Catalogues, parameters and distributions of orbital binaries
CCatalogues, parameters and distributions of orbital binaries
Oleg Malkov , Dmitry Chulkov Institute of Astronomy of the Russ. Acad. Sci., 48 Pyatnitskaya Street, Moscow 119017, Russia Faculty of Physics, Moscow State University, Moscow 119992, Russia
Abstract: The most complete list of visual binary systems with known orbital elements is compiled. It is based on OARMAC and ORB6 data and contains 3139 orbits for 2278 pairs. A refined subset of high quality orbits with available distance was also compiled. Relations between and distributions along different observational parameters are constructed, and an analysis of selection effects is made. Dynamical, photometric and spectral masses of systems are estimated, and reasons for discrepancies between them are discussed. d = M +M = a /(π P ) (1) where P is the orbital period (in years), M are the masses (in solar mass), a and π are the semi-major axis and the parallax (both in arcsec), respectively. The masses of the components can also be estimated from observed photometry, trigonometric parallax, and a mass-luminosity relation. The appropriate formula is M = f MLR (m + 5 lg π + 5 - A(l,b,π)), (2) where m are the apparent magnitudes, A is the interstellar extinction value, and f MLR is the mass-luminosity relation depending on the stellar luminosity class (LC). It is worth noting that the conformity of components to the mass-luminosity relation (in V band, in particular) is likely not exact, especially at the lower part of the main sequence, because of the stellar evolution and chemical abundance variations (Bonfils et al. 2005). Another route requires spectral classification of both components and reliable mass-spectrum relation, f
MSR : M = f
MSR (SpType ). (3) Let us call the masses, determined with Eqs. (1), (2) and (3), dynamical (M d ), photometric (M ph ), and spectral (M sp ), respectively. We computed the photometric mass for every pair and dynamical mass for every orbit. In the case of multiple orbits, we compared their values to choose the orbit that yields the most reliable mass. For photometric mass estimation, we used the MLR of Malkov (2007) for upper-MS, Henry and McCarthy (1993) for lower-MS, and Henry et al. (1999) for lowest masses. Subgiants and early-type (O-F6) giants were considered to be 1 mag brighter than dwarfs (Halbwachs 1986). Lastly, for the few remaining late-giant and supergiant stars, photometric masses were estimated with Tables II and VI of Straizys and Kuriliene (1981). Pairs with unknown LC were considered to be MS-systems. The LC of secondary component, when unknown, was considered to be the same as for the primary. If the secondary magnitude was unknown (35 of 652 systems, see Appendix A), equal brightness of components was ssumed, and, consequently, an upper limit for photometric mass was estimated. The photometric mass is absent in 17 cases, where the estimated absolute stellar magnitude is inappropriate for a given LC. We also calculated the dynamical mass uncertainty. OARMAC does not contain uncertainties of P and a, and we estimated the following: σ P to be 5%, 10%, 20% for A, B, C quality grades, respectively, and σ a to be 3%, 6%, 12% for A, B, C quality grades, respectively. As can be seen from Eq. (2), the absolute magnitude uncertainty depends mainly on the parallax uncertainty, and for the vast majority of our systems it does not exceed 0. m
3. However, the main factor that contributes to the photometric mass uncertainty for a given absolute magnitude is the mass-luminosity relation accuracy. For intermediate mass MS-stars, the MLR slope and its standard deviation value (see, e.g., Malkov 2007) produce about 0.06 for the log M uncertainty (i.e. mass uncertainty is about 15%). For stars of other LCs, however, we estimate that uncertainty to be 2-3 times worse, mainly because of higher MLR standard deviation values. Spectral masses were estimated from Table VI of Straizys and Kuriliene (1981). If the secondary spectral type was unknown, the listed spectral mass is the primary mass (i.e., it represents a minimum mass of the system). Main sequence is assumed if the luminosity class was unknown, which seems to be a reasonable assumption for our relatively nearby orbital binaries. Spectral mass is not available for six entries, which include Am, Be, L stars, and stars with inexact classification, e.g. M:. We estimated the spectral mass uncertainty to be about 20% for MS-stars (we refer to Straizys and Kuriliene (1981) figure 1) and about 30% for stars of other luminosity classes (see, e.g., Piskunov (1977) and the corresponding discussion in Straizys and Kuriliene (1981)).
Fig. 1 (a,b,c). Dynamical (Eq.1) versus photometric (Eq.2) mass of orbital binaries in different scales. ig. 2 (a,b,c). Dynamical (Eq.1) versus spectral (Eq.3) mass of orbital binaries in different scales.
In Figs 1-2 the stars we investigated have been plotted on the planes of the dynamical vs. photometric vs. spectral mass. First of all, we note that some systems exhibit obviously incorrect dynamical masses and/or a large discrepancy between dynamical, photometric and spectral masses (lists of such systems are given in Appendices C and D). The most obvious reason for that are poorly known parallaxes and/or orbital elements as well as the presence of a third still-undetected body (or subcomponents) that lead to dynamical mass overestimation (Tamazian et al. 2006). Among other reasons for a discrepancy between dynamical and photometric masses, we can mention incorrect spectral classification, interstellar extinction underestimation, and variability of components. These systems need further study. It should be noted that, for example, parallaxes of distant visual binaries can be refined by using relevant orbital and spectral data (Docobo et al. 2008). We should note also that Cvetkovic and Ninkovic (2010) in their study of Main Sequence binaries from ORB6 also found large deviations of the dynamical masses from the photometric ones for some systems. All reasons for a discrepancy between masses, determined with Eqs. (1), (2) and (3), are listed in Table 1. An influence of input parameters to the determined masses is indicated. Observing a discrepancy between particular types of masses (dynamical, photometric or spectral) in Figs 1-2, one can estimate from Table 1, which input parameters have incorrect values. parameter M d M ph M sp parameter M d M ph M sp Spectral (temperature) class no no yes Semi-major axis, period yes no no Luminosity class no yes yes Variability no yes yes/no * Interstellar extinction no yes no Unresolved binarity yes yes no Parallax yes yes no Third body yes no no
Table 1. Influence of various parameters and stellar characteristics on the values of dynamical, photometric and spectral masses. (*) – depends on variability type.
4. Statistical relations, parameter distributions and selection effects The final list (so-called system list) contains 652 systems. This list is large enough for a reliable statistical analysis of this class of binary systems. It is instructive to obtain relations between different observational parameters of orbital binaries. They can be used for statistical purposes. A period-axis relation is presented in Fig. 3.
Fig. 3. Statistical period – angular semi-major axis relation for orbital binaries.
It can be seen from Fig. 3 that the majority of systems satisfy the following approximate relation: log P (y) = 1.35 log a” + 2, and it is valid for systems with a<1”. These systems are relatively distant and, consequently, represent a mixture of masses, which broadens the relation. Systems with larger angular semi-major axes are nearby and the least massive systems, and both these circumstances lead to the flattening of the relation. In zeroth approximation one can consider all systems with a>1” to have periods of the order of 100 years. It is well known that observational selection limits the possibilities of observation of binary systems of different types and can seriously distort the results of the analysis. Here we analyze the main selection effects that determine the sample of orbital binaries. ig. 4. Distribution of stars with respect to the angle of orbital inclination I (red). Catalogued values in the 90-180 degrees range are converted in the 0-90 degrees range. The smooth green curve corresponds to the distribution in the case of random orientation of the planes of the orbits: dN ~ sin I di. Binary with an edge-on orbit is harder to discover, and it leads to a deficiency of highly inclined orbits (i around 90 degrees) in the figure.
Fig. 5. Distribution of stars with respect to the angle of longitude of the ascending node Ω (red). The horizontal green curve corresponds to the distribution in the case of random orientation of the planes of the orbits: dN ~ Ω dΩ. Number of orbits with Ω>180 in the figure is small due to the following reason. In the case of orbital binary without additional (e.g., radial velocity) data it is not possible to determine which node is ascending and which is descending, and in this case Ω is recorded in the 0-180 degrees (rather than 0-360 degrees) range. This tendency is apparently increasing toward Ω=180, as it can be seen in the figure (a slight increasing of the distribution between 0 and 180 degrees). ig. 6. Distribution of stars with respect to the angle of argument of periapsis ω (red). The horizontal green curve corresponds to the distribution in the case of random orientation of the planes of the orbits: dN ~ ω dω. Maxima at ω around 90 and 270 degrees mean that binary observed from a “pointy” end is easier to discover than one observed from a “blunt” end (in the latter case a wide enough binary can be mistakenly considered as an optical pair and excluded from the statistics). It does not concern low-e or low-I orbits. Moreover for circular orbits (with eccentricity equal to zero) ω is undefined and, by convention, set equal to zero. Such orbits contribute to a peak around ω=0 in the figure.
In Figs 4-6 we present distributions of 652 pairs from the system list along the Campbell elements: orbital inclination (i), longitude of the ascending node (Ω) and argument of periapsis (ω). It can be seen that i- and ω- distributions of catalogued orbital binaries are distorted by selection effects. Contrary, discoveries of orbits with different Ω are equiprobable events, however, Ω is more often recorded in the 0-180 degrees. Distributions of randomly aligned systems are given for comparison, they are modeled with the program described in (Malkov 2002). The overwhelming majority of binaries in the system list have apparent magnitudes of the primary components m ≤ 9. m
5. Also, the majority of systems have semi-major axes a" ~ 0".1 -1": closer systems are not resolved by instruments with aperture D ~ 100 cm, which are generally used to detect visual binaries, and wider systems usually do not exhibit significant orbital motion.
Fig. 7. Distribution of stars with respect to the magnitude difference of the components dm. Few pairs with dm>4m and with unknown secondary brightness (see Appendix A) are not shown.
The distribution with respect to the magnitude difference of the components dm (Fig. 7) gives, in principle, the possibility of estimating the mass ratio of the components. However, the dm distribution is also strongly distorted by selection. First, among the closest binaries, only pairs with comparable luminosities are detectable. Second, among binaries with the primary magnitude close to the limiting magnitude, only pairs with small dm are detected, and pairs with the secondaries fainter than the limiting magnitude appear as single stars and do not contribute to statistics.
Fig. 8. Difference between the magnitudes of the components vs the semi-major axis of the orbit (upper panel) and the magnitudes of the primaries (lower panel). The dashed lines mark the area that satisfies the definition of selected systems, a" ~ 0".1 -1", m ≤ 7. m m . In Fig. 8, the stars we investigated have been plotted on the plane of the semi-major axis of the orbit (upper panel) and the magnitudes of the primaries (lower panel) vs the difference between the magnitudes of the components. It can be seen that the mean value of dm increases with increasing a" and increases with increasing brightness. One can see that, for systems with a" < 0".1, or a" > 1", or m > 7. m
5, or dm > 2 m the set is definitely incomplete. Note, however, that the restricted sample (systems with a" ~ 0".1 -1", m ≤ 7. m
5, dm ≤ 2 m ) surely does not include all visual binaries with corresponding restrictions posed on their magnitudes and angular separations. For example, a distant (d ~ 2.5 kpc) wide (a ~ 500 A.U) gravitationally bound system can be observed like a visual binary, but its orbital motion is so slow that it would not exhibit evident orbital motions and, hence, would not contribute to the sample as orbital binary. For 207 systems of the 652 listed systems, a", m1 and dm satisfy these criteria. The distributions of these selected 207 systems, together with those of all 652 systems, among dynamical mass, period, semi-major axis, and eccentricity are shown in Figs. 9, 10, 11 and 12, respectively. Figs. 9-12. Dynamical mass, period, semi-major axis, and eccentricity distributions of all 652 listed systems (grey bars) and the selected 207 systems with a" ~ 0".1 -1", m ≤ 7. m m (black bars). The period distribution is shown in Fig. 10. Raghavan et al. (2010) surveyed the solar-type stars in the solar neighborhood and obtained log-normal distribution with a peak at about 300 years. Our sample is heavily biased for these long period binaries due to the lack of an extended set of observations. Opik (1924) found that the semi-major axis distribution of binaries follows the f(a) ~ 1/a law. Poveda et al. (2007) confirmed this distribution for wide binaries (a>100 A.U.). Our sample is, inevitably, restricted at large a (see Fig. 11) due to long orbital periods of wide binaries, hence the number of binaries significantly drops at a>50 A.U. The distribution has a broad, nearly flat peak at 10
5, and magnitude difference dm ≤ 2 m eclipsing binary components cannot be used to derive the stellar initial mass function, and, consequently, new observational data is needed, see Malkov (2003) for details), - be close enough (to facilitate RV observations); - have similar components (to be observed as an SB2 rather than an SB1 system). Selected orbital binaries, needing RV data: dynamical mass is 3 solar mass or more, π>20 mas, dm<3 m (format: WDS designation – dynamical mass – its error –visual magnitude of primary component – visual magnitude of secondary component – parallax – its error). WDS name M d σM d V1 V2 π σπ 00084+2905 5.19 0.16 2.22 4.21 33.62 0.35 02366+1227 19.08 0.98 5.68 5.78 28.79 0.43 04287+1552 4.46 0.28 3.74 4.86 21.69 0.46 06098-2246 3.13 0.32 6.56 6.57 20.39 0.69 07171-1202 4.05 1.71 8.61 6.96 21.60 2.11 07346+3153 5.43 0.96 1.93 2.97 64.12 3.75 09307-4028 3.70 0.75 3.91 5.12 53.15 0.37 10279+3642 3.37 1.05 4.62 6.04 21.19 0.50 11037+6145 5.56 0.33 2.02 4.95 26.54 0.48 11053-2718 3.79 0.14 5.70 5.70 23.13 0.29 11190+1416 3.04 1.02 7.01 7.99 32.53 1.39 12415-4858 7.30 3.02 2.82 2.88 25.06 0.28 13372-6142 3.39 1.41 5.98 7.22 27.99 0.58 15278+2906 3.08 0.24 3.68 5.20 29.17 0.76 15427+2618 4.18 0.28 4.04 5.60 22.33 0.50 17104-1544 7.05 0.49 3.05 3.27 36.91 0.80 18092-2211 4.04 1.32 9.58 9.92 30.49 1.20 19026-2953 5.24 0.37 3.27 3.48 36.98 0.87 21137+6424 3.85 0.23 7.21 7.33 23.39 0.42 . Orbital binaries where difference between photometric and dynamical masses exceeds three-sigma limits (format: WDS designation – dynamical mass – its error – photometric mass). WDS name M d σM d M ph WDS name M d σM d M ph d σM d M sp WDS name M d σM d M spsp