6 Combined solutions

  As was explained in Sect. 2.2, correlations exist between abscissa residuals on the same reference great circle, in particular for observations with differences between the abscissa values of less than 4 degrees. These correlations can only be fully accounted for through the use of the abscissa data in combined solutions for the astrometric parameters of groups of stars. Combined solutions are essential for solving mean parallax and proper motion values in a relatively small area of the sky (density of objects more than 0.2 per square degree). In addition, combined solutions offer the best possibilities for incorporating constraints obtained from other data.

 

Figure 4:  Definition of the angles and directions used to calculate the relative abscissae along the reference great circle

In a combined solution one starts by collecting all the abscissa data for the stars involved, and sort these on orbit number (there is one reference great circle, or RGC, per orbit). Covariance matrices are determined and applied per reference great circle. The reference great circle data file (Table 2.8.1 in Vol. 1) provides the pole positions for the NDAC and FAST RGCs, which are transformed to unit vectors in the equatorial reference system, ${\vec{p}}_N$ and ${\vec{p}}_F$. A unit length reference direction ${\vec{r}}$(Fig. 4) on each RGC can be obtained e.g. from its crossing with the Equatorial plane:  

$${\vec{p}} \equiv \left(\begin{array} {c} p_1 \\ p_2 \\ p_3\end{array}\right),$$ (15)

from which:  

$${\vec{r}} = {1\over\sqrt{p_1^2+p_2^2}}\left(\begin{array} {c} p_2 \\ -p_1 \\ 0\end{array}\right).$$ (16)

A vector ${\vec{t}}$ completes the triad $({\vec{r}},{\vec{t}},{\vec{p}})$: 

$${\vec{t}} = {1\over\sqrt{p_1^2+p_2^2}}\left(\begin{array} {c} -p_1p_3 \\ -p_2p_3 \\ p_1^2+p_2^2\end{array}\right).$$ (17)

Given the unit vector ${\vec{s}}$ directed towards the reference position of a star (the effect of aberration and other smaller effects have been removed in the great-circle reduction process and would have been almost identical for neighbouring stars), we calculate the angles $\zeta$ and $\eta$ (see Fig. 4):

$$\begin{eqnarray} \cos\zeta_j &= & {\vec{s}}^\prime {\vec{r}}_j\\ \cos\eta_j &= & {\vec{s}}^\prime {\vec{t}}_j\end{eqnarray}$$ (18) (19)

and $\psi$: 

$$\cos\psi_j = {\vec{s}}^\prime {\vec{p}}_j,$$ (20)

where j equals N or F, depending on the origin of the observation. The abscissa $\phi$ for this star is then derived from:

$$\begin{eqnarray} \cos\phi_j &= & \cos\zeta_j / \sin\psi_j \nonumber\\ \sin\phi_j &= & \cos\eta_j / \sin\psi_j .\end{eqnarray}$$ (21)

Although for large distances between stars the difference between the abscissae will be almost equal to the actual distance between those stars on the sky, this is not necessarily the case for small distances (less than a few degrees), where the correlations are the strongest.

The abscissa separation between two stars, $\Delta\phi_{jk}$, measured by the same consortium on the same RGC, is translated into a correlation coefficient, using the functions given in Table 1 (with n₅ as defined in Sect. 2.2 and $F\rm _c$ a flag indicating the relevant consortium):  

$$Q_{jk} = f(F_{\rm c},n_5,\Delta\phi_{jk}),$$ (22)

which produces a coefficient in the covariance matrix:  

$$V_{jk} = Q_{jk}\sigma_j\sigma_k.$$ (23)

Although no direct correlations exist between data for different stars on the FAST and NDAC RGCs, secondary correlations or covariances do occur as a result of the correlations between measurements of the same stars by NDAC and FAST. One of the differences between a natural correlation and a covariance due to a secondary correlation is that the first kind affects the total amount of information (``total weight'') of the observations, while the second kind does not.

The exact values of these covariances are difficult to estimate due to possible small correlations between attitude errors, but their approximate values can be derived as follows. Assume three sets of unit weight residuals, $\epsilon_{k,i}, (k=F,N)$ and $\epsilon_{k,j}, (k=F)$. Say that there are natural correlations between $\epsilon_{F,i}$ and $\epsilon_{N,i}$ (same observation, different reductions), such that $<\epsilon_{F,i}\epsilon_{N,i}\gt = Q_{FN_{ii}}$. Similarly, a natural correlation exist between $\epsilon_{F,i}$ and $\epsilon_{F,j}$ (same reference great circle, different stars), given by Q_FF<552>ij. No natural correlation exist between $\epsilon_{N,i}$ and $\epsilon_{F,j}$, but due to the other correlations, a covariance will occur, which can be approximated by substituting $\epsilon_{F,j} = Q_{FF_{ij}} \epsilon_{F,i} + \nu_{F,j}$, which results from the first correlation. Thus, as $\epsilon_{N,i}$ and $\nu_{F,j}$ are uncorrelated, we find for the covariance $<\epsilon_{N,i}\epsilon_{F,j}\gt\approx Q_{FF_{ij}}Q_{FN_{ii}}$,giving the following element in the covariance matrix:  

$$V_{FN_{ij}} = Q_{FF_{ij}}Q_{FN_{ii}}\sigma_{F_i}\sigma_{N_j}.$$ (24)

This covariance is only an approximation; in fact a slightly different value can be obtained by using a different link between the variables:  

$$V_{FN_{ij}} = Q_{NN_{ii}}q_{FN_{ij}}\sigma_{F_j}\sigma_{N_i}.$$ (25)

The effect of introducing these covariances in the covariance matrix is to preserve the proper weight reduction caused by the natural correlations: they cancel out when inverting the covariance matrix, leaving at the diagonal elements only the effects of the natural correlations. In the example given above the covariance matrix looks like:

$$\left(\begin{array} {ccc} 1 & q_{12} & q_{13} \\ q_{12} & 1 & q_{12}q_{13} \\ q_{13} & q_{12}q_{13} & 1 \end{array}\right),$$ (26)

where we substituted Q_FF<594>ij=q₁₂, Q_FN<596>ii=q₁₃, and assumed all variances to be equal to 1. A Gauss elimination produces:

$$\left(\begin{array} {ccc} 1 & q_{12} & q_{13} \\ 0 & 1-q_{12}^2 & 0 \\ 0 & 0 & 1-q_{13}^2\end{array}\right).$$ (27)

The inverted, variance matrix then reads as follows:

$$\left(\begin{array} {ccc} 1+{q_{12}^2\over 1-q_{12}^2} + {q_... ...}\over 1-q_{13}^2} & 0 & {1\over 1-q_{13}^2}\end{array}\right),$$ (28)

from which is obtained the Cholesky square root, as defined by Eq. (5), and by means of which the observations are weighted:

$$\left(\begin{array} {ccc} 1 & 0 & 0 \\ & & \\ {-q_{12}\ove... ...-q_{13}^2}} & 0 & {1\over \sqrt{1-q_{13}^2}}\end{array}\right).$$ (29)

Thus, in the final weighting of the observations, the covariances as produced by the secondary correlations (products of q₁₂ and q₁₃) have disappeared, and only the influences of the natural correlations remain.

After inverting ${\bf V}$ and taking its Cholesky square root, it can be applied to the original observations to obtain a set of uncorrelated and properly weighted observation equations. This is done for the data in each RGC, covering the observations of both consortia (when available). These de-correlated observations can then be incorporated in a classical least-squares solution.

The observational equations used for each star can either be the original equations, in which case corrections to the individual parameters are found, and a set of uncorrelated parameters is determined, or they can contain common parameters such as a common proper motion and/or a common parallax. When solving for a common parameter, it is essential that all abscissa residuals are corrected to represent a reference solution referring the abscissa residuals for all stars involved in the combined solution to the same parallax and/or proper motion values. These corrections are obtained using Eq. (10). Thus, for the LMC-stars example, the residuals are corrected to reference values of zero proper motion and a parallax of 0.02 mas, and then solved together for a common proper motion. The combined solution provides corrections to the reference values. It is also possible to use all observations, ignoring rejections from the standard processing, and determine new rejections under the new conditions, based on the residuals relative to the combined solution. When determining the mean proper motion of a star cluster, it is possible to incorporate precise differential proper motions obtained on the ground as constraints for the solution. In all cases, the degrees of freedom available will be strongly reduced, thus improving the reliability of the solution.

In the solutions for a star cluster the space velocity rather than the proper motion should be considered constant. Without the presence of internal motions this condition can be used to determine individual distances and radial velocities of cluster members, as was shown for the Hyades cluster by Dravins et al. (1997). For a cluster like the Pleiades, however, with a higher internal velocity dispersion but a larger distance and a small radial velocity, the differential distance variations and the projection of the radial velocity can be ignored, and the shared space velocity of cluster members can be expressed as a function of the proper motion of the centre of the cluster (indicated by the subscript ``c'') and the position of each object on the sky, relative to the cluster centre (see VL98 for a derivation):

$$\begin{eqnarray} \mu_\alpha\cos\delta &\approx & (\mu_\alpha\cos\delta)_{\rm c} ... ...u_\alpha\cos\delta)_{\rm c} \sin\delta\sin(\alpha-\alpha_{\rm c}).\end{eqnarray}$$ (30)

The effects are, as one would expect, most noticeable close to the equatorial poles ($\sin\delta\approx 1$ and relatively large variations in $\alpha$for relatively small angular separations) and for clusters covering large parts of the sky. There are different ways to solve for ($(\mu_\alpha\cos\delta)_{\rm c}$,$(\mu_\delta)_{\rm c}$). One way is to first transform all abscissae residuals back to zero proper motion, and then implement Eqs. (30) in the solution for each star. This then gives, as the proper motion part of the solution, the following equation (where $a_4=\mu_\alpha\cos\delta$ and $a_5=\mu_\delta$):

$$\begin{eqnarray} {\rm d}{\vec{v}}& =& (\mu_\alpha\cos\delta)_{\rm c}\left({\part... ...\alpha-\alpha_{\rm c}){\partial{\vec{v}}\over\partial a_4}\right).\end{eqnarray}$$ (31)

Alternatively, a first-order approximation is made for the cluster proper motion, from which the local projection corrections are calculated. In a subsequent iteration, further adjustments of the cluster proper motion will have negligible influence on these projection corrections. Only when the cluster is spread out over a large part of the sky, like the Hyades, must these and other corrections be fully taken into account.

Other possibilities of combined and constrained solutions may not involve small fields, and can be simpler to apply as they would not require de-correlating observations of different stars. Accidental correlations that could occur because of two stars appearing on the same reference great circle will be small and rather rare, and therefore of very little influence. Correlations between the FAST and NDAC reductions of the same observations are removed using Eq. (9). This kind of solution can be used for luminosity calibrations.

As an example, consider determining the mean absolute magnitudes of RR Lyrae stars in a single solution, using the parallax information contained in the abscissa residuals and the reddening-corrected apparent mean magnitudes. The procedure goes as follows. First assume a reasonable value for the absolute magnitude $\hat M_V(RR)$, related in a well defined manner (passband) to the reddening corrected mean apparent magnitude <m_v> of each star. Translate the difference into a predicted parallax:  

$$\hat\pi = 100\times {\rm e}^{-0.4605(m_v - \hat M_V(RR))},$$ (32)

where $\hat\pi$ is measured in mas. All abscissae are corrected for the difference $\Delta\pi = \hat\pi-\overline\pi$ between the published ($\overline\pi$) and the calculated predicted parallax (Eq. 10). The estimated absolute magnitude $\hat M_V(RR)$ will differ by a small amount from the best estimated value indicated by the observations:  

$$\tilde M_V(RR) = \hat M_V(RR) + \Delta M_V(RR).$$ (33)

This correction to the absolute magnitude will in the first approximation result in a scaling correction of the assumed parallaxes:  

$$\tilde\pi = \hat\pi\times (1-0.4605\Delta M_V(RR))\equiv\hat\pi (1-\nu),$$ (34)

giving corrections to the assumed individual parallaxes of $-\hat\pi\nu$,where $\hat\pi$ is the estimated parallax of an individual RR Lyrae star, and $\nu$ a correction factor which is the same for all RR Lyrae, if our basic assumption that the absolute magnitudes of these stars are equal is indeed correct. Thus, we can now create one solution for all RR Lyrae, solving for each star the position and proper motion corrections, and solving for one overall parallax correction factor $\nu$.The solution provides an accuracy of the estimated value $\tilde\nu$ and thus for $\tilde M_V(RR)$, and will tell how good the model is (at the available accuracy of the observations) through its $\chi^2$ value. By using this method rather than the direct interpretation of the published parallaxes, a more reliable and better to interpret solution is obtained, involving fewer degrees of freedom and deriving the critical quantity of the distance scale correction directly from the actual observations: the abscissa data. However, as the correction factor $\nu$ is multiplied by the estimated parallax, most of the weight in the solution will still come from the nearest stars. As long as the accuracy of the corrected apparent magnitudes is high, any bias on the derived parallaxes will be very small, and no bias is expected from the solution. This was confirmed by a simple simulation. A more general description of luminosity calibrations using this method can be found in VL98. A different method for obtaining unbiased luminosity estimates has been presented by Luri et al. (1996), and is based on a maximum likelihood estimate for a set of probability distributions.

An estimate of the expected accuracy of $\tilde\nu$ can be obtained from:  

$$\sigma_\nu \approx \sqrt{ {1\over\sum_i\hat\pi_i^2 \left({\b... ...eft({\bf T}_i {\partial {\vec{v}}_i\over\partial\pi}\right)} },$$ (35)

where the index i represents the different stars, and ${\bf T}_i$ is the Cholesky square root of the variance matrix for star i. This can be approximated with the following simplification to:  

$$\sigma_\nu \approx \sqrt{{1\over\sum_i\left({\hat\pi_i\over\sigma_{\pi_i}} \right)^2}},$$ (36)

where $\sigma_{\pi_i}$ is the parallax accuracy given in the final catalogue, but $\hat\pi_i$ is the estimated parallax value as defined above, and NOT the value given in the catalogue.