this short paper is to outline the basic principles of scanning space astrometry. .... is because the parallax factor (â sin θ, where θ is the angle of the star from the.
GAIA: At the Frontiers of Astrometry C. Turon, F. Meynadier and F. Arenou (eds) EAS Publications Series, 45 (2010) 109-114 www.eas.org
BASIC PRINCIPLES OF SCANNING SPACE ASTROMETRY L. Lindegren 1 and U. Bastian 2 Abstract. We outline the basic principles of scanning space astrometry, such as represented by Hipparcos, Gaia, and some other astrometric satellites planned or proposed. We explain the need for large-angle measurements, why these are essentially one-dimensional, how it is possible to determine absolute parallaxes, and why a Hipparcos-type scanning law is favourable. We discuss the choice of the basic angle between the two viewing directions, the principle of self-calibration, and why the resulting numerical problem must be difficult to solve.
1
Introduction
ESA’s Hipparcos mission was the first space experiment dedicated to astrometry. Its tremendous success in terms of the delivered accuracy and scientific results (Perryman 2009) convincingly demonstrated an entirely new way of doing astrometry, here called scanning space astrometry. It is radically different from all kinds of astrometric observations from the ground, both optical and radio, and also quite different from space astrometry with the HST, JMAPS or SIM. The next missions to implement this mode of observation are Nano-JASMINE and Gaia. The aim of this short paper is to outline the basic principles of scanning space astrometry.
2
How Gaia works
A continuously scanning telescope such as Hipparcos or Gaia transforms positional information into timing data. That is, it determines the precise time when the centre of a star image has some well-defined position in the field of view (FoV). This position could be defined by a group of slits, as for Hipparcos, or by the pixel layout on the CCDs in Gaia. The resulting “observation time” is a onedimensional (along-scan, AL) measurement of the stellar position relative to the 1
Lund Observatory, Lund University, Box 43, 22100 Lund, Sweden Astronomisches Rechen-Institut, Zentrum f¨ ur Astronomie der Universit¨ at Heidelberg, M¨ onchhofstr. 12–14, 69120 Heidelberg, Germany 2
c EAS, EDP Sciences 2011 ⃝ DOI: 10.1051/eas/1045018 Article published by EDP Sciences
110
GAIA: At the Frontiers of Astrometry
Fig. 1. Systematic errors in an astrometric catalogue: the maps show positional differences between the ground-based PPM Star Catalogue (R¨ oser & Bastian 1991; Bastian & R¨ oser 1993) and the Hipparcos Catalogue (ESA 1997). Each arrow is the mean difference in a cell of about 2 deg2 . To the left the whole sphere; to the right a region of 40◦ × 40◦ .
instrument axes (although it obviously depends as well on many other factors). One gets at the same time an approximate position of the star in the across-scan (AC) direction, but for Gaia this is much less precise due to the elongated shape of the pixels, the lower optical resolution across-scan, and the way the pixels are read out. But this is fine because it is almost only the AL measurements that matter. The astrometric catalogue is built up from a very large number of such observation times, by a process that involves also a precise reconstruction of the instrument pointing (attitude) as a function of time and of the optical mapping of the CCDs through the telescope onto the celestial sphere. In the following we explain why the measurements should be one-dimensional as well as some other important facts about scanning space astrometry.
2.1
Why are the measurements one-dimensional?
One of the aims of scanning space astrometry is to build a globally consistent reference system. By this we mean the following: if the catalogue is used to compute the angle θ between any two objects, at any time, then the uncertainty of θ is roughly independent of the size of θ. This condition is violated in the presence of significant “zonal” (systematic) errors such as those shown in Figure 1. The size of each astrometric field (about 0.7◦ ) is too small to give a handle on the zonal errors, but the differential effect between the fields is telling. As shown in Figure 2 (left), the differential error can only be measured along-scan, since across-can it cannot be distinguished from a tilt of the instrument around the x and y axes. The AC coordinates consequently need not be measured very accurately.
L. Lindegren and U. Bastian: Scanning Space Astrometry
111
Fig. 2. Left: positional errors in the preceding (P ) and following (F ) fields are decomposed into their components along (AL) and across (AC) scan. The differential AL error PAL − FAL is usefully measured while the differential AC error PAC − FAC is inextricably mixed up with attitude errors around the x and y axes. Right: why a basic angle of 90◦ (a simple fraction of 360◦ ) is bad for getting accurate star positions along a great circle.
2.2
Why is the basic angle 106.5 ◦?
We have seen that (at least) two FoVs are needed, with a large “basic angle” Γ between them. Hipparcos used Γ = 58◦ ; Gaia will have Γ = 106.5◦. How were these numbers chosen? When scanning more or less along a great circle (as during a day or so), the accuracy by which we can determine the one-dimensional positions of the stars along the great circle depends strongly on the choice of Γ (Fig. 3, left). It turns out that Γ = 360◦ × (m/n) is a bad choice for small integers m, n. This can be understood in terms of the connectivity of stars along the circle (Fig. 2, right). Both Hipparcos and Gaia avoid the really bad angles, although Hipparcos was uncomfortably close to 60◦ . Interestingly, these “bad choices” for Γ disappear when a global solution on the whole celestial sphere is considered (Fig. 3, right). This has been known for some time (Makarov 1998), but it was nevertheless thought prudent to avoid these values for Gaia. This allows to make great-circle solutions from just one day of data as a powerful first-look test of data quality. The choice of 106.5◦ rather than (say) 99.4◦ was dictated by the physical accommodation of the optical elements.
2.3
How is it possible to measure absolute parallaxes?
Traditional small-field astrometry, whether from the ground or using the HST, measures only relative parallaxes of stars within a field of less than a degree. This is because the parallax factor (≃ sin θ, where θ is the angle of the star from the Sun) is virtually the same for all the stars in the field. The relative parallaxes require an additive correction that must be determined by other methods, for example by estimating the parallaxes of the background stars using photometric or spectroscopic methods (e.g., Benedict et al. 2007). Although scanning space astrometry also makes purely differential measurements, absolute parallaxes are obtained since the relative shifts can now be measured between stars that are widely separated on the celestial sphere. For an
112
GAIA: At the Frontiers of Astrometry
Fig. 3. The relative variance of positions along a great circle (left) and on a sphere (right), as obtained from differential measurements between two randomly positioned fields separated by a given basic angle Γ. In the 1D case (left) certain values of Γ should be avoided. In the 2D case (right) there are no particularly bad angles although Γ ∼ 90◦ is preferable. (Calculations and diagrams by courtesy of S. Nzoke, Lund Observatory.)
observer at 1 AU from the Sun, the apparent shift of a star due to its parallax ϖ equals ϖ sin θ and is directed on a great circle from the star towards the Sun. As shown in Figure 4 (left), the AL parallax shift of the star at F is ϖF sin θ sin ψ = ϖF sin ξ sin Γ, where ξ = 45◦ is the constant solar aspect angle (between the Sun and the spin axis). At the same time the AL parallax shift of the star at P is zero. The AL measurement of F relative to P therefore depends on ϖF but not on ϖP , while the reverse is true at a different time, as shown in the right diagram. The sensitivity to parallax is proportional to sin ξ sin Γ, which should therefore be maximized. The choice of ξ is discussed below. While Γ = 90◦ is optimal for the basic angle according to this analysis, we have seen that other considerations led to a slightly larger value being adopted for Gaia.
Fig. 4. The measured along-scan (AL) angle between the stars at P, F depends on their parallaxes ϖP , ϖF in different ways depending on the position of the Sun. This allows to determine their absolute parallaxes rather than just the relative parallax ϖP − ϖF .
L. Lindegren and U. Bastian: Scanning Space Astrometry
113
Fig. 5. The spin axis z makes loops around the Sun, which must overlap as in the left and middle diagram in order to provide good observing conditions. The star at point a may be scanned whenever z is 90◦ from a, i.e., on the great circle A at z1 , z2 , z3 , etc.
2.4
Why does the scanning law look the way it does?
The first element of the so-called scanning law is a prescription for how the direction of the spin axis, z, should evolve as a function of time. It can be expressed, for example, by the functions αz (t), δz (t) which should be continuous and smooth. This prescription is complemented by the fixed spin rate (60′′ s−1 for Gaia) and the phase of the spin at some initial epoch. For parallaxes we want to make sin ξ as large as possible, where ξ is the solar aspect angle. ξ = 90◦ is not possible because the Sun would then enter the FoVs on every spin of the satellite. Considerations of straylight and the size of the sunshield have led to the practical constraint ξ ≤ 45◦ for Gaia (though an earlier design had ξ = 55◦ ). The conclusion is that the solar aspect angle should be kept constant at its maximum feasible value, or 45◦ for Gaia. A constant angle is also good for minimizing variations of the solar thermal impact on the instrument. Given the apparent path of the Sun on the celestial sphere and the fixed ξ, the first element of the scanning law reduces to the specification of ν(t), where ν is the inclination of the Sun-z arc to the ecliptic. A continuously increasing (or decreasing) ν(t) represents a precession-like, or revolving, motion of z around the Sun, resulting in a series of loops on the sphere (Fig. 5). The areas of the sky scanned in successive spins should preferably overlap, so that no gaps occur. This ˙ ≤ Φ, if P = 6 hr is the spin period and Φ = 0.69◦ the AC size of the requires |z|P FoV. Actually, this condition is not quite satisfied for Gaia, so there will be gaps; ˙ minimizes the nonbut in any case a roughly constant inertial precession rate |z| uniformity of the sky coverage. Thus ν(t) is uniquely defined by the initial angle and adopted precession rate. The resulting “uniform revolving scanning law” is the baseline for both Hipparcos and Gaia, albeit with slightly different parameters. The choice of ξ and precession rate (or, equivalently, K = the number of loops per year) determines the overall pattern of scanning. The resulting number and geometry of scans across an arbitrary point can be visualized as in Figure 5. From this it can be seen that the loops of z should overlap slightly as in the left diagram, in which case there are at least six distinct epochs of observations per
114
GAIA: At the Frontiers of Astrometry
year for any object (although a fraction of them may be missed due to the gap between successive spins). Moreover, the scans intersect at a large angle, which allows the two-dimensional coordinates of the objects to be determined from the one-dimensional measurements. K = 5.8 was selected for Gaia. Combined with the large basic angle, and given the technical constraints, the resulting scanning law is close to optimal for the global determination of stellar positions, proper motions and parallaxes – at least, nobody has yet come up with a better idea!
3
Gaia as a self-calibrating instrument
The observation times tobs of each object’s transit over a CCD depend not only on the astrometric parameters si of the observed object (i), but also on the attitude parameters aj for the relevant time interval (j), the calibration parameters ck for the relevant combination of CCD, FoV, etc. (k), and possibly on several global parameters g, such as the Parametrized Post-Newtonian parameter γ. Although dedicated calibration procedures are required for many other characteristics of the instrument, the full accuracy of the attitude and geometric calibration parameters can only be obtained in a global data reduction process, based on the “regular” astrometric observations. The idea is to make a single least-squares solution for all the parameters – including the “nuisance parameters” (attitude and calibration) – using as much data as possible to strengthen the solution. The basic equation for one observation is ∂t ∂t ∂t ∂t ∆g ≃ tobs − tcalc (si , aj , ck , g), ∆si + ∆aj + ∆ck + ∂si ∂aj ∂ck ∂g where the last term is the observation model and ∆si etc. are corrections to the initial parameter values in the linearised problem. Every observation thus carries information on the nuisance parameters in addition to the wanted (astrometric and global) parameters, and they must therefore be determined together. This is the principle of self-calibration. When applied to scanning space astrometry it leads to an extremely large system of equations in which the unknowns are connected in a very intricate way. Although it makes the solution difficult to compute in practice, it is precisely this feature that allows to build a globally consistent reference system and determine absolute parallaxes.
References Bastian, U., & R¨ oser, S., 1993, PPM Star Catalogue (South) (Heidelberg: Spektrum) Benedict, G.F., MacArthur, B.E., Feast, M.W., et al., 2007, AJ 133, 1810 ESA, 1997, The Hipparcos and Tycho Catalogues, SP–1200 (Noordwijk: ESA) Makarov, V.V., 1998, A&A, 340, 309 Perryman, M., 2009, Astronomical Applications of Astrometry: Ten years of exploitation of the Hipparcos satellite data (Cambridge: Cambridge University Press) R¨ oser, S., & Bastian, U., 1991, PPM Star Catalogue (North) (Heidelberg: Spektrum)
