1 and 7) as well as for the probability that the object is a (fastmoving) halo star, avoiding the bias in the Gaussian criterion discussed in the previous bullet.
Fig. 6 Predicted HTPM propermotion error as function of the Hipparcos broadband magnitude following Eqs. (23), (23). We require perspectiveaccelerationinduced propermotion errors to be an order of magnitude smaller (factor of safety = FoS = 10; see Sect. 7.2). 

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6.3. Results of the application
We applied the Gaussian and robust criteria, as described in Sects. 5.1 and 5.2, respectively, to the XHIP catalogue. The results are presented in Table A.1 (Appendix A). The run time for N = 10 000 MonteCarlo simulations is typically ~0.7 s per star and processing the full set of 117 955−11 171 = 106 784 XHIP entries with significant parallaxes (Sect. 4.4) hence takes about one day.
Fig. 7 The fraction of stars with XHIP literature radial velocities which are contained in the radialvelocity interval [m − R,m + R] as function of R, with m = −2.21 km s^{1} the mean v_{r} for the Gaussian criterion and m = −2.00 km s^{1} the median v_{r} for the robust criterion (Sects. 2.2 and 6.2). For the Gaussian criterion, we represent the histogram of literature radial velocities by a Gauss with standard deviation σ = 22.44 km s^{1} (Fig. 1). The dashed lines represent the classical limits 1σ = 68.27%, 2σ = 95.45%, and 3σ = 99.73%. The fraction of stars with the robust criterion builds up more slowly as a result of the nonGaussian broad wings as well as outliers representing halo and runaway stars. Since the Gaussian criterion ignores these features, it returns biased results for stars without literature radial velocity (see Sect. 6.3). 

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Fig. 8 Comparison of the robust and Gaussian criteria for all stars with σ_{ϖ}/ϖ better than 5%. The top panel compares the confidence levels while the bottom panel compares the radialvelocity standard errors σ_{vr} required to reach a confidence level c = 68.27%. The top panel shows two branches of data points: the linear, onetoone branch corresponds to stars with a measured radial velocity, whereas the lower, curved branch corresponds to stars without measured radial velocity in the XHIP catalogue. As explained in Sect. 6.2, the latter objects suffer from a bias in the Gaussian confidence level c_{Gauss}. 

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Figures 9, 10 and Tables 2, 3 show the results for the confidence levels of the literature radial velocities contained in the XHIP catalogue. We find, not surprisingly since perspectiveaccelerationinduced propermotion errors are relevant only for nearby, fastmoving stars – which are relatively rare – that the majority of stars have confidence levels exceeding c = 99.73%. This indicates that, at the c = 99.73% confidence level, the available radial velocity is sufficiently precise or, for stars without literature radial velocity, that the absence of a literature radial velocity, and hence the assumption v_{r} = −2.00 km s^{1} for the robust criterion or v_{r} = −2.21 km s^{1} for the Gaussian criterion, is acceptable. This holds for more than 100 000 stars using the robust criterion (Table 3) and more than 85 000 stars using the Gaussian criterion (Table 2). The large difference between the two criteria does not come unexpectedly:

1.
We already argued in Sect. 5.2 that the Gaussiancriterion is biased for distant stars, with distant meaning that theparallax probability distribution has an associated asymmetricdistance probability distribution. For such stars, the Gaussiancriterion systematically underestimates the mean value ofΣ and hence returns too conservative (small) values for σ_{vr} for a given value of the confidence level c and too pessimistic (small) values of c for a given value of σ_{vr}.

2.
We already argued in Sect. 6.2 that the Gaussian criterion is biased for stars without literature radial velocity. For such stars, the Gaussian criterion systematically returns too optimistic (large) values of the confidence level c since it ignores the broad wings of the observed distribution of radial velocities (Figs. 1, 7, and 8) and also ignores the probability that the object is actually a halo (or runaway) star.
The robust criterion does not suffer from these biases and hence, being more reliable, is prefered for all stars. The Gaussian criterion, nonetheless, provides a useful and also easily interpretable reference test case and we hence decided to retain it. Figure 8 shows that, for nearby stars with literature radial velocities, the Gaussian and robust criteria return equivalent results.
For a small but nonnegligible number of stars, Table 3 indicates insatisfactory results: 206 stars have a confidence level c < 68.27%: 97 of these do have a literature radial velocity in the XHIP catalogue but one which is insufficiently precise. The remaining 109 stars do not have a spectroscopically measured radial velocity (at least not one contained in the XHIP catalogue). New spectroscopy is hence required for these stars to guarantee a confidence level of at least 68.27%. For increased confidence levels, the numbers obviously increase: if requiring a c = 99.73% confidence level for all objects, for instance, the number of “problem stars” increases to 6562, split into 382 with insufficientlyprecise known radial velocity and 6180 without known radial velocity. We conclude that, depending on the confidence level one wants to achieve, hundreds to thousands of stars need to be spectroscopically remeasured.
Figure 11 shows the robust confidence level versus Hp magnitude. One can see that the typical star which needs a highpriority spectroscopic measurement (i.e., c < 68.27%) has Hp in the range 8–12 mag. Figure 12 shows the radialvelocity precision required to reach c = 68.27% (computed with the robust criterion) versus magnitude. Precisions vary drastically, from very stringent values well below 1 km s^{1} to very loose values, up to several tens of km s^{1}.
Fig. 9 Histograms of the Gaussian confidence level c from Sect. 5.1 for all stars combined, stars without measured radial velocity, and stars with measured radial velocity as function of radialvelocity quality grade Q_{vr} (see Sect. 2.2 and Appendix A). The vast majority of objects have c > 95.45% (see Table 2); they have been omitted from the histograms to improve their legibility. 

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7. Discussion
7.1. HTPM bright limit
The HTPM catalogue will contain the intersection of the Hipparcos and Gaia catalogues. Whereas the Hipparcos catalogue (ESA 1997b), which contains 117 955 entries with astrometry (and 118 218 entries in total), is complete to at least V = 7.3 mag (ESA 1997a, Sect. 1.1), the Gaia catalogue will be incomplete at the bright end. Gaia’s bright limit is G = 5.7 mag (de Bruijne 2012), where G is the whitelight, broadband Gaia magnitude, which is linked to the Hipparcos Hp, the Cousins I, and the Johnson V magnitudes through (Jordi et al. 2010): (26)For stars in the Hipparcos catalogue, the colour G − Hp ranges between −0.5 and 0.0 mag, with a mean value of −0.3 mag. This means that G = 5.7 mag corresponds roughly to Hp ≈ 6.0 mag. In practice, therefore, we are not concerned with the brightest ~4509 stars in the sky. The expected number of HTPMcatalogue entries is therefore ~117 955−4509 ≈ 113 500. The number of entries with significant parallax measurements equals 117 955−11 171 = 106 784, of which 106 784−4468 = 102 316 have G > 5.7 mag.
7.2. Propermotionerror threshold
For both the Gaussian and the robust criteria, we adopt, somewhat arbitrarily, the rule that the perspectiveaccelerationinduced HTPM propermotion error caused by radialvelocity errors shall be an order of magnitude smaller than the predicted standard error of the HTPM proper motion itself (Sect. 6.1). The adopted Factor of Safety (FoS) is hence 10. Some readers may find that this “rule” is too stringent. Unfortunately, there is no easy (linear) way to scale our results if the reader wants to adopt a different value for the FoS. Clearly, the value of Σ in each MonteCarlo run is linearly proportional to the FoS (see, for instance, Fig. 2). However, the robust confidence level c_{robust} and the radialvelocity standard error σ_{vr} required to meet a certain value of c_{robust} do not linearly depend on Σ but on the properties of the, in general asymmetric, distribution of the N = 10 000 values of Σ resulting from the MonteCarlo processing. To zeroth order, however, one can assume a linear relationship between σ_{vr} and Σ and hence the adopted FoS, as is also apparent from the “errorfree” criterion derived for c = 68.27% in Sect. 4.1. This is in particular a fair approximation for small variations around the default value (FoS = 10) in combination with nearby stars, which are most interesting because these are most sensitive to perspective acceleration. The MonteCarlo distribution of Σ values for these objects is generally well behaved, i.e., symmetric and with σ_{ϖ} ≪ ϖ and hence σ_{Σ} ≪ μ_{Σ} (see, for instance, Fig. 4; see also Sect. 5.1). Figure 13 shows, as an example for the nearby star HIP 57367 (see Table 4), how σ_{vr} (required for c_{robust} = 68.27%) and c_{robust} vary as function of the FoS. Linear scaling around the default FoS = 10 provides a decent approximation, at least over the range 10/2 = 5 < FoS < 20 = 10 × 2.
Fig. 11 Robust confidence level c versus Hipparcos broadband magnitude. The red box in the bottomright corner denotes the approximate area for highpriority followup spectroscopy: stars with confidence level c < 68.27% and mag. The latter restriction roughly reflects Gaia’s – and hence HTPM’s – bright limit (Sect. 7.1). 

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Fig. 12 Radialvelocity precision (standard error) for stars with robust confidence level c < 68.27% required to upgrade their confidence level to c = 68.27%. 

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7.3. Number of MonteCarlo simulations
To take measurement errors in the XHIP astrometry and literature radial velocities into account, we adopt a MonteCarlo scheme in which we run a number N of MonteCarlo simulations for each star in which we randomly vary the astrometric and spectroscopic data within their respective error bars (see Sect. 4.4). Clearly, the higher the value of N is, the more reliable the results are. We adopted N = 10 000 as a practical compromise, resulting in an acceptable, typical run time of ~1 s per star as well as smooth distributions of Σ (see, for instance, Fig. 4 or 5). To investigate the repeatability and hence reliability of our robust confidence levels c_{robust} and radialvelocity errors σ_{vr}, we have repeated the entire processing with N = 10 000 runs 100 times for the 206 stars with c_{robust} < 68.27% (Sect. 6.3) and find that the typical variation of the confidence level and the radialvelocity error, quantified by the standard deviation divided by the average of the distribution containing the 100 results, is less than 0.1% and 0.2%, respectively; the maximum variation among the 206 objects is found for HIP 107711 and amounts to 0.5% and 0.6%, respectively.
7.4. Highpriority and challenging stars
Table 4 shows the ten stars with the lowest robust confidence level. These stars are the highestpriority targets for spectroscopic followup. Nine of the ten entries do have a literature radial velocity but one which is insufficiently precise. The highestpriority object (HIP 57367) does not yet have a spectroscopic radial velocity and needs a measurement with a standard error better than 1 km s^{1}. For this particular object, this challenge seems insurmountable since it is one of the 20 white dwarfs with Hipparcos astrometry (Vauclair et al. 1997), objects for which it is notoriously difficult to obtain – even lowprecision – spectroscopic radial velocities.
Table 5 shows the ten stars, among the subset of stars with unacceptablylow robust confidence level (c_{robust} < 68.27%), with the smallest radialvelocity standard errors required to raise the robust confidence level to c_{robust} = 68.27%. Since c_{robust} < 68.27%, these stars do clearly need spectroscopic followup. However, the radialvelocity standard errors reach values as small as 0.04 km s^{1}, which is a real challenge, not only in terms of the required signaltonoise ratio of the spectroscopic data but also in view of the definition of the radialvelocity zeropoint at this level of precision (Crifo et al. 2010) as well as potential systematic errors in the radial velocities, both with instrumental origin and with astrophysical causes such as radialvelocity differences between various absorption lines etc. (see Lindegren & Dravins 2003, for a detailed discussion of this and other effects).
Fig. 13 Dependence, for HIP 57367 (see Table 4), of σ_{vr} (required to reach c_{robust} = 68.27%) and the robust confidence level c_{robust} on the Factor of Safety (FoS), i.e., the minimum factor between the predicted HTPM standard error and the perspectiveaccelerationinduced HTPM propermotion error caused by radialvelocity errors. The default FoS value adopted in this study is 10. The object is representative for a nearby star with a welldetermined parallax. The straight, red lines indicate linear scaling relations starting from the default value FoS = 10. 

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The ten stars with the lowest robust confidence level c_{robust} in the XHIP catalogue.
The ten stars, among the subset of stars with unacceptablylow robust confidence level (c_{robust} < 68.27%), with the most stringent radialvelocityerror requirements σ_{vr} needed to reach c_{robust} = 68.27%.
Table 2 in Dravins et al. (1999) shows the top39 of stars in the Gliese & Jahreiss (1995) preliminary third catalogue of nearby stars ranked according to the magnitude of the perspective acceleration (which is propertional to ϖμ). Similarly, Table 1.2.3 in ESA (1997a) shows the top21 of stars in the Hipparcos catalogue for which the magnitude of the perspective acceleration is significant enough to have been taken into account in the Hipparcos data processing (the accumulated effect on position is proportional to ϖμ  v_{r}  ). On the contrary, the top10 Tables 4 and 5 have been constructed based on the sensitivity of the perspective acceleration to radialvelocity errors (Sect. 4.3) and the associated confidence level of the available literature radial velocity. Hence, although there is a significant overlap of stars between the various tables, they are understandably not identical.
7.5. Objectbyobject analyses and other literature sources
In general, and in particular for the most interesting, delicate, or border cases, it will be useful to perform a more indepth literature search for and study of radial velocities and other available data before embarking on groundbased spectroscopy. For instance, we found a SIMBAD note on the Hipparcos catalogue (ESA 1997a,b, CDS catalogue I/239) for HIP 114110 (c_{robust} = 71.23%) and HIP 114176 (c_{robust} = 60.00%) that they are nonexisting objects: “HIP 114110 (observed with HIP 14113^{5}) and HIP 114176 (observed with HIP 114177) are noted as probable measurements of scattered light from a nearby bright star. The nonreality of 114110 and 114176 (traced to fictitious entries in the WDS and INCA) has been confirmed by MMT observations reported by Latham (priv. comm., 8 May 1998), and confirmed by inspection of the DSS [J. L. Falin, 12 May 1998]”. In addition, the completeness and coverage level of the XHIP radialvelocity compilation is not known. We did query SIMBAD as well as the GenevaCopenhagenSurvey (GCS, CORAVEL) database (Nordström et al. 2004, CDS catalogue V/117) and the RAVE database (Siebert et al. 2011, CDS catalogue III/265, with 77,461 entries with a mean precision of 2.3 km s^{1}) for radial velocities for the 109 stars without XHIP radial velocity and with confidence level below 68.27% but did not find new data. Unfortunately, the treasure contained in the full CORAVEL database (45 263 latetype Hipparcos stars with precisions below 1 km s^{1}), the public release of which was announced in Udry et al. (1997) to be before the turn of the previous millennium, remains a mystery to date. All in all, dedicated studies for individual objects might pay off by reducing the needs for spectroscopic followup.
7.6. Urgency of the spectroscopic followup
Mignard (2009) already acknowledges that, since the perspectiveaccelerationinduced propermotion error can be calculated as function of radial velocity, a factor – effectively the sensitivity coefficients C_{α} and C_{δ} from Eqs. (13), (14) in Sect. 4.3 – can be published to correct the HTPM proper motion for a particular star a posteriori when v_{r} becomes known or when a more precise v_{r} becomes available. Therefore, both the reference radial velocity v_{r} and parallax ϖ used in the HTPM derivation will be published together with the propermotion values themselves. This means that the spectroscopic followup identified in this paper is not timecritical: the HTPM catalogue can and will be published in any case, even if not all required spectroscopic followup has been completed. Of course, the implication for stars without the required radialvelocity knowledge will be that their HTPM proper motions will include a (potentially) significant perspectiveaccelerationinduced error.
8. Conclusions
We have conducted a study of the requirements for the availability of radial velocities for the HundredThousandProperMotion (HTPM) project (Mignard 2009). This unique project will combine Hipparcos astrometry from 1991.25 with earlyrelease Gaia astrometry (~2014.5) to derive longtimebaseline and hence precise proper motions. For the nearest, fastmoving stars, the perspective acceleration of the objects on the sky requires the presence of radial velocities for the derivation of the proper motions. We have quantitatively determined, for each star in the Hipparcos catalogue, the precision of the radial velocity that is required to ensure that the perspectiveaccelerationinduced error in the HTPM proper motion caused by the radialvelocity error is negligible. Our method takes the Hipparcos measurement errors into account and allows the user to specify his/her own prefered confidence level (e.g., 68.27%, 95.45%, or 99.73%). The results are available in Table A.1 (Appendix A). We have compared the radialvelocityprecision requirements to the set of 46 392 radial velocities contained in the XHIP compilation catalogue (Anderson & Francis 2012) and find that, depending on the confidence level one wants to achieve, hundreds to thousands of stars require spectroscopic followup. The highestpriority targets are 206 objects with a confidence level below 68.27%; 97 of them have a known but insufficiently precise radial velocity while the remaining 109 objects have no literature radial velocity in the XHIP compilation catalogue at all. The typical brightness of the objects requiring their radial velocity to be (re)determined is Hp ≈ 8−12 mag and the radialvelocity precisions vary drastically, ranging from 0.04 km s^{1} for the most extreme case (HIP 87937, also known as Barnard’s star) to a few tens of km s^{1}. With only few exceptions, the spectral types are K and M; 73% of them are in the south. Gaia’s RadialVelocity Spectrometer (RVS; Cropper & Katz 2011) will deliver radial velocities for all stars in the HTPM catalogue with Gaiaendofmission precisions below a few km s^{1} (and ~10 km s^{1} for earlytype stars; de Bruijne 2012); however, these performances require full calibration of the instrument and data and hence will most likely only be reached in the final Gaia data release, at which time the HTPM proper motions will be superseded by the Gaia proper motions. Fortunately, the spectroscopic followup is not timecritical in the sense that the HTPM catalogue will be published with information (sensitivity coefficients and reference parallax and radial velocity) to correct the proper motions a posteriori when (improved) radial velocities become available.
We finally note that the spectroscopic followup requirements for the HTPM proper motions quantified in this work will be dwarfed by the requirements coming from the endofmission Gaia proper motions, to be released around ~2021: for instance for the stars in the HTPM catalogue, for which the HTPM propermotion standard errors are 30−190 μas yr^{1}, the Gaia propermotion standard errors reach the brightstar floor around 3–4 μas yr^{1} (de Bruijne 2012), which means that the spectroscopic requirements for the correction of perspective acceleration in the Gaia astrometry, with a 5year baseline, will be a factor ~2–10 more demanding.
The HTPM proper motions will be limited in precision by the Hipparcos parallax uncertainties, which are typically ~1 mas (the typical HTPM propermotion standard error is hence 1 mas/23 yr ≈ 40 μas yr^{1}). The first intermediaterelease Gaia catalogue is based on just ~12 months of data, which is generally insufficient to unambiguously lift the degeneracy between proper motion and parallax for all stars. The underlying astrometric global iterative solution (Lindegren et al. 2012) will hence be based on a two rather than fiveparameter source model, fitting for position (α,δ) at the mean Gaia epoch only. The Hipparcos parallax is hence needed to correct the Gaia transit observations for the parallactic effect allowing to transform apparent directions into barycentric positions.
These authors also describe two other methods to derive astrometric radial velocities, namely by measuring changing annual parallax or by measuring changing angular extent of a moving group of stars (Madsen et al. 2002). The latter method also provides, as a bonus, improved parallaxes to movinggroup members (e.g., de Bruijne 1999; de Bruijne et al. 2001).
This is a typo and must be HIP 114113.
Acknowledgments
It is a pleasure to thank Mark Cropper for discussions about radialvelocity surveys and the referee, François Mignard, for his constructive criticism which helped to improve our statistical methodology. This research has made use of the SIMBAD database and VizieR catalogue access tool, both operated at the Centre de Données astronomiques de Strasbourg (CDS), and of NASA’s Astrophysics Data System (ADS).
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Appendix A: The results data file
Table A.1 describes the results data file, which is available at the CDS.
Description of the finalresults data file.
All Tables
Propermotion errors accumulated over t = 25 yr as function of declination due to truncation of the full model up to and including first, second, and thirdorder terms in time for an “extreme”, i.e., nearby, fastmoving star: ϖ = 500 mas (r = 2 pc), μ_{α} = μ_{δ} = 2000 mas yr^{1}, and v_{r} = 50 km s^{1}.
The ten stars with the lowest robust confidence level c_{robust} in the XHIP catalogue.
The ten stars, among the subset of stars with unacceptablylow robust confidence level (c_{robust} < 68.27%), with the most stringent radialvelocityerror requirements σ_{vr} needed to reach c_{robust} = 68.27%.
All Figures
Fig. 1 Distribution of all 46 392 radial velocities contained in the XHIP catalogue. The smooth, red curve fits the histogram with a Gaussian normal distribution. The bestfit mean and standard deviation are μ = −2.21 km s^{1} and σ = 22.44 km s^{1}, respectively. 

Open with DEXTER  
In the text 
Fig. 2 Schematic diagram showing how to quantify the sensitivity of the proper motion to a change in (read: measurement error of) radial velocity. A change in the radial velocity Δv_{r} introduced before the backward solution leads to a certain (HTPM) propermotion error Δμ. The linear dependence is commented on in Sect. 4.3. Since the magnitude of the propermotion error does not depend on the sign but only on the magnitude of the radialvelocity variation, the sensitivity curve is symmetric with respect to the true radial velocity. The dashed horizontal line denotes the maximum perspectiveaccelerationinduced propermotion error we are willing to accept in the HTPM proper motion. The distance Σ between the intersection points of the dashed horizontal line and the solid sensitivity curves determines the tolerance on the radialvelocity error. 

Open with DEXTER  
In the text 
Fig. 3 Sensitivity of the HTPM proper motion in right ascension to radial velocity for HIP 70890 (Proxima Centauri). The sensitivity is linear and has a value C_{α} = −74.59 μas yr^{1} per km s^{1} (Sect. 4.3). The dashed horizontal line indicates the maximumtolerable perspectiveaccelerationinduced propermotion error caused by an incorrect radial velocity. Since the expected HTPM standard error in right ascension is 97 μas yr^{1} for this star, we set this threshold to 97/10 = 9.7 μas yr^{1}. This implies, for a confidence level c = 68.27%, that the maximumacceptable radialvelocity standard error σ_{vr} for this object is km s^{1}. 

Open with DEXTER  
In the text 
Fig. 4 Histogram of the distribution of Σ in the N = 10 000 MonteCarlo simulations for star HIP38 (ϖ = 23.64 ± 0.66 mas, so 3% relative error). The smooth, red curve is a Gaussian fit of the histogram; it provides a good representation. 

Open with DEXTER  
In the text 
Fig. 5 Histogram of the distribution of Σ in the N = 10 000 MonteCarlo simulations for star HIP8 (ϖ = 4.98 ± 1.85 mas, so 37% relative error). The smooth, red curve is a Gaussian fit of the histogram; it provides a poor representation and does not account for the tail in the distribution. 

Open with DEXTER  
In the text 
Fig. 6 Predicted HTPM propermotion error as function of the Hipparcos broadband magnitude following Eqs. (23), (23). We require perspectiveaccelerationinduced propermotion errors to be an order of magnitude smaller (factor of safety = FoS = 10; see Sect. 7.2). 

Open with DEXTER  
In the text 
Fig. 7 The fraction of stars with XHIP literature radial velocities which are contained in the radialvelocity interval [m − R,m + R] as function of R, with m = −2.21 km s^{1} the mean v_{r} for the Gaussian criterion and m = −2.00 km s^{1} the median v_{r} for the robust criterion (Sects. 2.2 and 6.2). For the Gaussian criterion, we represent the histogram of literature radial velocities by a Gauss with standard deviation σ = 22.44 km s^{1} (Fig. 1). The dashed lines represent the classical limits 1σ = 68.27%, 2σ = 95.45%, and 3σ = 99.73%. The fraction of stars with the robust criterion builds up more slowly as a result of the nonGaussian broad wings as well as outliers representing halo and runaway stars. Since the Gaussian criterion ignores these features, it returns biased results for stars without literature radial velocity (see Sect. 6.3). 

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In the text 
Fig. 8 Comparison of the robust and Gaussian criteria for all stars with σ_{ϖ}/ϖ better than 5%. The top panel compares the confidence levels while the bottom panel compares the radialvelocity standard errors σ_{vr} required to reach a confidence level c = 68.27%. The top panel shows two branches of data points: the linear, onetoone branch corresponds to stars with a measured radial velocity, whereas the lower, curved branch corresponds to stars without measured radial velocity in the XHIP catalogue. As explained in Sect. 6.2, the latter objects suffer from a bias in the Gaussian confidence level c_{Gauss}. 

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In the text 
Fig. 9 Histograms of the Gaussian confidence level c from Sect. 5.1 for all stars combined, stars without measured radial velocity, and stars with measured radial velocity as function of radialvelocity quality grade Q_{vr} (see Sect. 2.2 and Appendix A). The vast majority of objects have c > 95.45% (see Table 2); they have been omitted from the histograms to improve their legibility. 

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In the text 
Fig. 11 Robust confidence level c versus Hipparcos broadband magnitude. The red box in the bottomright corner denotes the approximate area for highpriority followup spectroscopy: stars with confidence level c < 68.27% and mag. The latter restriction roughly reflects Gaia’s – and hence HTPM’s – bright limit (Sect. 7.1). 

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In the text 
Fig. 12 Radialvelocity precision (standard error) for stars with robust confidence level c < 68.27% required to upgrade their confidence level to c = 68.27%. 

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In the text 
Fig. 13 Dependence, for HIP 57367 (see Table 4), of σ_{vr} (required to reach c_{robust} = 68.27%) and the robust confidence level c_{robust} on the Factor of Safety (FoS), i.e., the minimum factor between the predicted HTPM standard error and the perspectiveaccelerationinduced HTPM propermotion error caused by radialvelocity errors. The default FoS value adopted in this study is 10. The object is representative for a nearby star with a welldetermined parallax. The straight, red lines indicate linear scaling relations starting from the default value FoS = 10. 

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In the text 