The paper discusses various stellar parameters and the ultimate fate of the binary star system CW Canis Major. Using observational data of the radial velocities and apparent visual magnitudes the individual masses and radii have been determined. These values have allowed us to estimate the spectral classification of the stars.
Abstract: The following paper discusses various stellar parameters and the ultimate fate of the binary star system CW Canis Major. Using observational data of the radial velocities and apparent visual magnitudes the individual masses and radii have been determined. These values have allowed us to estimate the spectral classification of the stars.
I. INTRODUCTION
Binary star systems have interested astronomers since the beginning of telescopic astronomy. The first visual binary star system, Mizar( U Ma) was discovered by the Italian astronomer J. B. Riccioli around 1650. For the next century only six more visual binary systems were observed: Ori(Huygens, 1656), Ari(Hooke, 1664), Cru (Fontenay, 1685), Cen (Richaud, 1689) Vir, and Castor (Bradley and Pound, 1718, and 1719 respectively). The aforementioned stars are amongst many others discovered in the 17th and 18th centuries as double stars, that is optical doubles; they are particularly singled out because they were later discovered to be true binary systems. Although Newton’s theory of universal gravitation allowed for the possibility of true binary systems their existence was not widely accepted until the early 19th century due in large part to the work of English astronomer William Herschel. Herschel himself did not seem to believe that binary systems existed, he was interested in double stars because he thought that their different parallaxes could give an idea of their distances, and help his goal of understanding structure of the heavens. In his Catalogue of 500 New Nebulae, Nebulous Stars, Planetary Nebulae, and Clusters of Stars; With Remarks on the Construction of the Heavens Herschel describes a binary system as distinct from a mere optical double:
“If a certain star should be situated at any, perhaps immense distance behind another, and but very little deviating from the line in which we see the first, we should then have the appearance of a double star. But these stars, being totally unconnected, would not form a binary system. If, on the contrary, two stars really should be situated very near each other, and at the same time so far insulated as not to be materially affected by the attractions of neighbouring stars, they will then compose a separate system, and remain united by the bond of their own mutual gravitation towards each other. This should be called a real double star; and any two stars that are thus mutually connected, form the binary sidereal system which we are now to consider.”
He later comments: “... as I shall soon communicate a series of observations made on double stars, whereby it will be seen, that many of them have actually changed their situation with regard to each other, in a progressive course, denoting a periodical revolution round each other; and that the motion of some of them is direct, while that of others is retrograde.” These observations are very important in the history of science, as put by Virpi S. Niemela in his A Short History and Other Stories of Binary Stars: “[binary stars are] are probably the first observational evidence of the universality of Newton’s law of gravitational attraction.” By the turn of the 19th century binary stars were a very important object of astronomical research, and in fact the discovery of binary systems became something of a competition amongst astronomers.7,9
Since the only way to directly measure the mass of stellar objects is to consider their gravitational interactions with other objects observations of binary systems yield the only methods of directly measuring the mass of a star. Binary systems are broken up into several subtypes: visual binaries: where both stars can be resolved separately, astrometric binaries: where one resolvable star revolves around an unseen companion, eclipsing binaries: where the two companions eclipse each other periodically, spectrum binaries: where the two companions with independent spectra. Armed with mass measurements from binary systems we can compare other stellar quantities with those of non-binary stars and make educated predictions about their masses.
The star studied in the following pages, CW Canis Majoris, is an eclipsing binary. Using plots of apparent visual magnitude and radial velocities of both primary and secondary as a function of orbital phase the individual masses and apparent magnitudes can be calculated, as well as the individual radii. Using these values a variety of quantities can be estimated and even the ultimate evolutionary fate of the system. The light curve data comes from All Sky Automated Survey (ASAS) taken at Fernbank Science Center Observatory throughout 1974-1976 as mentioned in Photometric Study of CW Canis Majoris by Richard M. Williamon , the calculation of the orbital period is taken from the same paper, and the radial velocity data were taken using the Reticon and Digicon at the 2.7 m telescope of McDonald Observatory as mentioned in Absolute Masses and Dimensions of Eclipsing Binaries. III. CW Canis Majoris by Claud H. Lacy. Many of the quantities calculated in this paper by our elementary methods are also calculated by Lacy, and Williamon using far more precise and sophisticated methods, therefore these values will be taken as accepted values to compare with. As already mentioned the methods of analysis that will be used are elementary, and are explained in An Introduction to Modern Astrophysics 2nd Edition by Bradley W. Caroll and Dale A. Ostlie.3,8,11
II. ANALYSIS OF LIGHT CURVE AND RADIAL VELOCITY MEASUREMENTS
As mentioned before the methods of analysis we will use are elementary and we will predicate our calculations on a couple assumptions. First we will assume that the primary eclipse is total, that is that only the secondary (the star that is eclipsing) contributes light to the light curve. We will assume that the angle of inclination of the entire system is 90°, which is a reasonable assumption since we are dealing with a binary system.
According to Williamon the primary eclipse occurs at HJD 2452743.58, and the orbital period is 2.11797737 days long. With this information along with the apparent visual magnitude data we can plot the light curve for CW Canis Majoris, shown below in Figure 1. We can find the orbital phase of the primary eclipse and using our assumption that the eclipse is total we can find the visual apparent magnitude of the secondary to be . We can also find the apparent magnitude of the system outside of eclipses. As it is apparent from the graph there is variability in the measurements of the apparent magnitude of the system. To correct this we will take as our value of the apparent magnitude of the system the average of all the points from orbital phase 0 to 0.3; the values for all the points in the light curve are shown in Table 1(found at the end). By taking such a large sample size (98 data points) we can ensure the relative accuracy of the calculation; after taking this average we find . With these two values we can easily calculate the visual magnitude of the secondary. For any two arbitrary stars we have the following relations:3,11
With these relations we can calculate the flux ratios of secondary and system with that of the sun, and then since fluxes, unlike magnitudes, are additive we can find the apparent magnitude of the primary; we have:
Abbildung in dieser Leseprobe nicht enthalten
Figure 1.
Abbildung in dieser Leseprobe nicht enthalten
Table 2.
The radial velocity measurements provided by Lacy allow us to find the center of mass of the binary system. This is accomplished by finding the intersection of the two radial velocity curves. This method yields a center of mass radial velocity, of17.6 . We can determine the velocity semi-amplitudes of each star, and (refer to figure 2),and using these values find the mass ratio:8
Abbildung in dieser Leseprobe nicht enthalten
Furthermore, since this is an eclipsing binary we can reasonably assume that the inclination is essentially 90°, and neglect the eccentricity of the orbits. If we do we can find the semi-major axes of each star, and , and therefore the total semi-major axis, , using the following equations:
Abbildung in dieser Leseprobe nicht enthalten
If we consider the general form of Kepler’s Third Law we can solve for the sum of the masses and using this value along with the mass ratio, computed above, we can find the masses of the individual stars:1
(Kepler’s Third Law, General Form)
Abbildung in dieser Leseprobe nicht enthalten
Figure 2.
Utilizing both sets of data we can find estimations for the individual radii. These are estimations because we will treat the tangential velocities of the stars as if they are independent of orbital period. We are confident in this approximation because of the regular sinusoidal shape of our radial velocity curves. This suggests that the eccentricities of the orbits are relatively low; the significance of this is due to the conservation of angular momentum. Since angular momentum must be conserved the tangential velocities of orbiting bodies periodically speed up and slow down according to their distance from the center of mass of the system. Since we are treating the eccentricities of these stars as nearly zero we can assume that the radial velocity semi amplitudes equal the tangential velocity of the stars as they go through their orbits. These values along with analysis of our light curve will allow us to find reasonable approximations for the individual radii; consider the following equations:3
Analyzing the light curve allows us to find the duration of the respective eclipses refer to figure 3 below.
Abbildung in dieser Leseprobe nicht enthalten
Figure 3.
[...]
-
Upload your own papers! Earn money and win an iPhone X. -
Upload your own papers! Earn money and win an iPhone X. -
Upload your own papers! Earn money and win an iPhone X. -
Upload your own papers! Earn money and win an iPhone X. -
Upload your own papers! Earn money and win an iPhone X. -
Upload your own papers! Earn money and win an iPhone X.