Marine Ségneré-Yter

Olivier Brun

Balakrishna Prabhu

A Markovian approach for improving end-to-end data rates in the Internet

We model the tradeo between the monitoring costs and gain in throughput for overlay-based routing in the Internet. A Markovian model is shown to t the real throughput traces quite well. The tradeo problem is formulated as Markov decision process and it is observed that the myopic policy that maximizes the immediate utility is close to optimal on the real traces.

Introduction

More than two decades ago, it was observed that the performance of network ows could be improved by choosing other paths than those computed by IP routing protocols (see, e.g., [START_REF] Savage | The end-to-end eects of internet path selection[END_REF]). Routing overlay networks were then proposed as a solution for achieving spectacular performance improvements, without the need to re-engineer the Internet (see [START_REF] Brun | Scalable, self-healing, and self-optimizing routing overlays[END_REF] and references therein). An overlay network is composed of Internet end-hosts which can monitor the quality of Internet paths between themselves by sending probe packets. Since all pairs of nodes are connected, the default topology of a routing overlay is that of a complete graph. Although the monitoring cost is highly variable depending on the metric to be probed, it is usually not possible to discover an optimal path by probing all links in large overlay networks (see [START_REF] Caro | Optimal path discovery problem with homogeneous knowledge[END_REF] for a graph-theoretic analysis of this issue). An alternative approach is to devise a parsimonious monitoring approach making the trade-o between the quality of routing decisions and the monitoring cost. Given a source and a destination node in the overlay, the idea is to probe only a small number of overlay paths between the two nodes at each measurement epoch, but to choose those paths so as to make the best routing decision.

Assuming known Markovian models for path delays, this trade-o problem was formulated as a Markov Decision Process (MDP) in [START_REF] Vaton | Joint minimization of monitoring cost and delay in overlay networks: Optimal policies with a markovian approach[END_REF] . Using delay data collected over the Internet, it was shown that the optimal monitoring policy enables to obtain a routing performance almost as good as the one obtained when all paths are always monitored, but with a modest monitoring eort.

In this paper, we adopt the theoretical framework introduced in [START_REF] Vaton | Joint minimization of monitoring cost and delay in overlay networks: Optimal policies with a markovian approach[END_REF], but focus on data throughput rather than RTT. We note that ecient parsimonious monitoring strategies are even more important for the throughput metric. Indeed, although lightweight methods for estimating the available bandwidth between two Internet end-hosts were proposed in [START_REF] Melander | A new end-to-end probing and analysis method for estimating bandwidth bottlenecks[END_REF][START_REF] Johnsson | On the analysis of packet-train probing schemes[END_REF], in practice the only accurate method is to transfer a large le between the two endpoints. It turns out that the MDP formulation for maximizing the data throughput is equivalent to the MDP formulation for minimizing the RTT. The contribution of the present paper is therefore not on the theoretical side, but rather to investigate the applicability of the approach proposed in [START_REF] Vaton | Joint minimization of monitoring cost and delay in overlay networks: Optimal policies with a markovian approach[END_REF] for optimizing throughput in overlay networks. To this end, we use we use throughput measurements that were made between 9 AWS (Amazon Web Services) data centres.

MDP formulation

The problem formulation in this section is essentially the same as in [START_REF] Vaton | Joint minimization of monitoring cost and delay in overlay networks: Optimal policies with a markovian approach[END_REF][START_REF] Mouchet | Scalable monitoring heuristics for improving network latency[END_REF] except that the quantity of interest is bandwidth instead of delay. Consider a single origin-destination pair and {1, 2, . . . , P } a set of P paths between the origin and the destination. The network topology is thus that of parallel links. At time step t, path i is assumed to be have a bandwidth X i (t), where X i (t) is a discrete-time Markov chain taking values in a nite set. The transition matrix for path i will be denoted by M i .

At each time step, the routing agent has to decide on which path it should send data. For this, the agent has at its disposal the last observed bandwidth for each path. Further, it can choose to measure the bandwidth on one or more paths and update its state information before taking the routing decision. The agent incurs a cost of c i for probing path i independently of time step. The decision-maker must nd a compromise between paying to retrieve information from the system to get a higher bandwidth and not retrieving information leading to a lower bandwidth.

Let u(t) ∈ {0, 1} P whose ith component indicates whether path i is monitored in time step i or not. The total cost paid for action

u(t) is i|ui(t)=1 -c i = -c • u(t) with c = (c 1 , • • • , c P ). Let r(t)
be the path chosen in time step t. A policy θ can be dened by the sequence {(u(t), r(t))} t≥0 . Just as in [START_REF] Vaton | Joint minimization of monitoring cost and delay in overlay networks: Optimal policies with a markovian approach[END_REF], it can be seen that knowing only the last observed bandwidth for a link is not enough to determine the distribution the bandwidth that will be obtained in a given step. The state can be made Markovian by incorporating the age of the last observed bandwidth as well. That is, the pair (y i (t), τ i (t)), where y i (t) is the last observed bandwidth of link i at time t and τ i (t) is the age of the last observation is sucient as the state variable for a Markovian representation of path i. All this information is summarized in a vector s(t) = (s 1 (t), s 2 (t), . . . , s P (t)) where s i (t) = (y i (t), τ i (t)).

Since the state is now Markovian, the problem can be formulated as a Markov Decision Process (MDP). This MDP can be further simplied by noting that in the model, the routing decision does not have any impact on the evolution of the state. Thus, a locally greedy routing decision conditioned on u(t) and the current state will be optimal. In other words, for a given u(t), it will be optimal to choose the path that maximizes the expected bandwidth. With this in mind, the decision problem can be reduced to determining which paths to monitor in each time step. For a given state s ≡ (y, τ ) of path i, dene the belief on the bandwidth being z of this path as follows: b i (z|s) := P(X i (τ ) = z|X i (0) = y), which is just the probability of path i transitioning from y to z in τ steps, and can be computed by choosing the corresponding element of M τ i . If path i is measured, then its actual bandwidth, X i (t), will be known and can be used in the routing decision. Otherwise, it is its expected conditional bandwidth

E[X i |s i ] = x∈Xi x • b i (x|s i )
that will be used. The locally greedy routing decision will be to choose r(t) that maximizes

(u i X i +(1 -u i )E[X i |s i ]).
Note that this decision is taken after performing the action of monitoring the subset of selected links. This leads to maximum bandwidth conditioned on s and u of B(X|s, u)

= max i (u i X i +(1 -u i )E[X i |s i ])
, and an expected maximum bandwidth of:

B(s; u) = x P i=1 b i (x i |s i ) B(x|s; u). (1)
Here the product measure is used because X i s evolve independently. Now that the routing decision is known, the nal MDP takes the form:

max θ E θ s0 ∞ t=0 ρ t B(s(t); u(t)) -c • u(t) .
(

) 2
where θ ≡ u(t) t≥0 , is limited to monitoring decisions only. We remark that the above problem formulation resembles the multi-armed bandit (MAB) framework. However, unlike standard MABs in which the cost function is decomposable in the individual costs of the bandits, in our problem the overall cost is not decomposable.

Numerical results

In order to validate our approach on real data, for which the Markovian assumption is not perfectly met, we use throughput measurements that were made between 9 AWS (Amazon Web Services) data centres located in several places around the world. In summer 2015, we measured the available throughput between all pairs of data centres every ve minutes, by transferring a 10 MB le through the Internet, for a period of four days. We thus collected some 8.3 × 10 4 measurement data over the 4 days period. Assuming that the available throughput over a path is the minimum of the throughputs of its constituent links, the analysis of these data revealed that the IP route is the maximum throughput route only in 23% of the cases, and that most of the time, the maximum throughput overlay route passes through 1 or 2 intermediate nodes (see [START_REF] Brun | Scalable, self-healing, and self-optimizing routing overlays[END_REF] for details).

We selected three origin-destination (OD) pairs: Virginia/Ireland, Virginia/Frankfurt and Frankfurt/Tokyo. For the rst two pairs, in addition to the IP path, we selected two alternative paths which were sometimes better than the IP path, whereas for the last example there was one alternative path.

For each path, we tted a Markov model using a clustering method called, Hierarchical Agglomerative Clustering [START_REF] James | An Introduction to Statistical Learning: With Applications in R[END_REF]. This method creates a hierarchy between clusters, like a tree. At the beginning, each value of bandwidth is a cluster. The algorithm agglomerate one by one the closest data (in term of a distance metric chosen) together in a new cluster, until it creates one big cluster. On our data, we use the Euclidean distance between the bandwidth values. After that, we decided where to cut the tree and obtain a certain number of clusters. Now that we have our dierent states, we have to determine the transition probability matrices, P i . We elaborate this matrix by counting the number of transition between each pair of states. Finally, we search the minimum value τ maxi which satisfy max(P lim i -P τmax i i ) < 10 -2 . It appears that, on real data, the τ max per link is lower than 10 and the number of states per link is between 2 and 12.

We evaluate the average utility (see ( 2)) for four policies : optimal, myopic policy that optimizes the immediate cost only, a receding horizon policy (with a horizon of 3) and a decomposition based heuristic. For a description of the last two policies, we refer the reader to [START_REF] Mouchet | Scalable monitoring heuristics for improving network latency[END_REF].

First, we check that the Markov models we tted are representative of the real traces. For this, for each OD pair, using the transition matrices, we generate a sample path of throughputs on each of the paths. On these sample paths, we apply the three heuristics (but not the optimal) and compute the average utility for each policy. We then apply the policies on the real traces and compute the average utilities. Table 1 shows the percentage relative error between the average utility computed on a sample path and that on the corresponding real trace. The relative error is less than 2% which indicates a good match. Finally, Table 2 shows the utilities of the four policies for varying monitoring costs. One surprising observation from these examples is that the myopic policy is almost optimal.

Conclusion and future work

The results indicate that Markovian models are a good t for throughput on paths in the Internet. Further, a myopic policy is nearly optimal for minimizing a linear combination of the throughput and monitoring costs.

As future work, we would rst like to understand why the myopic policy works well on these examples. It would be interesting to obtain conditions under which this is true. Next, we would like to generalize these models to multi-agent settings in which each node of the overlay can be seen as an agents. These agents can be either cooperative or be non-cooperative. Another possible improvement of the setting would be to allow the routing decision to inuence the future evolution of the bandwidth of the path and to get state information from the current routing decision.

Table 1 :

 1 Percentage relative error between utility computed using Markov model and on real trace.

	c	1	2	3	4	5	c	1	2	3	4	5
	Myopic 1.38 1.14 0.52 0.95 0.01	Myopic 1.36 0.91 0.6 1.07 0.03
	RH 1.38 0.98 0.50 0.62 0.01	RH 1.34 1.046 0.62 1.067 0.03
	H2 1.73 1.81 1.84 1.90 1.97	H2 1.63 1.90 1.92 1.97 2.06
		(a) Virginia/Ireland.		(b) Virginia/Frankfurt.
	c OPT Myopic RH H2		c OPT Myopic RH H2		c OPT Myopic RH H2
	1.0 41.98 41.98 41.98 40.45	1.0 41.91 41.91 41.91 40.77		1.0 58.44 58.44 58.44 57.83
	2.0 39.42 39.40 39.43 38.72	2.0 39.45 39.47 39.45 38.84		2.0 57.71 57.67 57.71 56.59
	3.0 37.90 37.89 37.90 36.98	3.0 38.02 38.03 38.03 36.97		3.0 57.20 57.20 57.20 55.35
	4.0 36.99 36.92 36.97 35.25	4.0 37.11 37.11 37.11 35.13		4.0 57.20 57.20 57.20 54.11
	5.0 36.43 36.43 36.43 33.52	5.0 36.38 36.36 36.36 33.30		5.0 57.20 57.20 57.20 53.09
	(a) Virginia/Ireland.			(b) Virginia/Frankfurt.		(c) Frankfurt/Tokyo.

Table 2 :

 2 Utilities for dierent policies as a function of the monitoring cost.

Acknowledgment

The work of Marine Ségneré was partially funded by a contract with Direction Générale de l'Armement (DGA), France.


C Gouy-Pailler
email: cedric.gouy-pailler@gipsa-lab.inpg.fr

M Congedo

C Jutten

C Brunner

G Pfurtscheller

MODEL-BASED SOURCE SEPARATION FOR MULTI-CLASS MOTOR IMAGERY

This paper presents a general framework to recover taskrelated sources from a multi-class Brain-Computer Interface (BCI) based on motor imagery. Our method gathers two common approaches to tackle the multi-class problem: 1) the supervised approach of Common Spatial Patterns and Sparse and/or Spectral variants (CSP, CSSP, CSSSP) to discriminate between different tasks; 2) the criterion of statistical independence of non-stationary sources used in Independent Component Analysis (ICA). Our method can exploit different properties of the signals to find the best discriminative linear combinations of sensors. This yields different models of separation. This work aims at comparing these models. We show that the use of a priori knowledge about the sources and the performed task increases classification rates compared to previous studies. This work gives a general framework to improve Brain-Computer Interfaces and to adapt spatial filtering methods to each subject.

INTRODUCTION

The ultimate goal of Brain-Computer Interfaces (BCIs) is to provide disabled people suffering from severe motor diseases with a tool to restore communication and movement [START_REF] Birbaumer | Brain-computerinterfaces (BCI): Communication and restoration of movement in paralysis[END_REF]. Although substantial progress has been made in the field, some improvements could help BCIs to move out of the laboratories. Brain-computer systems are always an association between an electrophysiological phenomenon and a signal processing algorithm. A typical example of a BCI is based on the imagination of movement, which results in somatotopic brain signal variations in specific frequency bands [START_REF] Pfurtscheller | Mu rhythm (de)synchronization and EEG single-trial classification of different motor imagery tasks[END_REF][START_REF] Congedo | Classification of movement intention by spatially filtered electromagnetic inverse solutions[END_REF].

On the one hand, Independent Component Analysis (ICA) has been widely used for analyzing and cleaning brain signals in electroencephalography (EEG). This approach, initiated in the early 90's by Jutten and Hérault [START_REF] Jutten | Blind separation of sources, part 1: an adaptive algorithm based on neuromimetic architecture[END_REF], aims at tackling the Blind Source Separation (BSS) problem (neither a priori information about the sources nor the mixing process is provided) by assuming mutual statistical independence between sources. Such models have proved useful to increase classification rates of BCIs [START_REF] Brunner | Spatial filtering and selection of optimized components in four class motor imagery EEG data using independent components analysis[END_REF][START_REF] Naeem | Seperability of four-class motor imagery data using independent components analysis[END_REF], but do not use a priori information about the tasks, namely the labels of tasks during the training step. Different separation principles can be used to tackle the BSS problem. They depend on the statistical properties of sources, and on how statistical independence is evaluated. When sources are assumed to be independent and identically distributed (iid), non-gaussianity of sources is used, which involves higher order statistics or mutual information. The non-gaussianity assumption case can be released, yielding other families of algorithms based on coloration or time-varying energy [START_REF] Pham | Blind separation of instantaneous mixtures of nonstationary sources[END_REF]. The efficient algorithm provided by Pham [START_REF] Pham | Blind separation of instantaneous mixtures of nonstationary sources[END_REF] to jointly diagonalize matrices allows to use different kinds of diversity in the time and frequency domains, therefore it is suitable for comparing different separation models.

On the other hand, the goal-oriented approach of Common Spatial Patterns (CSPs) has been developed and improved since its introduction in [START_REF] Müller-Gerking | Designing optimal spatial filters for single-trial EEG classification in a movement task[END_REF]. The idea of CSP is to find the linear combination optimizing the ratio between intra and inter-class covariance matrices. Many improvements were achieved: 1) by considering frequency-specific covariance matrices [START_REF] Lemm | Spatio-spectral filters for improving the classification of single trial EEG[END_REF] (CSSP); 2) the Sparse-Spatio Spectral version (CSSSP) aimed at jointly optimizing a temporal filter [START_REF] Dornhege | Combined optimization of spatial and temporal filters for improving brain-computer interfacing[END_REF]; 3) and the invariant CSP (iCSP) was able to reduce the negative effects introduced by non-stationary signals [START_REF] Blankertz | Invariant common spatial patterns: Alleviating nonstationarities in braincomputer interfacing[END_REF]. This approach proved useful to discriminate two motor imagery tasks but suffers from a lack of generalization to multi-class problems. A one-versus-rest (OVR) approach has been proposed [START_REF] Wu | One-Versus-the-Rest (OVR) algorithm: An extension of common spatial patterns(CSP) algorithm to multi-class case[END_REF] to generalize the approach to multi-class discrimination problems, but it suffers from some obvious limitations. For example, if one class is the exact barycenter of the two others, a one-versus-rest approach would fail. This paper presents an approach to use both the supervised idea of CSP to exploit our knowledge about performed tasks and the theoretical framework of ICA for nonstationary sources to recover task-related sources. Different properties of the signal can be used to perform the separation. The resulting different models are compared and quality of separation is assessed by classification rates in a 8-subject 4class motor imagery experiment (left hand, right hand, foot and tongue). The remainder of this paper is organized as follows: in section 2, we present the experimental paradigm, section 3 provides the reader with the different models and methods we compared; finally we present and discuss results.

SUBJECTS AND EXPERIMENTAL PARADIGM

In this study, the EEG data of eight subjects (three females and five males with a mean age of 23.8 years and a standard deviation of 2.5 years, [START_REF] Naeem | Seperability of four-class motor imagery data using independent components analysis[END_REF][START_REF] Brunner | Spatial filtering and selection of optimized components in four class motor imagery EEG data using independent components analysis[END_REF]), recorded during a cue-based four-class motor imagery task, was analyzed. Two sessions on different days were recorded for each subject, each session consisting of six runs separated by short (a couple of minutes) breaks. One run consisted of 48 trials (12 for each of the four possible classes), yielding a total of 288 trials per session.

The subjects were sitting in a comfortable armchair in front of a computer screen. As mentioned above, the paradigm consisted of four different tasks, namely the imagination of movement (motor imagery) of the left hand, right hand, foot, and tongue, respectively. At the beginning of each trial (t = 0 s), a fixation cross appeared on the black screen. In addition, a short acoustic warning tone was presented at this time instant. After two seconds (at t = 2 s), a cue in the form of an arrow pointing either to the left, right, down or up (corresponding to one of the four classes left hand, right hand, foot or tongue) appeared for 1.25 s, prompting the subjects to perform the target motor imagery task. No feedback (neither visual nor acoustic) was provided. The subjects were asked to carry out the mental imagination until the fixation cross disappeared from the screen at t = 6 s. A short break followed, lasting at least 1.5 s. After that, the next trial started. The paradigm is illustrated in Figure 1 (left). Figure 1: Timing scheme of the BCI paradigm (left) and electrode setup of the 22 channels with inter-electrode distances of 3.5 cm. Some locations corresponding to the international 10-20 system are labeled (right).

22 Ag/AgCl electrodes (with inter-electrode distances of 3.5 cm) were used to record the EEG, the setup is depicted in Figure 1 (right). Monopolar derivations were used throughout all recordings, where the left mastoid served as reference and the right mastoid as ground. The signals were sampled at 250 Hz and bandpass-filtered between 0.5 and 100 Hz. An additional 50 Hz notch filter was enabled to suppress power line noise.

Although a visual inspection of the raw EEG data was performed by an expert, no trials were removed from the subsequent analysis in this study in order to evaluate the robustness and sensitivity to outliers and artifacts of each model. Three EOG channels and one ECG channel were also used to measure electrophysiological activity of the subjects. Those channels were added to the analysis of the data according to the method described below.

METHODS

Method Overview

Figure 2 shows an overview of the whole processing stage during the training and the test step. The wavelet box in the training step processing overview is represented in gray because its use depends on the model employed for source separation (see below). The spatial filter computation is done according to the different methods described below. The training step is used to fix some of the parameters of the method whereas the test step consists in applying the procedure to unseen data with previously fixed parameters. The dashed lines represent information, which is shared between learning and test steps. Trainings are dependent from subjects and sessions.

Preprocessing

General Source Separation Framework

The main goal of this work is to find the best spatial filters using some statistical models of source separation and a priori knowledge about the tasks performed. We know that source separation is impossible if sources are modeled as Gaussian, identically and independently distributed. Whereas some algorithms like Infomax try to maximize independence, therefore using non-gaussianity of the sources; one other way to move from the Gaussian iid case is to consider simple time structures within the data. The goal of our work is to study the performance of some simple time structures by assuming that, in some basis (Fourier, wavelets, time), the coefficient of the process are uncorrelated with a smoothly varying variance. Such a general model leads to separation principles based only on second-order statistics.

The general framework is that we are trying to recover sources s(t) related to each task by assuming the simplest source separation model for linear or convolutive mixtures of sensor measurements x(t):

x(t) = As(t) ↔ X(t, f ) = AS(t, f ) (1) x(t) = A ⋆ s(t) ↔ X(t, f ) = A( f )S(t, f ) (2)
with A unknown and independent rows in s. The arrow means that a linear transform is applied, typically a short Fourier transform or a wavelet transform to take into account the time-frequency or time-scale properties of the signal. The separation principle given by Pham and Cardoso in [START_REF] Pham | Blind separation of instantaneous mixtures of nonstationary sources[END_REF] aiming at exploiting slow-varying variances of sources yields the joint diagonalization of covariance matrices either in the time or frequency/scale domain.

In the time domain, this principle leads to consider a partition of the observation interval into T q , with q ∈ [1..Q]. For each time interval, we define the covariance matrix:

C xx (T q ) = E t∈T q (x(t)x(t) T )
Then, the estimation of the separation matrix B = A -1 is done by approximately jointly diagonalizing the set

S = {C xx (T q )|q ∈ [1..Q]}
This joint diagonalization is done by the Pham's algo-rithm [START_REF] Pham | Blind separation of instantaneous mixtures of nonstationary sources[END_REF] by minimizing the following criterion:

C(B) = ∑ Q q=1 w q log det diag BC XX (T q )B T -log det BC XX (T q )B T
where w q is used to normalize covariance matrices to unit traces. Using the Hadamart inequality [START_REF] Pham | Blind separation of instantaneous mixture of sources via the Gaussian mutual information criterion[END_REF], it can be shown that this criterion is zero if and only if every matrices are diagonal.

The same procedure can be applied in the frequency/scale domain by previously projecting the signals onto a Fourier or wavelet basis. In this case it exploits the time-frequency diversity of sources. The model remains the same in such a basis due to linearity of the transform (wavelet basis).

In such a case covariance matrices are defined as:

C XX (T q , f ) = E t∈T q (X(t, f )X(t, f ) T )
Then, the estimation of the separation matrix B is done by approximately jointly diagonalizing the set

S = {C XX (T q , f )|q ∈ [1..Q]}
This joint diagonalization is done by the Pham's algorithm [START_REF] Pham | Blind separation of instantaneous mixtures of nonstationary sources[END_REF]. The criterion is the same as above except that the sum is over q and f . Thus the general procedure of this framework is summarized by: 1. for a partition {T 1 , • • • , T Q } of the observation interval in Q subintervals, compute local sample covariance matrices C xx (T q ), 2. jointly diagonalize the local normalized covariance matrices using the Pham's algorithm:

A priori knowledge about the performed tasks during training is included by considering only task-specific covariance matrices. This makes our approach close to CSPs but our advantage is that the multi-class generalization problem due to the Rayleigh quotient in CSP is avoided by our approach based on independence.

Model-Based source separation for Spatial Filtering

In the following, k will represent the index of the mental tasks k ∈ [1..4]. Frequency domains will be denoted as mu or beta (µ or β ), which represent wavelet transforms of the signals at scales 4 and 3 respectively. Those bands are known to be involved in motor imagery [START_REF] Pfurtscheller | Mu rhythm (de)synchronization and EEG single-trial classification of different motor imagery tasks[END_REF]. Discrete wavelet transform was used as frequency basis (Daubechies). E k (.) will denote the average across all trials related to class k. C xx (t ∈ [t 1 ,t 2 ], k) will denote the set of covariance matrices for every trial of one session of a subject for task k computed with EEG in the time domain between t 1 and t 2 . And finally C XX (t ∈ [t 1 ,t 2 ], k, f ) will denote the set of covariance matrices for every trials of one session of a subject for task k computed with EEG in the wavelet basis domain between t 1 and t 2 .

Different kinds of diversities are to be considered in the following models: 1. Inter-class diversity (ICD): sources related to motor imagery have a varying energy among classes. We exploit the fact that a source active for one mental task is active with a different energy (or not active at all) for another mental task. This kind of diversity is exploited by considering task-specific covariance structures, it is used by CSP to find discriminative linear transforms of sensors.

2. Time-varying energy (TVE): as motor tasks are known to be a succession of activations in different brain areas, it can be assumed that sources related to a mental task realization can be active with different energies across the task. Joint diagonalization covariance matrices computed using successive time windows will help recovering sources [START_REF] Pham | Blind separation of instantaneous mixtures of nonstationary sources[END_REF]. 3. Coloration (Col): sources are known to be non-white in the brain. This information can be used in the frequency domain by jointly diagonalizing matrices in the frequency domain. Pham showed [START_REF] Pham | Blind separation of instantaneous mixture of sources via the Gaussian mutual information criterion[END_REF] that sources can be recovered using this criterion if they have nonproportional power-spectra. Lastly signals can be considered either in the time domain (TD) or in the frequency domain (FD).

Exploiting inter-class diversity in the time domain

This first model uses ICD. We recall that this kind of diversity is exploited by considering task-specific covariance structures. For each trial of one specific task, we compute the covariance matrix of the EEG from t = 2.5 s to t = 7.5 s. Then we average across every trials of one specific task. As this is done for every mental task, the procedure leads to a joint diagonalization of 4 covariance matrices (one for each task):

S = {E k (C xx (t ∈ [2.5, 7.5], k)) | k ∈ [1..4]}

Exploiting inter-class diversity and time-varying energy in the time-domain

This second model aims at exploiting the idea that sources are active with different energies between different tasks and/or that the energy of a source is time-varying inside one task. This information is used by partitioning the previous interval to 4 subintervals, [2.5, 7.5] = ∪ 4 i=1 T i . Thus the diagonalization set consists of 16 covariance matrices:

S = {E k (C xx (t ∈ T i , k)) | i ∈ [1..4], k ∈ [1..4]}

Exploiting inter-class diversity and coloration in the frequency domain

As brain waves variations resulting from mental tasks are frequency-specific, we also designed a model to exploit diversity of frequency distribution of sources. This assumes that sources have non-proportional power spectra and/or a coloration diversity. The discrete wavelet transform is used to project signals onto wavelet basis (Daubechies). The set of diagonalization consists of 8 covariance matrices in the frequency domain, 2 scales for each task:

S = {E k (C XX (t ∈ [2.5, 7.5], k, f )) | k ∈ [1..4], f ∈ {µ, β }}

Exploiting separately inter-class diversity in the frequency domain, convolutive model

The last method consists in assuming that the structure of the mixing process has to be considered separately in each band (different mixing models in µ and β ). This assumption implies that sources in mu band have to be found separately from sources in beta band. This yields to 2 joint diagonalizations of 4 frequency-specific covariance matrices (4 tasks):

S µ = {E k (C XX (t ∈ [2.5, 7.5], k, µ)) | k ∈ [1..4]} S β = {E k (C XX (t ∈ [2.5, 7.5], k, β )) | k ∈ [1..4]}
Such a model results in two separation matrices B µ and B β , and thus two different sets of frequency-specific sources.

Artifacts removal

As EOG and ECG channels was available for this experiment, we added those channels into the separation and rejected the sources that were highly correlated with non-EEG channels (correlation coefficient greater than 0.5). Only columns of the mixing matrix corresponding to clean channels were kept for the test step.

Features Extraction

Our convolutive model yields frequency-specific sources, whereas the sources related to the other models are not frequency-specific. Due to the elimination of contaminated sources (high correlation with EOG channel), the number of sources can vary. We detected that, whereas the convolutive model could theoretically result in twice the number of sources we get for other models, our elimination step removes much more sources for the convolutive model than for any other model. Thus we observed that the number of viable sources remains almost the same for each model (around 20 sources).

In line with physiological knowledge of movement imagination, our feature extraction evaluates frequency-specific band power in the µ and β band. Power spectral density estimations were computed using Discrete Wavelet Transform (3 seconds of signal is used to compute the discrete wavelet transform). As the frequency sampling rate of the data is 250 Hz, only scales four and three (corresponding to the µ and β bands) are considered. Features from our frequencyspecific sources (convolutive model) were considered differently: µ band power is computed from µ-specific sources and β band power is computed from β -specific sources. This finally results in around 40 features for every model.

General Procedure

For each subject and each session, a 10 × 10-fold crossvalidation is performed. For each cross-validation, 90 % of the trials are used as training trials and the remaining of the data is used as test trials. A linear discriminant analysis (LDA, [START_REF] Naeem | Seperability of four-class motor imagery data using independent components analysis[END_REF]) is used to classify features.

RESULTS

Cross-Validated Results

We present in Table 1 the mean classification accuracy across subjects and sessions for cross-validated data. These results do not show any significant superiority of one model among the others. A high diversity of skills among subjects for controlling a BCI accounts for this result. Nevertheless, methods based on frequency domain seems to perform slightly worse than methods based on time domain. Overall performance of our methods is very satisfying regarding the difficulty of the considered task. Our method do not benefit from any feature selection algorithm, therefore it can be compared with [START_REF] Naeem | Seperability of four-class motor imagery data using independent components analysis[END_REF] in which the best classification rate is achieved with Infomax (about 65 %). Results obtained in [START_REF] Brunner | Spatial filtering and selection of optimized components in four class motor imagery EEG data using independent components analysis[END_REF] outperforms the results presented here (ranging from 65 to 75 %) but used a feature selection (sequential forward selection) to range about 1300 features from the feature extraction step.

High variability of classification rates across subjects (ranging from 40 to 80 percents) leads us to consider subjectspecific results. Table 2 presents results for each subject and each session. The classification rate (percentage and standard deviation) is considered in the second column of the table. The third column of the table gives the corresponding best model (notations are those introduced in section 3). From basic probability calculus we can derive our specific significance test: probability that one method appears 7 or more times as the best method if methods are randomly ranked is 0.08 (cumulative binomial distribution with parameters p = 0.25 and n = 16). Thus the table 2 shows that our method based on the use of inter-class diversity and timevarying energy in the time-domain is significantly best than the others (p = 0.08). It appears clear that the method based on inter-class diversity and coloration in the frequency domain performs worse than the others (best method for only one session).

Lastly we note an overall superiority of the time domain based methods over the frequency domain based methods, respectively the two first and two last methods. This fact appears obvious by considering the number of session for which time based methods appears as best models ( and each subject (S1 to S8) given by the best model. The standard deviation is given between round brackets.

DISCUSSION

Different a priori information were considered in this paper, namely we used Inter-Class Diversity, Time-Varying En-ergy and Coloration either in the time (for Time-Varying Energy) or in the frequency (Coloration) domain. First of all, we showed in section 3 that finding multi-class spatial filters can benefit from the use of simple a priori knowledge. Then some simple tests of significance showed that exploiting Inter-Class Diversity and Time-Varying energy in the time domain proved better than the other methods considered here. It was quite obvious that using a priori knowledge about the tasks performed would improve classification rates. But improvements due to a priori knowledge about time-varying energy was quite surprising. This result supports the hypothesis that different sources appears during the performance of the tasks and that their time course is not constant. Time interval partitioning was very simple and we think that some refined partitioning of intervals could result in significant improvements of the classification rates. We pointed out a disadvantage of such a refined framework by showing that none of the presented methods could be considered as best for every subjects. Unsurprisingly, the design of optimal spatial filters have to cope with inherent difficulties of studying brains and real subjects: methods have to be subject-dependent to optimal results. The set of methods presented here raises the classical issue of Occam's Razor: fitting data with the most refined model could be dangerous if subjects' imagined tasks are significantly modified by training. These two considerations have to be tackled in two different ways to make such signal processing algorithm available for daily life use: 1) an automatic procedure has to be designed to select subject-specific methods; 2) adaptive algorithms will be necessary to adapt systems to subjects.

In summary, we presented here an efficient framework for increasing classification rates of multi-class BCI paradigms. Our framework is well grounded on the Pham's theoretical work about joint approximate diagonalization and provides natural a priori knowledge that can be used to gather advantages of both Independent Component Analysis (we assume statistical independence of sources) and Common Spatial Patterns.

CONCLUSION

We presented different models to use a priori knowledge about the performed task. Those models are based on the mathematical theory of non-stationary source separation. Our models do not suffer from any multi-class limitations and yield good results regarding the complexity of our treatment.

Figure 2 :

 2 Figure 2: Training and test steps. Overview of the method.

Table 1 :

 1 12 over 16) against the number of frequency-based methods ranked as best (4 over 16 sessions). Summary of the cross-validated performance

		Mean (%) Std Deviation
	ICD, TD	69.3	17.4
	ICD, TVE, TD	70.2	16.2
	ICD, col., FD	66.8	16.8
	Convolutive	68.8	17.2
	Correct (%) Best Model
	S1 ses1 77.2 (1.27)	ICD, TD
	S1 ses2 78.8 (1.32)	ICD, TD
	S2 ses1 52.3 (2.00)	ICD, TVE, TD
	S2 ses2 56.0 (2.08)	ICD, TVE, TD
	S3 ses1 82.3 (1.08)	ICD, TVE, TD
	S3 ses2 84.3 (0.86)	ICD, TD
	S4 ses1 83.4 (1.24)	ICD, col., FD
	S4 ses2 80.7 (2.11)	ICD, ,TVE, TD
	S5 ses1 77.1 (1.38)	ICD, TD
	S5 ses2 88.4 (1.31)	Convolutive
	S6 ses1 55.8 (2.07)	ICD, TVE, TD
	S6 ses2 66.1 (1.83)	Convolutive
	S7 ses1 39.1 (1.44)	ICD, TVE, TD
	S7 ses2 43.6 (2.31)	ICD, TD
	S8 ses1 81.4 (0.84)	Convolutive
	S8 ses2 86.7 (0.90)	ICD, TVE, TD

Table 2 :

 2 Classification rates for each session (ses1 or ses2)

Acknowledgments

This work was supported by the French Ministry of Defense (Délégation Générale pour l'Armement -DGA), the Open-ViBE project (Open-Source platform for a Virtual Brain Environment, Agence Nationale de la Recherche -ANR), and by the European program of Cooperation in the Field of Scientific and Technical Research (COST B27 -Electric Neuronal Oscillations and Cognition).


J.-L Prieur

M Scardia

L Pansecchi

R W Argyle

M Sala

Speckle observations with PISCO in Merate: XI. Astrometric measurements of visual binaries in 2010

Keywords: binaries, close -binaries, visual -astrometry -techniques, interferometric -stars, individual (ADS 5469,

We present relative astrometric measurements of visual binaries, made in 2010 with the speckle camera PISCO at the 102 cm Zeiss telescope of Brera Astronomical Observatory, in Merate. Our observing list contains orbital couples as well as binaries whose motion is still uncertain. We obtained new 207 measurements of 191 objects, with angular separations in the range 0 ′′ .2 -4 ′′ .7, and an average accuracy of 0 ′′ .01. The mean error on the position angles is 0 • .6. Most of the position angles were determined without the usual 180 • ambiguity with the application of triple-correlation techniques and/or by inspection of the long integration files. We also present revised orbits for ADS 5469, 8128, 8239, 10345, and new orbits for ADS 10850 and 11170, partly derived from those observations, and infer estimated values for the masses of those systems that are compatible with the spectral types.

INTRODUCTION

This paper deals with the results of speckle observations of visual binary stars made in Merate (Italy) in 2010 with the Pupil Interferometry Speckle camera and COronagraph (PISCO) on the 102 cm Zeiss telescope of INAF -Osservatorio Astronomico di Brera (OAB, Brera Astronomical Observatory). It is the eleventh of a series [START_REF] Scardia | [END_REF], 2006, 2007, 2008a, Prieur et al. 2008, Scardia et al. 2009, Prieur et al., 2009, Scardia et al., 2010, Prieur et al., 2010, Scardia et al., 2011, herein: Papers I to X), whose purpose is to contribute to the determination of binary orbits. PISCO was developed at Observatoire Midi-Pyrénées (France) and first used at Pic du Midi from 1993 to 1998. It was moved to Merate in 2003 and used there since.

We briefly describe our observations in Sect. 2. Then we present and discuss the astrometric measurements in Sect. 3. Finally in Sect. 4 we propose revised orbits for ADS 5469, 8128, 8239, 10345, and new orbits for ADS 10850 and 11170, partly derived from those observations, and discuss the estimated values for the masses of those systems.

OBSERVATIONS AND DESCRIPTION OF THE OBSERVING LIST

The observations were carried out with the PISCO speckle camera and the ICCD (Intensified Charge Coupled Device) detector belonging to Nice University (France). This instrumentation is presented in Prieur et al. (1998) and our observing procedure is described in detail in Paper VI.

The description of our observing list can be found in our previous papers (e.g., Paper VI). It basically includes all the visual binaries for which new measurements are needed to improve their orbits, that are accessible with our instrumentation.

The distribution of the angular separations measured in this paper is displayed in Fig. 1a and shows a maximum for ρ ≈ 0 ′′ .7. The largest separation of 4 ′′ .7 was obtained for ADS 6175. The smallest separation was measured for ADS 4229), with ρ = 0 ′′ .20. The diffraction limit is ρ d = λ/D ≈ 0 ′′ .13 for the Zeiss telescope (aperture D = 1.02 m) and the R filter (λ = 650 nm).

Likewise, the distribution of the apparent magnitudes mV and of the difference of magnitudes ∆mV between the two components are plotted in Figs. 1b and1c, respectively. The telescope aperture and detector sensitivity lead to a limiting magnitude of about mV = 9 (Fig. 1b) and a limiting ∆mV of about 4.0 (Fig. 1c).

ASTROMETRIC MEASUREMENTS

The 207 astrometric measurements obtained with the observations made in 2010 are displayed in Table 1. They concern 191 binaries. For each object, we report its WDS name (Washington Double Star Catalogue, Mason et al. 2011, hereafter WDS Catalogue) in Col. 1, the official double star designation in Col. 2 (sequence is "discoverer-number"), and the ADS number in Col. 3 [START_REF] Aitken | New General Catalogue of Double Stars[END_REF] when available. For each observation, we then give the epoch in Besselian years (Col. 4), the filter (Col. 5) and the focal length of the eyepiece used for magnifying the image (Col. 6), the angular separation ρ (Col. 7) with its error (Col. 8) in arcseconds, and the position angle θ (Col. 9) with its error (Col. 10) in degrees. In Col. 11, we report some notes and some infor-mation about the secondary peaks of the auto-correlation files (e.g. diffuse, faint or elongated). For the systems with a known orbit, the (O-C) (Observed minus Computed) residuals of the ρ and θ measurements are displayed in Cols. 13 and 14, respectively. The corresponding authors are given in Col. 12, using the bibliographic style of the "Sixth Catalogue of Orbits of Visual Binary Stars" (Hartkopf & Mason, 2011, hereafter OC6). Unless explicitly specified, the measurements refer to the AB components of those systems. In Col. 14, the symbol Q indicates that there was a quadrant inconsistency between our measures and the orbital elements published for this object. The characteristics of the V , R, RL and I filters used for obtaining those measurements are given in Table 2. Some objects were observed with no filter because they were too faint. This is indicated with W (for "white" light) in the filter column (Col. 5 of Table 1). In that case, the bandpass and central wavelength correspond to that of the ICCD detector (see Prieur et al., 1998).

As for the other papers of this series, we interactively processed all the auto-correlation files computed in real time during the observations and obtained a series of measurements with different background estimates and simulated noise. The measures and errors displayed in Table 1 were derived from the mean values and the standard deviation of those multiple measurements (see Paper III for more details). The average values of the errors of the measurements reported in this table are 0 ′′ .013 ± 0 ′′ .009 and 0 • .64 ± 0 • .50 for ρ and θ, respectively.

There is only one unresolved object: HO 215. We observed this object in 2004 (Paper I), 2006 (Paper V) and 2010 (this paper) (see Fig. 3). In 2006, close secondary peaks appeared in the autocorrelation after removal of a model of an unresolved star. This detection was suspicious since those peaks were nearly superimposed on the first Airy rings. Indeed those secondary peaks do not appear when processing with the same method the autocorrelations obtained in 2004 and 2010. Therefore we can now conclude that HO 215 appeared as unresolved with our instrumentation during the period 2004-2010.

ADS 1860: this object is known as a wide triple system (see Fig. 2). The C component (mV = 9.1) lies just outside the PISCO field of view at a distance of 7 ′′ .1. In 2010, we Note: In column 9, the exponent * indicates that the position angle θ could be determined without the 180 • ambiguity.

(mm) ( ′′ ) ( ′′ ) ( • ) ( • ) ( ′′ ) ( • ) 00093+7943 STF2 2010.053
(O-C) ∆θ(O-C) (mm) ( ′′ ) ( ′′ ) ( • ) ( • ) ( ′′ ) ( • ) 10205+0626 STF1426
(mm) ( ′′ ) ( ′′ ) ( • ) ( • ) ( ′′ ) ( • ) 13346+3308 BU933 8958
In column 14, the exponent Q indicates discrepant quadrants between our measurements and the published orbits. measured the Aa,B and Aa,Ab pairs, with ρ = 2 ′′ .801 and ρ = 0 ′′ .584, respectively. The closest pair (CHR 6 Aa) is an orbital couple with a period of 46 years and a very large difference of luminosity in V : ∆mV ∼ 4. The primary is an A5 star and the companion must be a cool star. In order to reduce the contrast between the two components, we observed this object with the I filter.

Quadrant determination

As our measurements were obtained from the symmetric auto-correlation files, the θ values first presented a 180 • ambiguity. To resolve this ambiguity and determine the quadrant containing the companion, we have used [START_REF] Aristidi | [END_REF]'s method, by analysing the restricted triple correlation (RTC hereafter) files that were computed in real time during the observations. For the couples with the largest separations, a straightforward determination could be done when the companions were visible in the long integration files.

As a result, in Table 1, we are able to give the unambiguous (i.e. "absolute") position angles of 165 out of 207 measurements, i.e. 80% of the total. They are marked with an asterisk in Col 9. Otherwise, our angular measurements were reduced to the quadrant reported in the WDS catalogue, which is extracted from the Fourth Catalogue of Interferometric Measurements of Binary Stars (Hartkopf et al. 2011, hereafter IC4).

Our "absolute" θ values are consistent with the values tabulated in WDS for all objects except for ADS 2253, 4208, 6963 and 9324. We display some parameters of all those objects in Table 3. In Col. 2, we indicate the quadrant (Q) we found with the RTC, using the usual convention of numbering them from 1 to 4 to indicate the North-East, South-East, South-West and North-West quadrants, respectively. In Col. 3 we recall the filter we have used (W indicates the absence of filter: the corresponding central wavelength is close that of the R filter). In Col. 4, we give the differences of magnitude between the two components from the IC4, and in Col. 5 the global spectral types found in the SIMBAD astronomical data base. For all those objects, the small value of ∆mV can account for the difficulty of measuring the quadrant for those binaries. Moreover, most of the measurements from other observers reported in IC4 were all obtained in V , whereas we observed in R or in W (which is similar to R). A quadrant inversion between V and R is likely, when the two stars have a different spectral type (e.g. blue primary and red secondary).

Comparison with published ephemerides

The (O -C) (Observed minus Computed) residuals of the measurements for the 72 systems with a known orbit in Table 1 are displayed in Cols. 13 and 14 for the separation ρ and position angle θ, respectively.

Those residuals were computed with a selection of valid orbits mainly found in the "master" file of the OC6 catalogue. We did not always use the most recent orbits since sometimes older orbits led to equivalent or even smaller residuals.

For ADS 5469, 8128, 10345 and 10850 we also reported the residuals obtained with our new orbits presented in Sect. 4.

Fig. 4 shows that the residuals have a rather large scatter that can be explained by the (old) age of many orbits. The mean values computed with the residuals of Table 1 are < ∆ρO-C >= -0 ′′ .01 ± 0 ′′ .06 and < ∆θO-C >= -0 • .9 ± 2 • .3. 1 computed with the published orbits.

In the following, we examine the cases of ADS 1860 Aa-B, 2436 and 8128, that appear with the largest residuals in Fig. 4.

ADS 1860 (STF262 Aa-B) and ADS 2436: The orbits of Heintz (1996) for STF262 Aa-B and Heintz (1998) for ADS 2436 are preliminary. The arcs of orbit are still short (58 • and 97 • , for STF262 Aa-B and ADS 2436, respectively) and have a small curvature. Hence, their orbital periods are likely to be very long (about 1400 and 400 years, respectively) and it is still too premature to revise Heintz' orbits. Clearly new measurements are needed to improve the quality of those orbits.

ADS 8128: its periastron passage occurred in the beginning of 2010 and our last orbit (Prieur et al, 2009) leads to rather large residuals in theta (close to -7 • ). We propose a new orbit in Sect. 4.2.

REVISED ORBITS OF ADS 5469, 8128, 8239, 10345, AND NEW ORBITS FOR ADS 10850 AND 11170

In this section we present the new orbits we have computed for ADS 5469, 8128, 8239, 10345, 10850 and 11170.

We have followed the same method for those six objects. Using our last measurements with PISCO and the other available observations contained in the data base maintained by the United States Naval Observatory (USNO), we first computed the preliminary orbital elements with the analytical method of [START_REF] Kowalsky | Procès-verbaux de l'Université Impériale de Kasan Maedler[END_REF]. We then used them as initial values for the least-squares method of Hellerich (1925). When convergence was achieved, Hellerich's method led to an improvement of the orbital elements (with the exception of the major axis) and to an estimation of the corresponding errors. The final value of the major axis was then set to the value that minimized the residuals in separation of Hellerich's solution.

The final orbital elements are presented in Table 4. The description of the format of the tables contained in this section can be found in Papers VI and VII. The errors reported for ADS 5469, 8128, 8239 and 10345 were obtained by Hellerich's least-squares method. For ADS 10850 and 11170, this method failed to converge, which explains the absence of errors for those orbits.

The corresponding (O -C) residuals, restricted to the last observations for reasons of space, are given in Tables 5,6, 7, 8, 9 and 10 for ADS 5469, 8128, 8239, 10345, 10850, and 11170 respectively. In the last column, we report the name of the observer, using the US Naval Observatory convention.

The ephemerides for 2012-2021 are presented in Table 11. The apparent orbits are shown in Fig. 5a,b,c and 6a,b,c as solid lines. The observational data used for the calculation of the orbital elements are plotted as small crosses. The orientation of the graphs conforms to the convention adopted by the observers of visual binary stars. For each object, the location of the primary component is indicated with a big cross. The straight line going through this point is the line of apsides. An arrow shows the sense of rotation of the companion.

In Fig. 5a',b',c' and 6a',b',c' we have plotted the theta measurements (small crosses) and the corresponding values of our computed orbits (continuous lines, in green for the on-line edition). The new data obtained after the last publication are plotted as small × (in red for the on-line edition) and the previous data as small + (in blue). The orbits of ADS 10850 and 11170 have been computed for the first time and all the plotted measurements can therefore be considered as new data. For the other objects, there are 8 new observations in [START_REF] Scardia | [END_REF].942-2010.201 for ADS 5469, 17 new observations in 2007.290-2010.395 for ADS 8128, 5 new observations in 1998.392-2009.260 for ADS 8239, and 175 new observations in 1980.565-2010.865 for ADS 10345.

In Table 12, we present some physical parameters of those systems. The visual magnitudes (Col. 3), the difference of magnitude between the components (Col. 4) and the spectral types (Col. 5) were extracted from the IC4 and the SIMBAD data bases. The dynamical parallaxes are presented in Col. 6. Except for ADS 10850 (see in Sec. 4.5), those parallaxes were derived from our orbital elements using Baize & Romani (1946)'s method, with our revised formulae presented in Scardia et al. (2008b). In Col. 7, we report either the Hipparcos parallaxes from ESA (1997) or the revised values from van Leeuwen (2007a), as indicated in Col. 11. In Cols. 8, 9 and 10, we give the corresponding angular and linear sizes of a and M total , respectively, that were computed from our orbital elements and the Hipparcos parallaxes.

Note that we found a small difference between the parallax values reported in van Leeuwen (2007a)'s paper and those printed in his book (van Leeuwen, 2007b). In this paper, we adopted the values from van Leeuwen (2007a) which correspond to his most recent results (van Leeuwen, private communication), and were selected by the SIMBAD data base. [START_REF] Muller | [END_REF]. This error was soon discovered and the interest for this object disappeared. ADS 5469 was then neglected by the observers for the next two decades. In 1944, it was then a great surprise to discover the companion in the first quadrant, beyond the periastron, at about 260 • from the position measured by Aitken. The available measures are then nearly exclusively located in the first quadrant. The only measurements of good quality obtained before the periastron passage (which occurred in 1935) are the two Aitken measures. As a result all the orbit calculations of ADS 5469 rely on the validity of those two observations. Three orbits have been published yet: [START_REF] Muller | [END_REF], [START_REF] Docobo | IAU Comm. 26 Inf. Circ[END_REF] and Heintz (1998).

Heintz' orbit leads to rather large residuals in ρ (-0 ′′ .11) with our measurement of 2010.201. The other measures made in the last years indicate the same trend. We also noticed that Heintz (1998) did not give any information on the errors and derived masses for this orbit. We thus decided to compute a new orbit with the 39 available observations.

Using the procedure described above, we obtained a straightforward convergence, although the uncertainties on the elements are rather large (see Table 4).

The mean rms residuals are 0 ′′ .066 and 1 • .841 for ρ and θ, respectively. The dynamic parallax is in good agreement with the Hipparcos measurement (especially with van Leeuwen's revised value) and the systemic mass is consistent with the theoretical expected value for a F8V+G3V system (see Table 12).

Note finally that in the literature, this object is also known as G 108-29, a high proper motion star, with a proper motion of -59.89 ± 0.95 mas/yr in right-ascension, and -376.61 ± 0.75 mas/yr in declination, according to Hipparcos (van Leeuwen, 2007a 6).

The first orbit was computed by Hopmann (1960), who proposed a long period orbit (P = 1148 yr). The next measurements suggested a shorter period.

Our new orbital elements reported in Table 4 well fit the observations (see Fig. 5b and 5b'), with mean rms errors of 0 ′′ .16 and 1 • .5 for the residuals in ρ and θ, respectively. Compared to our previous orbit computed in 2008 (Paper VII), we have now a better constraint on the periastron epoch which occurred in the beginning of 2010. It seems that PISCO observations are the only ones that were made in that critical period. Although this orbit is still preliminary (grade 3-4) since only about half of it has been monitored yet, we do not think it will change much in the future.

The spectral type is F9V according to [START_REF] Harlan | [END_REF]. The companion is probably of type G3V since ∆mV = 1.2 (see Table 12). Using the Hipparcos parallax of ESA (1997), the total mass M total = 2.0 M⊙ is compatible with the theoretical values for a system of two stars of type F9V and G3V, and the dynamical parallax of 31.2 mas is in very good agreement with this Hipparcos measurement of 31.17 mas (see Table 12). The agreement is also good with the revised Hipparcos values published by van Leeuwen (2007a). WDS 11368-1221 -BU 456 -ADS 8239 ADS 8239, BU 456 (HIP 56641): this couple was discovered by S.W. Burnham in 1877 in his own small backyard observatory in Chicago, using an excellent 15-cm aperture refractor built by Alvan Clark in 1870 (Burnham, 1877). This instrument had no micrometer and the first measure was soon done by A. Hall with the 66-cm aperture Clark refractor of the U.S. Naval Observatory. This couple has been regularly observed since, but its declination of -12 • makes it difficult to observe from Merate.

New orbit of

The first orbit was calculated by [START_REF] Zagar | [END_REF], and then followed successively by the orbits of Abrami (1954), [START_REF] Scardia | IAU Comm. 26 Inf. Circ. n[END_REF], Baize (1991) and Olevic & Popovic (2000). The most recent orbit of Olevic & Popovic (2000) begins to show systematic errors in separation, whereas our orbit of [START_REF] Scardia | IAU Comm. 26 Inf. Circ. n[END_REF] is still valid and leads to smaller residuals. Nevertheless we decided to take advantage of about 30 years of observations to revise our orbit of 1982.

Using the 74 available measurements we obtained the elements reported in Table 4. The corresponding standard deviations of the residuals are 2 • .2 and 0 ′′ .06 for θ and ρ, respectively. This new orbit is still provisory, with a large uncertainty on the period, because the orbit has only been monitored close to the periastron.

The small magnitude difference of ∆mV = 0.2 suggests a companion of G-K spectral type. The sum of the masses computed using the Hipparcos parallax of 1.1 M⊙ is smaller than the theoretical values, but is compatible with theoretical values when taking into account its large uncertainties of 60%. The dynamical parallax obtained with Baize-Romani (1946)'s method is in fair agreement with Hipparcos measurement. The total dynamic systemic mass is slightly small, which was already noticed by Baize with his own determination and he suggested a spectral type later than K for both components. Scardia (1980) does not represent the observations any longer, showing a systematic discrepancy both in separation and in position angle, indicating that the period and the semi major axis were underestimated. Heintz' orbit (1981) computed at the same epoch has a much longer period and semi major axis. This orbit is in better agreement with the latest observations, but now leads to systematic residuals. Taking advantage of the good quality measurements obtained since our last determination of 1980, we computed the new orbital elements presented in Table 4 with the 766 available measurements. The mean rms residuals are 0 ′′ .08 and 0 • .9, for ρ and θ, respectively. The uncertainties on the orbital elements are small, if we except the period, which is poorly known because the orbit has been monitored yet only close to the periastron (see Fig. 6a). The true orbital period must be very long, likely in the range 700-900 years.

New orbit of WDS 17053+5428

Its spectral type is F5V according to SIMBAD data base. The very small magnitude difference ∆mV = 0.03 (see Table 12) indicates that the two components have the same spectral classification. The systemic mass of 2.7 M⊙ derived from our orbital elements and Hipparcos parallax is in very good agreement with the expected value for two F5V stars, which is 2.8 M⊙ according to [START_REF] Drilling | Allen's Astrophysical Quantities[END_REF]. Similarly, the dynamic parallax of 36.3 mas well agrees with the parallax measured by Hipparcos (see Table 12). Note that the B component is itself a spectroscopic binary (P = 2270 days, V0 = -16.6 km/s).

New orbit of WDS 17520+1520 -STT 338AB

-ADS 10850 ADS 10850, STT 338AB (HIP 87427): this couple was discovered by O. Struve between August 26th 1841 and December 7th 1842 with the 38 cm Merz-Mahler refractor of Pulkova Central Observatory [START_REF] Struve | Catalogus Novus Stellarum Duplicium et Multiplicium[END_REF]. The first measure was made by Maedler with the 25 cm refractor of Dorpat Observatory, on April 25th 1843 (Maedler, 1845).

The observations are numerous with 314 measures available for θ and 300 measures for ρ, but the old data are scattered both for θ and ρ. Our orbit is the first ever computed for this object. For this computation, we took advantage of the 8 measures obtained by our group since 1998 (which represents 50% of the total measures reported in the WDS after 1998). The orbital elements presented in Table 4 were obtained using Kowalsky's method. The mean rms error was 0 ′′ .069 and 1 • .927 for ρ and θ, respectively. It is still preliminary (i.e., grade 4 orbit) because the orbital period is long and the arc of orbit covered by the observations is short (see Fig. 6b).

The spectral type is K0III from SIMBAD data base. The small magnitude difference indicates that the two components have a similar spectral type. The systemic mass of 2.3 M⊙ obtained with the parallax measured by Hipparcos (ESA, 1997), is in very good agreement with the theoretical value of 2.2 M⊙ expected for a system of two K0III stars [START_REF] Drilling | Allen's Astrophysical Quantities[END_REF]. As our orbit is only preliminary, this can be considered as very satisfactory. Conversely, the value of 12 M⊙ obtained with the revised parallax of van Leeuwen (2007a) is much too large.

Since the primary star is a giant star, the dynamical parallax cannot be computed with Baize-Romani's method which is based on the mass-luminosity relation. So we derived it in Table 12 from Kepler's third law:

̟ dyn = a (M1 + M2) 1/3 P 2/3
where M1 and M2 are the component masses in M⊙, P the period in years, and a the semi major axis in arcseconds. Assuming similar masses for the two components of 1.1 M⊙, we obtained ̟ dyn = 7.9 mas. This value is in good agreement with the Hipparcos parallax from ESA (1997), but in disagreement with van Leeuwen (2007a) revised values.

New orbit of WDS 18126+3836 -BU 1091 -ADS 11170

ADS 11170, BU 1091 (HIP 89237): this couple was discovered by S.W. Burnham on October 20th 1888 (Burnham, 1890) at the Lick Observatory with the excellent 30 cm refractor built by Alvan Clark and installed on Mount Hamilton in 1881.

Our new orbit was computed using our measures made in 2007, 2008 (see Table 10) and in 2011 (three measures not yet published), and the other measures available in the USNO data base. This led to a total number of 45 measures.

The observations of ADS 11170 made before 1920, were reported in the first quadrant, and are spread out both in ρ and θ. They would indicate a strange trajectory, in complete disagreement with the measures made after 1940 and for which the validity of the quadrant is well established (see the IC4 catalogue). We therefore assumed that this was an error, and moved those old observations from the first to the third quadrant. In this way, it was possible to compute a preliminary orbit with a long period and high eccentricity.

The spectral type is F8V according to SIMBAD. The magnitude difference suggests that the companion spectral type is about G5V. Hence the systemic mass is expected to be 2 M⊙. This is very close to what we obtained with the Hipparcos parallax revised by van Leeuwen (2007a), and still compatible with the value obtained with ESA (1997), when taking the uncertainties into account (see Table 12). The dynamical parallax derived with Baize-Romani (1946)'s method is in good agreement with the two values given for the Hipparcos parallax in ESA (1997) andvan Leeuwen (2007a).

CONCLUSION

In 2010, we obtained 207 new measurements of 191 visual binaries with PISCO in Merate, with an average accuracy of 0 ′′ .01 for the angular separation and 0 • .6 for the position angles. The total number of measurements made in Merate since 2004 now exceeds 2300. This work is thus a good contribution to the continuing monitoring of long period visual binary systems, which is important for refining systemic stellar masses.

We then presented new orbital elements computed for ADS 5469, 8128, 8239, 10345, 10850 and 11170. The periastron passage of ADS 8128 occurred in 2010 and to our knowledge, our six measurements are the only ones made in 2010. This shows the importance of our observations made in this critical part of the orbit. They now provide a good constraint on the orbit of this binary star. The next periastron passage should occur in more than five hundred years!

Figure 1 .

 1 Figure 1. Distribution of the angular separations of the 207 measurements of Table 1 (a), the total visual magnitudes of the 191 corresponding binaries (b) and the differences of magnitude between their two components (c).

Figure 2 .

 2 Figure 2. Autocorrelation of ADS 1860 obtained from PISCO observations made on January 19th 2010 with the I filter. The inner and outer secondary peaks correspond to the pairs Aa,Ab (ρ = 0 ′′ .6, θ = 41 • , ∆m V ∼ 4) and Aa,B (ρ = 2 ′′ .8, θ = 230 • ), respectively

Figure 3 .

 3 Figure 3. HO215: autocorrelations obtained with PISCO observations in 2004 (a), 2006 (b) and 2010 (d). The observations of 2006 suggested the existence of a companion when removing a model of unresolved star (c), but this was not confirmed by the observations of 2004 and 2010.

Figure 4 .

 4 Figure 4. Residuals of the measurements ofTable 1 computed with the published orbits.

Figure 5 .

 5 Figure 5. New orbits of ADS 5469 (a), ADS 8128 (b) and ADS 8239 (c). The corresponding temporal variation of the position angles is plotted in (a'), (b') and (c'). Online-edition: the measurements obtained after the computation of the previous published orbits are plotted in red (see text).

Figure 6 .

 6 Figure 6. New orbits of ADS 10345 (a), ADS 10850 (b) and ADS 11170 (c). The corresponding temporal variation of the position angles is plotted in (a'), (b') and (c'). Online-edition: the measurements obtained after the computation of the previous published orbits are plotted in red (see text).

Table 1 .

 1 PISCO measurements made in 2010 and corresponding O -C residuals (begin.)

	WDS	Name	ADS Epoch Fil. Eyep.	ρ	σρ	θ	σ θ Notes Orbit

∆ρ(O-C) ∆θ(O-C)

Table 1 .

 1 PISCO measurements made in 2010 and corresponding O -C residuals (cont.)

					R	20 0.820 0.008 17.3 * 0.4	Hei1997	-0.05	0.9
	00109+5750 KR1		2010.053 W	20 2.077 0.011 191.8 * 0.5 Elong.
	00209+1059 BU1093		2010.053 R	20 0.739 0.008 117.4 * 0.6	Lin2010c	-0.02	-0.5
	00444+3337 STF55		2010.050 W	20 2.185 0.013 330.1 * 0.3
	00546+1911 STT20		2010.050 R	20 0.578 0.026 183.5 * 0.3	Doc2007d	0.02	1.4
	00550+2338 STF73		2010.050 R	20 1.007 0.008 322.7 * 0.5	Mut2010b	-0.04	-0.6
	"	"	"	"	"	"	"	"	" "	Doc1990b	-0.06	-0.8
	01030+4723 STT21		2010.050 R	20 1.231 0.011 174.1 * 0.3	Hei1966	0.12	-1.5
	01037+5026 HU517		2010.053 R	20 0.594 0.016 31.1 0.7 Diffuse
	01095+4715 STT515		2010.053 R	10 0.505 0.003 119.6 * 0.4	Mut2010b	-0.01	-0.2
	"	"	"	"	"	"	"	"	" "	Sca2001d	0.00	-0.5
	01112+3743 HO215	-2010.053 R	10	-	-	--Unres
	01443+5732 BU870		2010.053 RL	20 0.569 0.014 331.7 * 0.6	Hrt2009	0.03	2.0
	01593+2450 STF194		2010.088 R	20 1.258 0.008 278.1 * 0.3
	02020+0246 STF202		2010.050 R	20 1.840 0.009 266.4 * 0.3	Sca1983f	0.07	1.8
	02039+4220 STT38BC		2010.086 R	10 0.219 0.005 95.7 * 0.9	Doc2007d	-0.02	-3.7
	02052-0058 BU516		2010.105 W	20 0.677 0.011 314.6 1.6
	02062+2507 STF212		2010.105 W	20 1.926 0.010 161.2 * 0.5
	02124+3018 STF227		2010.088 R	20 3.952 0.040 68.5 * 0.4
	02128+3722 HO497		2010.089 W	20 0.411 0.011 88.4 * 1.3 Elong.
	02174+6121 STF234		2010.089 W	20 0.701 0.012 223.8 * 1.0	Sca1981e	0.03	-1.3
	02231+7021 MLR377	-2010.053 R	20 0.651 0.017 140.1 1.3	Pal2005b	0.14	-2.6
	02291+6724 STF262Aa,B 2010.053 I	20 2.801 0.014 229.7 * 0.3	Hei1996b	0.19	0.4
	02291+6724 CHR6Aa,Ab 2010.053 I	20 0.584 0.020 41.1 * 1.3	Dru2003	-0.01	1.2
	02327+0620 STF276		2010.105 W	20 1.896 0.028 273.8 0.6
	02407+2637 STT43		2010.050 W	20 0.676 0.014 346.2 * 0.5	Sca2001d	0.02	2.2
	"	"	"	"	"	"	"	"	" "	Hrt2001a	-0.05	-0.0
	02589+2137 BU525		2010.050 W	20 0.554 0.023 92.1 * 0.6	Csa1978	0.05	-0.8 Q
	02592+2120 STF333		2010.050 R	20 1.398 0.008 209.5 * 0.3	FMR2009c	0.03	0.1
	03122+3713 STF360		2010.105 W	20 2.856 0.018 124.6 * 0.3	WSI2004a	0.03	-1.4
	03175+6540 STT52		2010.089 R	10 0.467 0.009 58.4 * 1.5	Hei1998	0.08	6.7
	03196+6714 HU1056		2010.088 R	20 1.082 0.013 257.7 0.4
	03344+2428 STF412		2010.050 R	10 0.735 0.004 353.8 0.3	Sca2002a	0.00	0.5
	03354+3341 STF413		2010.105 W	20 2.381 0.012 124.7 * 0.4
	03362+4220 A1535		2010.050 W	20 0.752 0.008 339.8 1.0	Hrt2008	0.03	-0.5
	"	"	"	"	"	"	"	"	" "	Mrl1970b	-0.03	-3.0
	03407+4601 STT59		2010.105 W	20 2.824 0.021 355.3 * 0.5
	03427+6950 STF419		2010.089 R	20 2.989 0.015 72.2 * 0.3
	03446+3210 BU880		2010.105 W	20 0.583 0.008 16.5 * 1.6
	03454+4952 HU103		2010.089 R	20 1.165 0.014 202.9 * 1.3 Elong.
	03463+2411 BU536		2010.053 W	20 0.926 0.015 178.0 * 0.5	Hrt2009	-0.06	-1.8
	03491+3216 STT516		2010.052 RL	20 2.221 0.011 43.4 * 0.3
	03521+4048 STT66		2010.089 W	20 0.975 0.023 144.4 * 0.7
	04020+6231 BRD1		2010.105 W	20 2.005 0.011 223.7 * 0.4
	04115+4152 BU546		2010.089 W	20 0.896 0.013 228.6 1.0
	04179+5847 STF511		2010.053 R	20 0.535 0.014 77.4 * 1.2	Hei1996c	0.03	-1.9
	05252+0155 STF708		2010.201 W	20 2.695 0.066 321.1 * 0.3
	05308+0557 STF728		2010.105 R	20 1.237 0.008 44.5 * 0.8	USN1999b	-0.03	-0.4
	05319+2141 COU268	-2010.201 W	20 0.732 0.008 165.1 1.0
	05351+3056 BU1267		2010.201 W	20 0.356 0.014 172.7 3.6
	05357+2054 COU270	-2010.201 W	20 0.710 0.012 41.9 * 2.1
	05371+2655 STF749		2010.105 R	20 1.146 0.008 140.1 * 0.4	Sca2007a	-0.02	-1.0 Q
	05381-0011 STF757		2010.195 W	20 1.454 0.014 239.4 * 0.4 Elong.
	05386+3030 BU1240		2010.195 R	10 0.200 0.003 327.2 1.4 Elong. FMR2008	-0.01	-0.1
	"	"	"	"	"	"	"	"	" "	Sca1982f	-0.01	-1.6

Table 1 .

 1 PISCO measurements made in 2010 and corresponding O -C residuals (cont.)

	WDS	Name	ADS Epoch Fil. Eyep.	ρ	σρ	θ	σ θ Notes Orbit	∆ρ

Table 1 .

 1 PISCO measurements made in 2010 and corresponding O -C residuals (cont.)

			7730 2010.270 R	20 0.911 0.008 310.6 0.5	Sca2008c	0.01	-0.8
	"	"	"	"	"	"	"	"	" "	Nov2006	-0.01	-0.8
	10279+3642 HU879	7780 2010.300 R	10 0.523 0.004 223.7 * 0.6 Elong. Msn2001c	0.00	0.8
	10397+0851 STT224	7871 2010.300 R	20 0.510 0.011 139.5 0.6 Faint	Hrt2010a	-0.00	-1.7
	10426+0335 A2768	7896 2010.324 R	20 0.613 0.018 250.0 * 0.6	Hrt2010a	0.02	0.2
	10480+4107 STT229	7929 2010.324 R	20 0.677 0.011 262.2 1.0	Alz1998a	0.02	1.0
	11000-0328 STF1500 8007 2010.387 R	20 1.370 0.015 299.9 0.3
	11037+6145 BU1077	8035 2010.264 V	10 0.586 0.003 17.8 * 0.3	Sca2010d	-0.01	-1.4
	"	"	" 2010.324 V	10 0.588 0.003 17.2 * 0.5	Sca2010d	-0.01	-1.6
	"	"	" 2010.327 V	10 0.597 0.004 17.7 * 0.3	Sca2010d	0.00	-1.1
	"	"	" 2010.384 V	10 0.592 0.006 17.7 * 0.7	Sca2010d	-0.00	-0.7
	11050+3825 HO378	8047 2010.387 R	20 1.058 0.012 237.2 0.3
	11154+2734 STF1521 8105 2010.392 R	20 3.678 0.035 96.8 * 0.3
	11158+4227 COU1904	-2010.327 W	20 0.459 0.016 213.4 1.0
	11190+1416 STF1527 8128 2010.264 R	10 0.310 0.003 186.0 * 0.3	Pru2009	0.08	-7.2
	"	"	"	"	"	"	"	"	" " "	This paper	-0.00	-0.4
	"	"	" 2010.300 R	10 0.326 0.003 187.0 0.4 Faint	Pru2009	0.10	-7.0
	"	"	"	"	"	"	"	"	" " "	This paper	0.01	0.2
	"	"	" 2010.324 R	10 0.317 0.003 187.5 0.8	Pru2009	0.09	-7.1
	"	"	"	"	"	"	"	"	" " "	This paper	0.01	0.4
	"	"	" 2010.327 R	10 0.318 0.005 188.2 0.3 NF	Pru2009	0.09	-6.4
	"	"	"	"	"	"	"	"	" " "	This paper	0.01	1.0
	"	"	" 2010.384 R	10 0.318 0.005 187.6 0.6	Pru2009	0.10	-8.4
	"	"	"	"	"	"	"	"	" " "	This paper	0.01	-0.3
	"	"	" 2010.395 R	10 0.323 0.003 186.5 * 0.4	Pru2009	0.10	-9.8
	"	"	"	"	"	"	"	"	" " "	This paper	0.01	-1.6
	11245+2037 STF1537 8149 2010.393 W	20 2.266 0.011 358.0 * 0.4
	11267+6654 HU1133	8159 2010.395 W	20 0.781 0.008 346.4 * 0.8
	11332+4927 HU727	8210 2010.300 W	20 1.256 0.008 203.5 0.5 Elong.
	11371+4040 A1996	8241 2010.393 W	20 2.011 0.023 189.5 0.7
	11406+2102 STF1566 8263 2010.393 W	20 2.382 0.012 349.7 * 0.4
	11428+2105 HU888	8275 2010.393 W	20 0.550 0.008 175.9 * 0.4
	11431+3715 HU1135	8276 2010.393 W	20 0.695 0.008 335.3 0.7
	11517+4449 HJ842	8324 2010.387 W	20 3.150 0.016 88.6 * 0.8 Elong.
	11547+0944 BRT1276	-2010.395 W	20 2.875 0.014 358.2 * 0.3
	11563+3527 STT241	8355 2010.387 R	20 1.802 0.009 144.7 * 0.6
	11598+5324 STT243	8378 2010.387 W	20 1.217 0.013	8.9 * 0.3
	12126+3546 STF1613 8460 2010.395 W	20 1.162 0.008	7.3 0.4
	12291+3123 STT251	8569 2010.327 W	20 0.724 0.008 58.3 * 0.8	Sca2003c	0.06	-2.0
	12329+5448 A1600	8594 2010.396 W	20 0.837 0.015	8.3 * 0.7
	12417-0127 STF1670 8630 2010.300 R	20 1.440 0.008 22.2 * 0.3	Sca2007c	-0.01	0.2
	"	"	" 2010.324 R	20 1.433 0.008 22.5 * 0.5	Sca2007c	-0.02	0.7
	"	"	" 2010.384 R	20 1.468 0.009 21.8 * 0.7	Sca2007c	-0.00	0.3
	"	"	" 2010.426 R	20 1.467 0.010 21.1 * 0.3	Sca2007c	-0.01	-0.2
	12438+0733 STF1674 8646 2010.387 W	20 2.322 0.013 173.5 * 0.7
	12533+1310 HU894	8696 2010.396 W	20 1.188 0.008 143.9 * 0.3
	12533+2115 STF1687 8695 2010.387 RL	20 1.132 0.008 192.9 * 0.5	Hei1997	0.10	-2.8
	12564-0057 STT256	8708 2010.327 R	20 1.072 0.008 99.0 0.3
	12574+3022 STF1696 8716 2010.324 W	20 3.631 0.056 203.7 * 0.4 Diffuse
	13007+5622 BU1082	8739 2010.324 RL	20 1.069 0.010 101.8 * 0.3	Sod1999	0.01	-2.2
	13100+1732 STF1728 8804 2010.445 R	10 0.633 0.003 12.3 0.3	Mut2010b	-0.01	0.1
	"	"	"	"	"	"	"	"	" "	WSI2006b	-0.01	0.0
	13101+3830 BU608BC 8805 2010.387 RL	20 1.288 0.015 267.4 * 0.6
	13120+3205 STT261	8814 2010.387 R	20 2.554 0.013 338.4 * 0.5
	13166+5034 STT263	8843 2010.393 W	20 1.743 0.009 136.1 * 0.3
	13207+0257 STF1734 8864 2010.387 R	20 1.106 0.012 174.4 * 0.8
	13235+2914 HO260	8887 2010.393 W	20 1.573 0.011 85.5 * 0.3	WSI2004a	-0.06	0.9
	13288+5956 STF1752 8919 2010.396 W	20 0.968 0.017 107.9 * 0.4
	13329+4908 STF1758 8940 2010.396 W	20 3.389 0.017 291.6 * 0.3 Elong.
	13340+0847 A1792	8946 2010.385 W	20 0.713 0.030 306.4 * 0.7

Table 2 .

 2 Characteristics of the filters used for the measurements of Table1.

	Name	Identification	λc	∆ λ
			(nm)	(nm)
	V	ORIEL/57581	530	57
	R	ORIEL/57621	644	70
	RL	ORIEL/57661	743	69
	I	ORIEL/57701	855	67
	W	ICCD alone	650	420

Table 3 .

 3 Objects with discrepant quadrants

	Name	Q Filter ∆m V	Spectral type
	ADS 2253	2	W	0.1	A3
	ADS 4208	2	R	0.05	B8 IV
	ADS 6963	1	W	0.04	A2
	ADS 9324	1	W	0.2	F2

Table 4 .

 4 New orbital elements of ADS 5469, 8128, 8239, 10345, 10850, and 11170.

	ADS	Ω 2000	ω	i	e	T	P	n	a	A	B	F	G
		( • )	( • )	( • )		(yr)	(yr) ( • /yr)	( ′′ )	( ′′ )	( ′′ )	( ′′ )	( ′′ )
	ADS 5469	106.3 146.3 41.3	0.891 1935.683 349.21 1.0309	1.06 -0.17657 -0.97044	0.80096 -0.37855
		±20.3 ±19.3 ±27.9 ±0.0428 ±4.82	±180 ±0.531 ±0.059
	ADS 8128 184.30	2.6	53.9	0.881 2010.223 551.4 0.6529 2.624 -2.60866 -0.26648	0.23450 -1.53119
		±1.42 ±6.0 ±2.5 ±0.015 ±0.194 ±151 ±0.179 ±0.037
	ADS 8239	42.3	327.1 53.3	0.748	1914.42 706.0 0.5099	1.69	1.39998	0.54176	0.10759	1.22864
		±1.32 ±2.3 ±1.9 ±0.046	±0.58	±191 ±0.138 ±0.09
	ADS 10345 282.85 193.31 142.2 0.5139 1946.19 812.0 0.4434	4.48 -0.17482	4.42758	3.58503 -0.23915
		±0.80 ±0.83 ±1.3 ±0.029	±0.72 ±70.5 ±0.039 ±0.03
	ADS 10850 28.2	111.1 124.2	0.582	1799.9 1276.6 0.2820	1.14 -0.07946 -0.72332 -1.05000 -0.30034
	ADS 11170 140.5	41.7 133.4	0.884	1930.1 1460.0 0.2466	1.15 -0.32820	0.95175	0.96556 -0.03138

Table 5 .

 5 ADS 5469: O-C residuals of our new orbit (after 1990.0).

	Epoch	∆ρ (O-C) ∆θ (O-C) Observer
		( ′′ )	( • )
	1991.093	-0.06	-3.7	WSI
	1991.250	0.05	-0.5	HIP
	1991.820	0.05	-0.1	TYC
	1993.120	-0.08	3.3	Hei
	1995.120	0.06	4.2	Hei
	1997.140	0.16	2.7	Hei
	2005.942	-0.00	0.2	Hor
	2005.942	-0.00	0.2	Hor
	2007.012	-0.01	-0.2	Hor
	2007.198	-0.00 P	-0.7 P Pru
	2007.826	-0.00	0.6	Hor
	2008.773	-0.01	-0.7	Tok
	2009.262	-0.02	-0.5	Tok
	2010.201	-0.05 P	-0.8 P Sca

Table 6 .

 6 ADS 8128: O-C residuals of our new orbit (after 2006.0).

	Epoch	∆ρ (O-C) ∆θ (O-C) Observer
		( ′′ )	( • )
	2007.277	-0.01 P	0.1 P Pru
	2007.288	-0.02 P	-0.3 P Pru
	2007.290	0.00	2.7	Alz
	2007.311	0.01	1.4	Hrt
	2009.207	0.01 P	-0.2 P Sca
	2009.260	0.01	-0.2	Tok
	2009.265	-0.00 P	1.1 P Sca
	2009.330	0.01 P	0.5 P Sca
	2009.349	-0.01 P	0.8 P Sca
	2009.382	0.01 P	0.1 P Sca
	2009.385	0.01 P	0.2 P Sca
	2009.390	0.00 P	-0.1 P Sca
	2009.396	0.01 P	0.3 P Sca
	2010.264	-0.00 P	-0.4 P Sca
	2010.300	0.01 P	0.2 P Sca
	2010.324	0.01 P	0.4 P Sca
	2010.327	0.01 P	1.0 P Sca
	2010.384	0.01 P	-0.3 P Sca
	2010.395	0.01 P	-1.6 P Sca

Table 7 .

 7 ADS 8239: O-C residuals of our new orbit (after 1990.0).

	Epoch	∆ρ (O-C) ∆θ (O-C) Observer
		( ′′ )	( • )
	1991.250	0.04	-0.2 HIP
	1991.860	0.09	-1.1 TYC
	1996.160	-0.04	-1.9 Pri
	1996.370	0.14	-3.0 Hei
	1998.392	-0.15	1.5 WSI
	2000.339	-0.01	-0.4 WSI
	2001.327	-0.04	2.0 WSI
	2007.317	-0.02	1.4 Hrt
	2009.260	0.04	0.7 Tok

Table 8 .

 8 ADS 10345: O-C residuals of our new orbit (after 2007.0). The symbol P indicates PISCO measurements.

	Epoch	∆ρ (O-C) ∆θ (O-C) Observer
		( ′′ )	( • )
	2008.369	-0.03	0.1 Ant
	2008.383	-0.02	-0.5 Ant
	2008.388	0.02	0.4 Ant
	2008.458	-0.15	0.3 WSI
	2008.564	0.01 P	0.2 P Sca
	2008.859	-0.03	0.6 Ary
	2009.304	-0.00	0.5 Ant
	2009.448	-0.02	-0.6 Los
	2009.541	-0.00 P	0.1 P Sca
	2009.608	0.00	-0.2 WSI
	2009.872	0.13	0.0 Ary
	2010.259	-0.02	0.0 Los
	2010.459	-0.00	0.0 Ant
	2010.538	0.00 P	0.2 P Sca
	2010.539	-0.05	0.3 WSI
	2010.723	0.00	0.1 Ant
	2010.865	-0.01	1.4 Ary
	4.1 New orbit of WDS 06487+0737 -A 2731 -
	ADS 5469

ADS 5469, A 2731 (HIP 32650): This couple was discovered in 1913 by Aitken with the 91-cm great refractor of Lick Observatory. In the years following this discovery, a significant motion was immediately suspected, but this ac-

Table 9 .

 9 ADS 10850: O-C residuals of our new orbit (after 2000.0). The symbol P indicates PISCO measurements.

	Epoch	∆ρ (O-C) ∆θ (O-C) Observer
		( ′′ )	( • )
	2000.489	-0.00	-0.2	Doc
	2000.521	-0.02	-0.7	WSI
	2001.610	-0.03	3.0	Ary
	2004.657	-0.01 P	0.0 P Sca
	2005.460	-0.06	2.7	WSI
	2007.655	-0.00 P	-0.4 P Sca
	2008.473	0.02	0.2	Thr
	2008.475	0.00	-0.5	Thr
	2008.659	-0.01 P	-0.1 P Sca
	2009.615	-0.02 P	-0.1 P Sca
	2010.459	-0.02	-1.1	Ant
	2010.669	-0.01 P	-0.9 P Sca
	2010.681	-0.00	-3.6	Ant

Table 10 .

 10 ADS 11170: O-C residuals of our new orbit (after 2000.0). The symbol P indicates PISCO measurements.

	Epoch	∆ρ (O-C) ∆θ (O-C) Observer
		( ′′ )	( • )
	2001.373	0.03	-2.2	Mor
	2005.550	-0.01	0.8	Slm
	2005.589	-0.07	-0.4	WSI
	2005.708	0.00 P	0.8 P Sca
	2007.611	0.01	0.6	Gii
	2007.668	-0.00 P	-0.5 P Sca
	2008.665	-0.01 P	-0.9 P Pru
	2009.431	-0.01	0.1	Orl
	2009.442	-0.01	0.1	Orl
	2010.500	-0.02	-0.2	Los
	2011.628	-0.00 P	1.0 P Sca
	2011.737	0.00 P	-0.5 P Sca
	2011.742	0.00 P	-0.3 P Sca
	tually came from a wrong measure reported between the
	two Aitken measures of 1913.97 and 1921.82 (see

  ).

	4.2 New orbit of WDS 11190+1416 -STF 1527 -
	ADS 8128
	ADS 8128, STF 1527 (HIP 55254): this couple was
	measured for the first time in 1822, by F.G.W. Struve in
	Dorpat with a Troughton meridian circle of 9.5 cm in diam-
	eter equipped since 1821 with an excellent filar micrometer
	built by J. Fraunhofer (Struve, 1827). It has been regularly
	observed since and 268 measurements are now available, if
	we add our six measurements made in 2010. However it
	should be mentioned that the 51 measurements made in
	the 19th century have a rather large scatter, especially in
	angular separation. It should be mentioned that 16 out of
	the 19 observations made after 2006 were obtained by our
	group (see Table

Table 11 .

 11 New ephemerides of ADS 5469, 8128, 8239, 10345, 10850, and 11170.

		ADS 5469	ADS 8128	ADS 8239	ADS 10345 ADS 10850 4.0 0.831 164.4	0.760	319.9
	2015.0	1.383	66.2	0.295	255.1	1.152	161.7	2.495	3.0	0.832	164.2	0.767	319.7
	2016.0	1.393	66.4	0.318	268.5	1.162	162.1	2.514	2.0	0.832	163.9	0.773	319.5
	2017.0	1.402	66.6	0.351	279.8	1.171	162.6	2.533	1.1	0.832	163.7	0.779	319.3
	2018.0	1.411	66.8	0.391	289.0	1.181	163.0	2.553	0.2	0.833	163.4	0.784	319.2
	2019.0	1.421	67.0	0.435	296.4	1.191	163.4	2.573	359.3	0.833	163.2	0.790	319.0
	2020.0	1.430	67.2	0.483	302.3	1.200	163.8	2.593	358.4	0.833	163.0	0.796	318.8
	2021.0	1.439	67.4	0.532	307.2	1.210	164.2	2.614	357.5	0.834	162.7	0.802	318.7

Table 12 .

 12 Physical parameters (̟ dyn , a and M total ) derived from the new orbital elements.

	Name	HIP	m V	∆m V	Spectral type	̟ dyn	̟ HIP		a	M total	Source of ̟ HIP
						(mas ) (mas )	( ′′ )	(AU)	(M ⊙ )
	ADS 5469	32650	7.93	1.3	F8V+G3V	17.1	19.96	1.06	53.1	1.2	ESA (1997)
							±1.57	±0.06	±5.1	±1.3
	"	"	"	"			18.82		56.3	1.5	van Leeuwen (2007a)
							±1.00		±4.3	±1.6
	ADS 8128	55254	6.95	1.2	F9V	31.2	31.17	2.62	84.2	2.0	ESA (1997)
							±1.55	±0.04	±4.4	±1.1
	"	"	"	"			32.53		80.7	1.7	van Leeuwen (2007a)
							±1.39		±3.6	±1.0
	ADS 8239	56641	8.97	0.2	G5	18.6	20.45	1.69	82.6	1.1	ESA (1997)
							±2.88	±0.09	±12.4	±0.8
	"	"	"	"			20.75		81.5	1.1	van Leeuwen (2007a)
							±1.44		±7.1	±0.7
	ADS 10345	83608	5.80	0.03	F5V	36.3	37.08	4.48	120.7	2.7	ESA (1997)
							±0.89	±0.03	±3.0	±0.5
	"	"	"	"			36.45		122.8	2.8	van Leeuwen (2007a)
							±0.46		±1.8	±0.5
	ADS 10850	87427	6.42	0.3	K0III	7.9	7.40	1.14	154.	2.2	ESA (1997)
							±2.0		±42.	±1.8
	"	"	"	"			4.27		267.	12.	van Leeuwen (2007a)
							±1.31		±82.	±11.
	ADS 11170	89237	8.19	0.6	F8V	6.0	5.77	1.15	199.	3.7	ESA (1997)
							±1.24		±43.	±2.4
	"	"	"	"			6.85		168.	2.2	van Leeuwen (2007a)
							±0.93		±23.	±0.9

RAS, MNRAS 000, 000-000

Acknowledgements:

We thank the members of the United States Naval Observatory, Washington DC, for kindly sending on request some lists of measurements of visual binaries. This work has made use of the "Fourth Catalogue of Interferometric Measurements of Binary Stars" (http://ad.usno. navy.mil/wds/int4), the "Sixth Catalogue of Orbits of Visual Binary Stars" (http://ad.usno.navy.mil/wds/orb6), and the Washington Double Star Catalogue (http://ad.usno.navy.mil/wds/wds) maintained at the U.S. Naval Observatory, and the SIMBAD astronomical data base (http://simbad.u-strasbg.fr/simbad) operated by the Centre de Données Astronomiques de Strasbourg (France).