Impact of basic angle variations on the parallax zero point for a scanning astrometric satellite
Alexey G. Butkevich, Sergei A. Klioner, Lennart Lindegren, David Hobbs, Floor van Leeuwen
AAstronomy & Astrophysics manuscript no. parallaxShiftBA c (cid:13)
ESO 2018October 4, 2018
Impact of basic angle variations on the parallax zero point for ascanning astrometric satellite
Alexey G. Butkevich , , Sergei A. Klioner , Lennart Lindegren , David Hobbs , and Floor van Leeuwen Pulkovo Observatory, Pulkovskoye shosse 65, 196140 Saint-Petersburg, Russiae-mail: [email protected] Lohrmann Observatory, Technische Universität Dresden, 01062 Dresden, Germanye-mail: [email protected] Lund Observatory, Department of Astronomy and Theoretical Physics, Lund University, Box 43, 22100 Lund, Swedene-mail: [lennart; david]@astro.lu.se Institute of Astronomy, Madingley Road, Cambridge CB3 OHA, UKe-mail: [email protected]
Received; accepted
ABSTRACT
Context.
Determination of absolute parallaxes by means of a scanning astrometric satellite such as H ipparcos or Gaia relies on theshort-term stability of the so-called basic angle between the two viewing directions. Uncalibrated variations of the basic angle mayproduce systematic errors in the computed parallaxes.
Aims.
We examine the coupling between a global parallax shift and specific variations of the basic angle, namely those related to thesatellite attitude with respect to the Sun.
Methods.
The changes in observables produced by small perturbations of the basic angle, attitude, and parallaxes are calculatedanalytically. We then look for a combination of perturbations that has no net e ff ect on the observables. Results.
In the approximation of infinitely small fields of view, it is shown that certain perturbations of the basic angle are observation-ally indistinguishable from a global shift of the parallaxes. If such perturbations exist, they cannot be calibrated from the astrometricobservations but will produce a global parallax bias. Numerical simulations of the astrometric solution, using both direct and iterativemethods, confirm this theoretical result. For a given amplitude of the basic angle perturbation, the parallax bias is smaller for a largerbasic angle and a larger solar aspect angle. In both these respects
Gaia has a more favourable geometry than H ipparcos . In the caseof
Gaia , internal metrology is used to monitor basic angle variations. Additionally, Gaia has the advantage of detecting numerousquasars, which can be used to verify the parallax zero point.
Key words.
Methods: data analysis – Methods: statistical – Space vehicles: instruments – Catalogs – Astrometry – Parallaxes
1. Introduction
The European Space Agency’s space astrometry mission
Gaia ,aiming to determine astrometric parameters for at least one bil-lion stars with accuracies reaching 10 microarcseconds (de Brui-jne 2012), was launched in December 2013 (Gaia Collaborationet al. 2016b).
Gaia is based on similar observation principles asthe highly successful pioneering astrometric mission H ipparcos (ESA 1997). In particular, both satellites use an optical systemproviding two viewing directions separated by a wide angle, re-ferred to as the basic angle (Perryman et al. 2001). The goal ofthe present paper is to show how certain time-dependent vari-ations of the basic angle can bias the parallax zero point of anastrometric solution derived from observations by such a scan-ning astrometric satellite.The data processing for an astrometric satellite should, as faras possible, be based on the principle of self-calibration (Linde-gren & Bastian 2011). This means that the same observationaldata are used to determine both the scientifically interesting as-trometric parameters and the so-called “nuisance parameters”that describe the instrument calibration, satellite attitude, andother relevant parts of the observation model (Lindegren et al.2012). The self-calibration is, however, of limited applicabilitywhen variations of di ff erent parameters do not produce fully in- dependent e ff ects in the observables. Such situations can occurwhen the variation of certain parameters leads to changes in theobservables that resemble the changes produced by the varia-tion of some other parameters. The more similar the changes inthe observables are, the stronger the correlation between the pa-rameters. If the changes are identical, the problem of parameterestimation is degenerate: the same set of observables is equallywell described by di ff erent sets of parameter values.As long as the degeneracy involves only nuisance parame-ters, it has no e ff ect on the astrometric solution and only leadsto an arbitrary but unimportant shift of the respective nuisanceparameters. However, if there is a degeneracy between the as-trometric and nuisance parameters, the self-calibration processwill in general lead to biased astrometry. The celestial referenceframe is an example of a complete degeneracy between the as-trometric and attitude parameters, which can only be lifted bymeans of external data, in this case the positions and propermotions of a number of extragalactic objects (Kovalevsky et al.1997; Lindegren et al. 2012). Concerning the instrument calibra-tion parameters, it is possible to formulate the calibration modelin such a way that (near-)degeneracies are avoided among itsparameters, as well as between the calibration and attitude pa-rameters. Article number, page 1 of 8 a r X i v : . [ a s t r o - ph . I M ] A p r & A proofs: manuscript no. parallaxShiftBA
However, it is still possible that the actual physical varia-tions of the instrument contain components that are degeneratewith the astrometric parameters. By definition, such variationscannot be detected internally by the astrometric solution (e.g.,from an analysis of the residuals), but only through a comparisonwith external data, e.g. astrophysical information or independentmetrology. In Gaia the latter is chosen as will be detailed in Sec-tion 4.1.It turns out that the basic angle is an important example of aquantity that could vary in a way that cannot be fully calibratedfrom observations. Already in the early years of the H ipparcos project it was realised that certain periodic variations of the ba-sic angle, caused by a non-uniform heating of the satellite by thesolar radiation, lead to a global shift of the parallaxes (Lindegren1977). Subsequent analyses (Arenou et al. 1995; van Leeuwen2005) concluded that the possible e ff ect on the H ipparcos paral-laxes was negligible, suggesting a very good short-term stabilityof the basic angle in that satellite.For Gaia the situation is di ff erent. The much higher accuracytargeted by this mission necessitates a very careful considera-tion of possible biases introduced by uncalibrated instrumentale ff ects, including basic angle variations. This is even more ev-ident in view of the very significant ( ∼ Gaia (Gaia Collaboration et al. 2016b; Lindegren et al. 2016). Inthis context the near-degeneracy between a global parallax zeropoint error and a possible basic-angle variation induced by solarradiation is particularly relevant. The theoretical analysis of theproblem presented here expands and clarifies earlier analyticalresults by Lindegren (1977, 2004) and van Leeuwen (2005).An analytical treatment of the problem is given in Sect. 2.Section 3 presents the results of numerical experiments confirm-ing the theoretical expectations. In Sect. 4 we consider the prac-tical implications of results. Some concluding remarks are givenin Sect. 5.
2. Theory
In this section we consider how small perturbations of variousparameters change the observed quantities. We first demonstratethat, to first order in the small angles, arbitrary variations of ob-servables are equivalent to certain variations of the basic angleand attitude (Sects. 2.1–2.4). Then we find the changes of ob-servables due to a common shift of all parallaxes (Sect. 2.6).Combining these results, we derive in Sects. 2.7–2.8 the specificvariations of the basic angle and attitude that are observationallyindistinguishable from a common shift of the parallaxes.
To study the coupling between the instrument parameters andparallax, it is convenient to make use of the rotating referencesystem aligned with the fields of view. This system, known asthe Scanning Reference System (SRS) in the
Gaia nomenclature(Lindegren et al. 2012), is represented by the instrument axes x , y , z (Fig. 1), with z directed along the nominal spin axis of thesatellite, x bisecting the two viewing directions separated by thebasic angle Γ , and y = z × x . The direction towards an object isspecified by the unit vector u = x cos ϕ cos h + y sin ϕ cos h + z sin h , (1)with the instrument angles ϕ and h describing the position of theobject with respect to the SRS (Fig. 1). For a star in the preceding zx yub PFoVFFoV satellite rotation (cid:75) /2 (cid:75) /2 (cid:160)(cid:49)(cid:106) ghgh Fig. 1.
Definition of the instrument axes x , y , z of the Scanning Ref-erence System (SRS), the basic angle Γ , and the field angles g and h specifying the observed direction to a star ( u ) in either field of view. ϕ is the along-scan instrument angle of the star. In the SRS the directionto the solar system barycentre, b , is specified by the angles ξ and Ω . field of view (PFoV) ϕ (cid:39) +Γ /
2, while in the following field ofview (FFoV) ϕ (cid:39) − Γ / An observation consists of a measurement of the coordinates ofa stellar image in the focal plane at a particular time. In practicethe measurement is expressed in detector coordinates (e.g. pix-els), but we consider here an idealised system providing a directmeasurement of the two field angles g and h in the relevant fieldof view. While the across-scan field angle h coincides with thecorresponding instrument angle, the along-scan field angle g isreckoned from the centre of the corresponding field of view inthe direction of the satellite rotation (Fig. 1). Projected on thesky, the field-of-view centre defines two viewing directions sep-arated by the basic angle Γ . Thus, g p = ϕ − Γ / g f = ϕ + Γ / (cid:41) , (2)where subscripts p and f denote values for the preceding and fol-lowing field of view, respectively. We assume that the instrumentis ideal except for the basic angle Γ , which can deviate from itsnominal (conventional) value Γ c by a time-dependent variation: Γ ( t ) = Γ c + δ Γ ( t ) . (3)It is important to note that the along-scan field angle g , as definedhere, is not the same as the along-scan field angle η normallyused in the context of the Gaia data processing (Lindegren et al.2012). While η is measured from a fixed, conventional originat ϕ = ± Γ c /
2, our g is measured from the actual, variable fieldcentre at ϕ = ± Γ ( t ) /
2. This di ff erence motivates the change innotation from η to g . For consistency, a corresponding change ismade in the across-scan direction, although our h is the same asthe across-scan field angle ζ used in the Gaia data processing.
Article number, page 2 of 8.G. Butkevich et al.: Basic angle variations and parallax zero point
Any increase or decrease of the basic angle makes the fields ofview move further from each other or closer together. This, inturn, changes the observed field angle g for a given stellar image.However, since the attitude (celestial pointing of the SRS axes)is unchanged, the value of ϕ for a given star is not a ff ected by thebasic angle. For example, let us consider the preceding field ofview. An increase of the basic angle causes the observed imageto be shifted with respect to the centre of the field of view so thatthe observed along-scan field angle g p is decreased. The oppositee ff ect takes place in the following field of view. The across-scanfield angles h p and h f are obviously not a ff ected. The variationsof the field angles caused by the basic-angle variation δ Γ aretherefore δg p = − δ Γ δg f = + δ Γ δ h p = δ h f = . (4)This agrees with Eq. (2) taking into account that δϕ = A quaternion representation is used to parametrise the attitudeof
Gaia (Lindegren et al. 2012, Appendix A). Here it is moreconvenient to describe small changes in the attitude by meansof three small angles δ x , δ y , and δ z representing the rotationsaround the corresponding SRS axes. Since the direction u to thestar is regarded here as fixed, the corresponding changes in theobserved field angles are found to be δg p = − δ z δg f = − δ z δ h p = cos( Γ c / δ y − sin( Γ c / δ x δ h f = cos( Γ c / δ y + sin( Γ c / δ x . (5)In these and following equations, we neglect terms of the secondand higher orders in δ x , δ y , δ z , and δ Γ . To this approximation wecan use Γ c instead of Γ in the trigonometric factors. Combining Eqs. (4) and (5) we see that a simultaneous changeof the basic angle by δ Γ and of the attitude by δ x , δ y , δ z gives thefollowing total change of the field angles: δg p = − δ Γ − δ z δg f = + δ Γ − δ z δ h p = cos( Γ c / δ y − sin( Γ c / δ x δ h f = cos( Γ c / δ y + sin( Γ c / δ x . (6)An exact inversion of this system of equations gives δ x =
12 sin ( Γ c / (cid:16) δ h f − δ h p (cid:17) δ y =
12 cos( Γ c / (cid:16) δ h p + δ h f (cid:17) δ z = − (cid:16) δg p + δg f (cid:17) δ Γ = δg f − δg p . (7) The first two equations in (7) show that arbitrary small changesin the across-scan field angles δ h p and δ h f to first order can berepresented as changes in the attitude by δ x and δ y . Similarly, thelast two equations show that arbitrary changes in the along-scanfield angles δg p and δg f can be represented as a combination ofa change of the basic angle δ Γ and a change in the attitude by δ z .In general, an arbitrary perturbation of the observed stellarpositions, being a smooth function of time and stellar position,clearly result in a smooth, time-dependent variation of δg p , δg f , δ h p , and δg f . From Eq. (7) it follows that such a perturbation isobservationally indistinguishable from a certain time-dependentvariation of δ x , δ y , δ z , and δ Γ . The position of the barycentre of the solar system with respect tothe instrument can be specified by a distance R (in au) and twoangular coordinates. We take the angular coordinates to be ξ and Ω defined as in Fig. 1. According to the scanning law, ξ is nearlyconstant while Ω is increasing with time as the satellite spins.The barycentric position of the satellite is R = R ( − x cos Ω sin ξ + y sin Ω sin ξ − z cos ξ ) . (8)The observed direction u to a star is given by Eq. (4) of Linde-gren et al. (2012) as a function of the astrometric parameters ofthe star. Linearisation yields the change in the direction causedby a small change of the parallax δ(cid:36) : δ u = u × ( u × R δ(cid:36) ) . (9)We now assume that the direction to a star is changed onlyfrom a change of its parallax, while the basic angle and attitudeare kept constant. The fixed basic angle implies δϕ = δg . Thefixed attitude means that x , y , and z are constant, so that Eq. (1)gives the change in direction δ u = u cos h δg + w δ h , (10)where u = − x sin ϕ + y cos ϕ w = − x cos ϕ sin h − y sin ϕ sin h + z cos h (cid:41) (11)are unit vectors in the directions of increasing ϕ and h , respec-tively. They are evidently orthogonal to each other and to u .Equating δ u from (9) and (10) and taking the dot product with u and w givescos h δg = − u (cid:48) R δ(cid:36)δ h = − w (cid:48) R δ(cid:36) (cid:41) . (12)Substituting Eqs. (8) and (11) we findcos h δg = − sin ( Ω + ϕ ) sin ξ R δ(cid:36)δ h = (cid:2) cos h cos ξ − cos( Ω + ϕ ) sin h sin ξ (cid:3) R δ(cid:36) (cid:41) . (13)Up to this point the derived formulae are valid throughout thefield of view to first order in the (very small) variations denotedwith a δ . To proceed, we now consider a star observed at thecenter of either field of view, so that g = h = ϕ = ± Γ c /
2. Inthis case the variations of the field angles caused by δ(cid:36) become δg p = − sin ( Ω + Γ c /
2) sin ξ R δ(cid:36)δg f = − sin ( Ω − Γ c /
2) sin ξ R δ(cid:36)δ h p = cos ξ R δ(cid:36)δ h f = cos ξ R δ(cid:36) . . (14) Article number, page 3 of 8 & A proofs: manuscript no. parallaxShiftBA
Considering only stars at the centre of either field of view e ff ec-tively means that we neglect the finite size of the field of view.In both H ipparcos and Gaia the half-size of the field of view is Φ < − rad. Since | g | , | h | < Φ , neglected terms in Eq. (14) areof the order of Φ × δ , where δ represents any of the quantities δ Γ , δ x , etc. Equation (14) is therefore expected to be accurate to <
1% at any point in the field of view. The implications of thisapproximation are further discussed in Sect. 4.4.
Substituting Eq. (14) into Eq. (7) we readily obtain a relationbetween the change in parallax and the corresponding changesin basic angle and attitude: δ x = δ y = cos ξ sec( Γ c / R δ(cid:36)δ z = sin Ω sin ξ cos( Γ c / R δ(cid:36)δ Γ = Ω sin ξ sin( Γ c / R δ(cid:36) . (15)These equations should be interpreted as follows: a change inparallax by δ(cid:36) is observationally indistinguishable (to order Φ × δ ) from a simultaneous change of the attitude by δ x , δ y , δ z andof the basic angle by δ Γ . The formulae were derived for one star,but if δ(cid:36) is the same for all stars, they hold for all observations ofall stars. Equation (15) therefore defines the specific variations ofthe attitude and basic angle that mimic a global shift in parallax.Remarkably, the rotation around the x axis is not a ff ected by theglobal parallax change, while the rotation around the y axis isindependent of Ω and therefore, in practice, almost constant.In a global astrometric solution all the attitude and stellarparameters (including (cid:36) ) are simultaneously fitted to the ob-servations of g and h . A specific variation in the basic angle ofthe form δ Γ ( t ) ∝ cos Ω sin ξ R will then lead to a global shift ofthe fitted parallaxes (together with some time-dependent attitudeerrors δ y , δ z ). Since the e ff ects of such a basic-angle variationare fully absorbed by the attitude parameters and parallaxes, thevariation is completely degenerate with the stellar and attitudemodel and cannot be detected from the residuals of the fit.For a satellite in orbit around the Earth (as H ipparcos ) or nearL (as Gaia ), R is approximately constant. In order to achieve astable thermal regime of the instrument for a scanning astromet-ric satellite one typically chooses a scanning law with a constantangle between the direction to the Sun and the spin axis – theso-called solar aspect angle. This means that angle ξ is nearlyconstant as well (see Sect. 4.5). In the next section we considerthe case when R and ξ are exactly constant. Nevertheless, in re-ality both R and ξ are somewhat time-dependent, and this case isdiscussed in Sect. 4.3.Since ξ and R are nearly constant, the degenerate compo-nent of the basic-angle variation is essentially of the form cos Ω ,which is periodic with the satellite spin period relative to thesolar-system barycentre. This leads to a fundamental design re-quirement for a scanning astrometry satellite, namely that thebasic angle should not have significant periodic variations witha period close to the period of rotation of the satellite, and espe-cially not of the form cos Ω . From Eq. (15) it is seen that a global parallax shift correspondsto variations of δ Γ and δ z proportional to cos Ω and sin Ω , respec-tively, while δ y and δ x are constant. These quantities correspond to terms of order k = δ x = (cid:88) k ≥ a ( x ) k cos k Ω + b ( x ) k sin k Ω δ y = (cid:88) k ≥ a ( y ) k cos k Ω + b ( y ) k sin k Ω δ z = (cid:88) k ≥ a ( z ) k cos k Ω + b ( z ) k sin k Ω δ Γ = (cid:88) k ≥ a ( Γ ) k cos k Ω + b ( Γ ) k sin k Ω . (16)Specifically, if a ( Γ ) k = b ( Γ ) k = a ( Γ )1 (cid:44)
0, we findthe following relations between the amplitude of the basic an-gle variation, the global shift of the parallaxes, and the non-zeroharmonics of the attitude errors: δ(cid:36) = R sin ξ sin( Γ c / a ( Γ )1 = . a ( Γ )1 a ( y )0 = ξ sin Γ c a ( Γ )1 = . a ( Γ )1 b ( z )1 =
12 tan( Γ c / a ( Γ )1 = . a ( Γ )1 . (17)The numerical values correspond to the mean parameters rele-vant for Gaia , that is Γ c = ◦ . ξ = ◦ , and R = .
01 au.It is not the purpose of this paper to investigate the possiblee ff ects of other harmonics of the basic-angle variation. However,it can be mentioned that a ( Γ )1 is the only harmonic parameter thatis degenerate with the attitude and stellar parameters. This meansthat a ( Γ )0 , b ( Γ )1 , and a ( Γ ) k , b ( Γ ) k for k >
3. Results of numerical simulations
In this section we present the results of numerical tests per-formed to check the above conclusions. To study di ff erent as-pects of the problem, we make use of two di ff erent, though com-plimentary, solutions: a direct solution, where inversion of thenormal matrix provides full covariance information, and an it-erative solution, with separate updates for di ff erent groups ofunknowns, similar to the method used in the actual processingof Gaia data (Lindegren et al. 2012). While the direct solutioncan only handle a relatively small number of stars, and thereforeis of limited practical use, it enables us to investigate importantmathematical properties of the problem and to examine the cor-relations between all the parameters. By contrast, the iterativesolution cannot provide this kind of information, but is more re-alistic in terms of the number of stars and has been successfullyemployed in the processing of real
Gaia data.
For the direct solutions a special simulation software was devel-oped. It simulates the observations of a small number of starsand the reconstruction of their astrometric parameters based onconventional least-squares fitting. The normal equations for theunknown parameters are accumulated and the normal matrixis inverted using singular value decomposition (Golub & vanLoan 1996). This decomposition allows us to study mathemat-ical properties of the problem, especially details of its degen-eracy. The simulations included 10 stars uniformly distributed Article number, page 4 of 8.G. Butkevich et al.: Basic angle variations and parallax zero point
Table 1.
The parallax shift and the attitude harmonics obtained in thesmall-scale direct solution (of type SA) with a ( Γ )1 = Quantity Predicted Computed[mas] [mas] δ(cid:36) a ( x )0 × − a ( x )1 × − b ( x )1 × − a ( y )0 a ( y )1 × − b ( y )1 − × − a ( z )0 − × − a ( z )1 − × − b ( z )1 a ( Γ )1 = ff ects of the basic angle variations in pure form. Thesolutions always include five astrometric parameters per star:two components of the position, the parallax, and two compo-nents of the proper motion. Additional parameters representingthe variations in attitude and basic angle were introduced as re-quired by various types of solutions.The first test is to check the theoretical predictions inEq. (17). To this end, a solution was made including only thestar (S) and attitude (A) parameters, where the latter were takento be the harmonic amplitudes of δ x , δ y , and δ z as given by thefirst three equations of (16) for k ≤
1, i.e. with a total of nineattitude parameters. The results of this solution, summarised inTable 1, are in very good agreement with the theoretical expecta-tions. Small deviations from the predicted values may be causede.g. by the limited number of stars, the finite size of the field ofview, and numerical rounding errors.Another test is the singular value analysis of the normal ma-trix. In addition to the solution described above (of type SA), wecomputed three other solutions with di ff erent sets of unknownsbut always using the same observations. Solution S includedonly the star parameters as unknowns, solution SB included alsothe harmonic coe ffi cients of δ Γ in the last equation of (16) for k ≤
1, and solution SBA included all three sets of unknowns.The results, summarised in Table 2, again confirm the the-oretical predictions. The problem is close to being degenerateonly in case SBA where all three kinds of parameters (star, basicangle, attitude) are fitted simultaneously. In this case there is onesingular value ( ∼ − ) much smaller than the second smallestvalue ( ∼ − ). This indicates that the problem has a rank defi-ciency of one, which is clearly caused by the (near-)degeneracybetween the global parallax shift and the specific basic-angle andattitude variations described by Eq. (17). In all other cases (SB,SA, S) no isolated small singular values appear: the problem isthen formally well-conditioned, although (as we have seen incase SA) the solutions may be biased by the basic-angle varia-tions.The circumstance that the smallest singular value in caseSBA is not zero is partly attributable to rounding errors in a so-lution involving ≥
50 000 unknowns. However, even in exact
Table 2.
The singular values σ i of the normal matrix for the di ff erenttypes of direct solutions. SBA SB SA S σ . × − σ ... ... ... ... ...σ max Notes.
The singular values are sorted lowest to highest. The di ff erentsolutions are denoted by the parameters included in the fit: S, B and Astand respectively for the astrometric (star) parameters, basic angle, andattitude angles. arithmetics the small-scale direct solution does not represent afully degenerate problem because (i) the strict degeneracy onlyoccurs if one neglects the finite size of the fields of view, and(ii) some parameters of the mission are slightly time-dependent,but were assumed to be constant by the harmonic representa-tions in Eq. (16). The question of the time dependence is furtheraddressed in Sect. 4.3.The attitude parameters used in the small-scale solutions arenot representative of any practically useful attitude model, butwere chosen solely to verify the expected degeneracy with thebasic angle variation and parallax zero point. In particular, theharmonic model of δ x , δ y , δ z in Eq. (16) cannot describe a solidrotation of the reference frame, which explains why Table 2 doesnot show the six-fold degeneracy between the attitude and stellarparameters normally expected from the unconstrained referenceframe. This simplification is removed in the large-scale simula-tions described below, which use a fully realistic attitude model. To test the e ff ect of the basic angle variation in an iterative so-lution, we make use of the Gaia
AGISLab simulation software(Holl et al. 2012, Appendix B). This tool allows us to simulateindependent astrometric solutions in a reasonable time, based onthe same principles as the astrometric global iterative solution(AGIS; Lindegren et al. 2012) used for
Gaia but employing asmaller number of primary stars and several other time-savingsimplifications.To investigate the parallax zero point we have done a set oftests for di ff erent values of the basic angle in the range from30 ◦ to 150 ◦ , including the H ipparcos and Gaia values of 58 ◦ and106 ◦ .
5, respectively. The nominal
Gaia scanning law and real-istic geometry of the
Gaia fields of view are assumed in thesesimulations which include one million bright ( G =
13) starsuniformly distributed on the sky and observed during 5 yearswith no dead time. The nominal along-scan observation noise of95 µ as per CCD is assumed, based on the estimated centroid-ing performance of Gaia for stars of G =
13 mag. According tode Bruijne (2012) this corresponds to an expected end of mis-sion precision of around 10 µ as for the parallaxes. Additionally,basic-angle variations with an amplitude a ( Γ )1 = Article number, page 5 of 8 & A proofs: manuscript no. parallaxShiftBA P a r a ll a x s h i ft [ m a s ] Basic angle [degrees]
Fig. 2.
Parallax zero point shift for a basic angle variation of the formcos Ω with amplitude 1 mas. The dots show results of the large-scaleiterative solution for several di ff erent basic angles, including the H ip - parcos and Gaia values; the solid curve shows the theoretical relationfrom Eq. (17). including a number of higher-order e ff ects neglected in our ana-lytical approach.The parallaxes obtained in the iterative solutions are o ff setfrom their "true" values (assumed in the simulation) by randomand systematic errors. Results for the mean o ff set δ(cid:36) are shownby the points in Fig. 2. The curve is the theoretical relation fromthe first equation in (17). The points deviate from the curve by atmost 1 µ as, or (cid:46) . ff sets of the individual parallaxes,were 7.3 µ as in all the experiments, practically independent ofthe basic angle and in rough agreement with the expected end ofmission precision.
4. Discussion
In the preceding sections it was shown that a global shift ofthe parallaxes is observationally indistinguishable from a certaintime-variation of the basic angle. The relevant relation, strictlyvalid at the centre of the field of view, is given by the last identityin Eq. (15). Here we proceed to discuss some practical implica-tions of this result. Ω dependence Elementary design principles have led to the choice of a nearlyconstant solar aspect angle ξ (Fig. 1) for both H ipparcos (43 ◦ )and Gaia (45 ◦ ). Moreover, for a satellite in orbit around the Earthor close to the second Lagrange (L ) point of the Sun-Earth-Moon system, the barycentric distance R is always close to 1 au.The form of basic-angle variation that is degenerate with paral-lax is then essentially proportional to cos Ω . This result is highlysignificant in relation to expected thermal variations of the in-strument. The oblique solar illumination of the rotating satellitemay produce basic-angle variations that are periodic with thespin period relative to the Sun, i.e. of the general harmonic formof the last line in Eq. (16). A nearly constant, non-zero coe ffi cient a ( Γ )1 could therefore be a very realistic physical consequence ofthe way the satellite is operated.Knowledge of the close coupling between a possible ther-mal impact on the instrument and the parallax zero point has resulted in very strict engineering specifications for the accept-able amplitude of short-term variations of the basic angle in bothH ipparcos and Gaia . In the case of
Gaia it was known alreadyat an early design phase that the basic angle variations cannot befully avoided passively and need to be measured. Therefore Gaiaincludes a dedicated laser-interferometric metrology system, thebasic angle monitor (BAM; Mora et al. 2014), to measure theshort-term variations. According to BAM measurements duringthe first year of the nominal operations of
Gaia , the amplitudeof the cos Ω term, referred to 1.01 au and epoch 2015.0, wasabout 0.848 mas (Lindegren et al. 2016). Uncorrected, such alarge variation would lead to a parallax bias of 0.741 mas ac-cording to Eq. (17). For Gaia
Data Release 1 (Gaia Collabora-tion et al. 2016a) the observations were corrected for the basicangle variations based on a harmonic model fitted to the BAMmeasurements. Γ c From Eq. (17) it is seen that the parallax shift is inversely propor-tional to sin( Γ c / Ω term inthe basic angle variation, the parallax shift therefore decreaseswith increasing basic angle (cf. Fig. 2). From this point of viewthe optimum basic angle is therefore 180 ◦ . This value wouldhowever be very bad for the overall conditioning and precisionof the astrometric solution (Makarov 1997, 1998; Lindegren &Bastian 2011), which instead favours a value around 90 ◦ . More-over, the sensitivity to the cos Ω variation is only a factor 1.4smaller at 180 ◦ than at 90 ◦ . The Gaia value Γ c = ◦ . Γ c can be understood from simple arguments. Con-sider the along-scan e ff ects of the parallax shift δ(cid:36) . As long asthe nominal basic angle Γ c is large, the e ff ects in the two fieldsof view are significantly di ff erent. However, the smaller the ba-sic angle is, the more similar are the e ff ects in the two fields ofview. This can be seen from Eq. (14) but is also obvious withoutany formula. We emphasise that it is the basic angle variationsthat induce field angle perturbations and that the solution tries tofind parallaxes and attitude parameters that fit the perturbed fieldangles. It is then obvious that a smaller basic angle will requirea larger parallax shift to absorb a basic-angle variation of givenamplitude. R and ξ As noted above, the barycentric distance R and the angle ξ arenot strictly constant but functions of time. In this case Eq. (15)gives the particular time dependences of δ Γ , δ x , δ y , and δ z thatare degenerate with a global parallax shift. In particular, δ Γ ( t ) = C R ( t ) sin ξ ( t ) cos Ω ( t ) , (18)where C is constant, is indistinguishable from a parallax shift of δ(cid:36) = C / sin( Γ c / δ Γ ( t ) is not completely de-generate with δ(cid:36) and may therefore contain components thatcan be detected by analysis of the residuals and subsequentlyeliminated by means of additional calibration terms. However,an arbitrary variation δ Γ ( t ) in general also contains a componentof the form (18), which will result in some parallax shift. Thisshift can be estimated by projecting the variation onto the func-tion on the right-hand side of Eq. (18) in the least-squares sense: Article number, page 6 of 8.G. Butkevich et al.: Basic angle variations and parallax zero point δ(cid:36) =
12 sin( Γ c / (cid:10) δ Γ ( t ) R ( t ) sin ξ ( t ) cos Ω ( t ) (cid:11)(cid:10) R ( t ) sin ξ ( t ) cos Ω ( t ) (cid:11) , (19)where the angular brackets denote averaging over time. If R and ξ are constant, the factor ( R sin ξ ) − can be taken out of the av-erages. If, in addition, δ Γ ( t ) is strictly periodic in Ω , we recoverthe first equality in Eq. (17).For Gaia, which operates in the vicinity of L , its barycen-tric distance R varies between approximately 0.99 and 1.03 auas a combination of the eccentric heliocentric orbit of the Earth,the Lissajour orbit around L and the time-dependent o ff set be-tween the Sun and the Solar system’s barycentre. As discussedin Sect. 4.5 below ξ varies by about 1% from its nominal value45 ◦ . In order to derive Eqs. (14) and (15) we neglected the finite sizeof the field of view by considering only observations at the fieldcentre ( g = h = Φ (cid:39) − times smaller than thebasic-angle variations, where Φ is half the size of the field ofview. As a consequence, a basic-angle variation of the form (18)is not strictly degenerate with δ(cid:36) and the attitude angles when afinite field of view is considered.However, if the instrument has also periodic optical distor-tions separately in each field of view that need to be calibrated,the corresponding, more complex calibration model may con-tribute to the degeneracy and, in worst case, restore completedegeneracy. The spherical coordinates R , ξ , Ω introduced in Sect. 2.6 definethe position of the solar system barycentre in the scanning refer-ence system (SRS). This is what matters for calculating the par-allax e ff ect, which depends on the observer’s displacement fromthe barycentre. On the other hand, the physical relevance of thecos Ω modulation is connected with the changing illuminationof the satellite by the Sun, which depends on the heliocentricdistance R h and the (heliotropic) angles ξ h , Ω h representing theproper direction towards the centre of the Sun at the time of ob-servation. The di ff erence between the heliotropic and barytropiccoordinates is at most about 0.01 au and 0.01 rad, respectively.This is small but should be taken into account for an accuratemodelling of the basic-angle variations. In this context it can benoted that the expected thermal impact on the satellite scales as R − , while the parallax factor scales as R . On the other hand,the scanning law is chosen to keep ξ h as constant as possible,while ξ can vary at the level of 1%. If the basic angle varies as R − sin ξ h cos Ω h it is no longer strictly of the form (18). The re-sulting parallax bias can be estimated by means of Eq. (19). If the basic angle varies as a consequence of the changing solarillumination of the rotating satellite we expect to see a parallax bias according to Eq. (19). However, as discussed above, the de-generacy with the parallax zero point is not perfect, and in princi-ple this opens a possibility to calibrate the basic angle variationsfrom the observations. At least three di ff erent e ff ects that con-tribute to breaking the degeneracy could be used: the finite sizeof the field of view (Sect. 4.4), the time dependence of R due tothe eccentricity of the Earth’s orbit (Sect. 4.3), and the di ff erencebetween the barytropic and heliotropic angles (Sect. 4.5). Unfor-tunately all three e ff ects only come in at a level of a few per centof the variation, or less, which makes the result very sensitive tosmall errors in the calibration model. Moreover, the finite fieldof view is of little use if we have to calibrate complex periodicvariations of optical distortions independently in each field ofview. The best chance may be o ff ered by the time variation of R ,where the ratio of the parallax e ff ect to the illumination strengthgoes as R and consequently varies by ±
5% over the year. Thusthe hope to break the degeneracy purely from the observationsthemselves, i.e. based on the self-calibration principle, is ratherlimited.
5. Conclusions
We have presented an analysis of the e ff ect of basic angle varia-tions on the global shift of parallaxes derived from observationsby a scanning astrometric satellite with two fields of view.The method of small perturbations was used to derive thechanges in the four observables (the across- and along-field an-gles in both fields of view) resulting from perturbations of fourinstrument parameters (the basic angle and three components ofthe attitude). Conversely, any given perturbation of the four ob-servables can equally be represented by a specific combinationof the instrument parameters. Applying this technique to the per-turbations induced by a change in the parallax, we derived thetime-dependent variations of the instrument parameters that ex-actly mimic a global shift of the parallaxes.These relations confirm previous findings that an uncorrectedvariation of the basic angle of the form a ( Γ )1 cos Ω , with Ω beingthe barycentric spin phase, leads to a global shift of the parallaxzero point of (cid:39) . a ( Γ )1 for the parameters of the Gaia design.Results of numerical simulations are in complete agreement withthe analytical formulae.In general, periodic variations of the basic angle can be ex-pected from the thermal impact of solar radiation on the spin-ning satellite (Lindegren 1977, 2004). Those periodic variationsare typically related to the heliotropic spin phase Ω h , which isclose to the barycentric spin phase Ω . If the thermally-inducedvariations contain a significant component that is proportionalto cos Ω h , their e ff ect on the observations is practically indis-tinguishable from a global shift of the parallaxes. Although thedegeneracy is not perfect, it is di ffi cult to break except by us-ing of other kinds of data or external information. In the case of Gaia this includes, in particular, direct measurement of the basicangle variations by means of laser metrology (BAM). The use ofastrophysical information such as parallaxes of pulsating stars(Windmark et al. 2011; Gould & Kollmeier 2016) and quasarsis vital for verifying the successful determination of the parallaxzero point.
Acknowledgements.
The authors acknowledge useful discussions with manycolleagues within the
Gaia community, of which we especially wish to men-tion Ulrich Bastian, Anthony Brown, Jos de Bruijne, Uwe Lammers, FrançoisMignard, and Timo Prusti. The authors warmly thank the anonymous referee forvaluable comments and suggestions. The work at Technische Universität Dres-den was partially supported by the BMWi grants 50 QG 0601, 50 QG 0901 and
Article number, page 7 of 8 & A proofs: manuscript no. parallaxShiftBA
50 QG 1402 awarded by the Deutsche Zentrum für Luft- und Raumfahrt e.V.(DLR).
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