4 Orbit calculation

The orbits were computed using the method of Kowalsky (1873) and Hellerich (1925). The error bars calculation is described in Scardia (1983) for errors on the orbital elements, and in Scardia (1984b) for errors on the sum of masses of the stars.

  

Figure 2: Separation and angular positions versus date of the couple BU 524 AB. Measurements are represented by the points. The last mesurements are interferometric ones. The solid line is the predicted position from our short-period orbit, dashed lin e is the solution given by our long-period one (see text)

The orbital elements are presented below for each couple. WDS identification (equinox 2000) (Mason 1998) is given for each star together with discover name and ADS catalog number. Orbits are plotted in Fig. 3 and Fig. 4. History of all measurements for each star are presented at the end of the paper. We computed eight orbits. Two of them may be considered as reliable (BU 524 and A 1777); a mass estimation and detailed dicussion is given below. Th six others are premature and the orbital elements are summarised in Table 2.

• WDS 02537+3820 - BU 524 - ADS 2200 AB: This close double star (separation $\simeq 0 \hbox{$.\!\!^{\prime\prime}$}1$) with $\Delta m \simeq 1$ is a difficult object for visual observations. This explains the large residuals on the orbit plot. 160 measurements have been published since 1878, 129 have been retained for the orbit computation. On the plot of the new orbit one can see the lack of data near the periastron where the predicted separation is under $0\hbox{$.\!\!^{\prime\prime}$}1$. In such a case this star is a good target dedicated for speckle interferometry techniques as demonstrated by the three points near periastron (two values comes from the CHARA, the third is ours). According to our orbit, the next periastron will occur in October 2027.

  
  

  
  
  

  
  Dynamic parallax (Couteau 1978) could not be computed since star A is not a main-sequence star (spectral class F4IV). This does not agree with published masses of stars A and B estimated from their spectral classes (Tokovinin 1997): ${\cal M}_A=1.3 8 \; {\cal M}_\odot$ and ${\cal M}_B=1.15 \; {\cal M}_\odot$. If we suppose that the spectral class F4IV for the primary is well estimated, it might be possible that this system contains other stars to be discovered.

  Van den Bos (1938) calculated two sets of orbital elements, a short period one (P=31.6 yr) and a long period one (P=63.1 yr), the second being justified by quadrant uncertainties and the fact that the observed separation was always greater than $0\hbox{$.\!\!^{\prime\prime}$}1$, whereas the short-period orbit predicted a very close periastron of about $0 \hbox{$.\!\!^{\prime\prime}$}002$. We then computed a long period orbit with the following orbital elements: T=1988.066, P=62.851 yr, e=0.0286, $a=0\hbox{$.\!\!^{\prime\prime}$}211$, $\Omega=129.79^\circ$, $\omega=51.47^\circ$, $i=133.73^\circ$. Residuals in separation and in position angle are plotted in Fig. 2 as a function of time. As can be shown on this figure and on the orbit plot, this long period orbit does not fit the observations close to the periastron when the star velocities are high (especially the last speckle measurements). From an astrophysical point of view, the total mass of the system given by this orbit is too small (0.89 $\; {\cal M}_\odot$). This long period orbit is then to be rejected.

  
  

  
  
  

  
  

• WDS 11551+4629 - A 1777 - ADS 8347 AB: Our orbit does well fit recent interferometric observations near periastron which present very small residuals. This is another example of the valuable contribution of speckle techniques. The star A is a spectroscopic binary of separation 0.222 mas ( Tokovinin 1997). The star B seems (up to now) single. There are 2 more stars in the quintuple system whose separations are (Tokovinin 1997): AB $\times$ C $\simeq 4\hbox{$.\!\!^{\prime\prime}$}$ and ABC $\times$ D $\simeq 63\hbox{$.\!\!^{\prime\prime}$}$.

  
  

  
  
  

  
  

  Due to large error bars of semi-major axis a and period P, the total mass is very uncertain. For information we found in Tokovinin (1997):

  • ${\cal M}_{Ab}= 1.9 \; {\cal M}_\odot$
  • ${\cal M}_{Aa}= 1.8 \; {\cal M}_\odot$
  • ${\cal M}_B= 1.5 \; {\cal M}_\odot$
  giving a total mass ${\cal M}_A + {\cal M}_B=5.2 \; {\cal M}_\odot$ which agrees with our dynamic parallax.

  
  

  
  
  

  
  

• Other orbits Orbital elements for the stars A 2681, A 2477, BU 1077, COU 612, COU 321 and STF 2556 are given in Table 2. These orbits are premature and may be considered as equally tentative as previous ones. Some are based upon a few number of data (17 measurements for A 2681, 20 for Cou 321), some have very small magnitude difference causing quadrant errors. These orbits have to be improved by further observations.

  

Figure 3: Orbit of BU 524AB, A 2681, A 2477 and BU 1077. Our orbit is plotted in solid line on the graph, previous orbits in dashed line. For BU 524AB the long-period solution is plotted as well (see text). As usual North is down and East is right. Line of apsides of the new orbit is drawn in solid line. Measurements are plotted as points connected to their predicted value ("O-C'' lines). Points corresponding the first and last measurement are labelled such as "1971'' and "1997''. The quadrant convention used for the compu tation of orbital elements of the star A 2681 is not the same for our orbit and the previous one; comparison has to be made carefully

  

Figure 4: Orbit of A 1777AB, Cou 612, Cou 321 and STF 2556. As for Fig. 3, our or bit is drawn in solid line and previous ones in dashed lines. As it was the case for A 2681 in the previous plots, quadrant convention used for the computation of orbital elements of Cou 321 is not the same for our orbit and the previous one

  

Table 2: Orbital elements for the 6 stars A 2681 (WDS 06575+0253), A 2477 (WDS 09245+1808), BU 1077 (WDS 11037+6145), Cou 612 (WDS 15390+2545), Cou 321 (WDS 19180+2012) and STF 2556 (WDS 19394+2215)