# Stars/Binaries
## Theoretical binaries

Def. "[t]wo stars that appear to be one when seen with the naked eye, either because they orbit one another (binary stars) or happen to be in the same line of sight even though they are separated by a great distance" is called a double star.
Def. "[a] star that appears as a double due to an optical illusion; in reality, the stars may be far apart from each other" is called an optical double.
Def. "[t]wo stars which form a stellar system, such that they orbit the point of equilibrium of their gravitational fields" is called a double star.
Def. "[a] stellar system that has two stars orbiting around each other" is called a binary star.
“A binary star is a star system consisting of two stars orbiting around their common center of mass. The brighter star is called the primary and the other is its companion star, ... or secondary. Many visual binaries have long orbital periods of several centuries or millennia and therefore have orbits which are uncertain or poorly known.”
Def. "[a] binary star whose components can be visually resolved" is called a visual binary.
"The overall frequency of occurrence of binary stars among the [pre-main sequence] PMS population is at least as large as has been documented for main-sequence stars (Duquennoy & Mayor 1991), that is, certainly greater than 50%."
"Young, stellar-mass binary systems have been found with semimajor axes ranging from 0.02 to 10$^3$ AU (orbital periods ranging from a couple of days to 10$^4$ years), with a binary frequency distribution as a function of semimajor axis that is qualitatively consistent with the log-normal–like distribution found for main-sequence stars. With this evidence in hand, Mathieu (1994) concluded that “binary formation is the primary branch of the star-formation process.”"

## Contact binaries

"[A] contact binary is a binary star system whose component stars are so close that they touch each other or have merged to share their gaseous envelopes. ... Almost all known contact binary systems are eclipsing binaries; eclipsing contact binaries are known as W Ursae Majoris variables, after their type star, W Ursae Majoris."

## Zero-age contacts

“Zero-age contact must be a consequence of star fission under critical angular momentum.” When the angular momentum is too large, the star breaks into a detached binary. When the angular momentum is too small, the star remains as a single star. BH Centauri and V1010 Ophiuchi have zero-age radii and are zero-age contact systems. BH Centauri is an overcontact system.

## Common envelope binaries

"[A] common envelope (CE) refers to a short-lived (months to years) phase in the evolution of a binary star in which the largest of the two stars (the donor star) has initiated unstable mass transfer to its companion star. Breach of the common envelope crosses the Lagrange Point L1 with the donor-star mass beyond the Roche Lobe acting as the third dynamic point in a formerly binary system. Mass transfer is unstable when the radius of the donor star expands more rapidly or shrinks less rapidly than does the binary orbit. Hence, the donor will start mass transfer when it overfills its Roche lobe and as a consequence the orbit may shrink while the star expands, causing it to overflow the Roche lobe even more, which accerelates the mass transfer, causing the orbit to shrink faster and the donor to expand faster, etcetera. This leads to the run-away process of dynamically unstable mass transfer. The result will be the fast expansion of the donor's stellar envelope, which will then engulf the companion star. Hence the name common envelope."

## Overcontact systems

When stars share an envelope the pair may be called an overcontact binary.
BH Centauri is a zero-age overcontact binary system with primary and secondary masses equal to (9.4±5.4) and (7.9±5.4) M$_ʘ$. With later analysis including more recent data, "the mass ratio went from 0.97 to 0.84, and the degree of overcontact went from 21% to 48%."
V1010 Ophiuchi and V701 Scorpii are both overcontact systems. For V1010 Ophiuchi "the masses are 2 and 1 M$_⊙$ while the radii are 2.1 and 1.5 R$_⊙$. The location in the HR diagram suggests that they are zero-age stars, as do the radii".
At the right is a computer generated picture of the overcontact system V701 Scorpii. The overcontact in V701 Scorpii appears to be almost twice as much as in V1010 Ophiuchi.
"Most people would agree that fission is the most probable way to form binary systems, especially the close systems. The angular momentum must be the deciding factor as to whether a gas cloud becomes a single star or a binary system."

## Close binaries

"The [close-binary supersoft source] CBSS model invokes steady nuclear burning on the surface of an accreting white dwarf (WD) as the generator of the prodigious super soft X-ray flux. As of 1999, eight SSXSs have orbital periods between ~4 hr and 1.35 d: RX J0019.8+2156 (MW), RX J0439.8-6809 (LMC), RX J0513.9-6951 (LMC), RX J0527.8-6954 (LMC), RX J0537.7-7034 (LMC), CAL 83 (LMC), CAL 87 LMC), and 1E 0035.4-7230 (SMC)."

## Symbiotic binaries

"A symbiotic binary star is a variable binary star system in which a red giant has expanded its outer envelope and is shedding mass quickly, and another hot star (often a white dwarf) is ionizing the gas. Three symbiotic binaries as of 1999 are SSXSs: AG Dra (BB, MW), RR Tel (WD, MW), and RX J0048.4-7332 (WD, SMC)."

## Binary star formations

"Binary and multi-star systems consist of two or more stars that are gravitationally bound, and generally move around each other in stable orbits. When two such stars have a relatively close orbit, their gravitational interaction can have a significant impact on their evolution." "[T]he masses of stars [are] determined from computation of the orbital elements."
"Large lengths, such as the radius of a giant star or the semi-major axis of a binary star system, are often expressed in terms of the astronomical unit (AU)—approximately the mean distance between the Earth and the Sun (150 million km or 93 million miles)."
"The most common multi-star system is a binary star, but systems of three or more stars are also found. For reasons of orbital stability, such multi-star systems are often organized into hierarchical sets of co-orbiting binary stars. Larger groups called star clusters also exist. These range from loose stellar associations with only a few stars, up to enormous globular clusters with hundreds of thousands of stars."
"If components in binary star systems are close enough they can gravitationally distort their mutual outer stellar atmospheres. In some cases, these close binary systems can exchange mass, which may bring their evolution to stages that single stars cannot attain. Examples of binaries are Sirius and Cygnus X-1 (of which one member is probably a black hole). Binary stars are also common as the nuclei of many planetary nebulae, and are the progenitors of both novae and type Ia supernovae."
"The observation of binaries consisting of pre-main sequence stars, supports the theory that binaries are already formed during star formation."
"The question of binary star formation is now regarded as a central unsolved issue in star formation, given the observational evidence that the majority of stars are in binary systems both during the main sequence ... and pre-main sequence stages". "[T]he 'theory gap' currently separating theoretical models from the observations they purport to describe is the major unsolved problem in this area."

## Bifurcations

Def. a "division into two" is called bifurcation.
"Bifurcation means the splitting of a main body into two parts. ... Bifurcation theory [is] the study of bifurcation in dynamical systems ... [T]he forking of a river into its tributaries [is referred to as river bifurcation] ... [A bifurcation can be a] [f]alse dilemma ... in which two alternative statements are held to be the only possible options when there are more options ... [In cybernetics, a bifurcation is] when a system switches from one stable state to another, [where] minor fluctuations may play a crucial role in deciding the outcome".
“About a century ago, Liapounov and Poincaré found that the sequence of Jacobi’s ellipsoids branches towards pear-shaped configurations for sufficiently high rotation.”
Equatorial break-up and instability occur at the point of bifurcation.
“Bifurcation of protostars can occur because of excessive angular momentum either during hydrodynamic collapse (‘fragmentation’), or else after the star has arrived on the HR diagram as a visible, slowly contracting star (‘fission’).”
Spectral types B2-B5 and F3-G2 binaries with orbital periods shorter than 10 or 100 yr may result from the bifurcation of rapidly rotating protostars.
Binary formation by ... bifurcation is difficult to achieve theoretically for compressible viscous gases and may not occur frequently or ever.
Systems of spectroscopic binaries of periods less than 3.6 days "involve separations that are less than 2.5 times the sum of the radii of the components."
These systems "probably have already exchanged mass and no longer have their original mass ratios."
For these systems, we are unable to give their mecahnisms of origin.
Provisionally, "most or all binaries [are] formed in capture processes (including initial star formation in bound systems) and that bifurcation or fission need not have occurred frequently."
For the inclusion of compressibility and viscosity in theoretical calculations, "it is very difficult to produce binaries by fission and only under special circumstances."
“In view of the high frequency of spectroscopic binaries, the formation mechanism must be a frequent one, not a rare occurrence."
For most binaries to result from fission of a star into at least two stars, the expanding separation between the binaries must result in capture rather than expulsion from the system.

## W Ursae Majoris

The W Ursae Majoris "system consists of a pair of stars in a tight, circular orbit with a period of 0.3336 days, or eight hours and 23 seconds. During every orbital cycle, each star eclipses the other, resulting in a decrease in magnitude. The maximum magnitude of the pair is 7.75. During the eclipse of the primary, the net magnitude drops by 0.73, while the eclipse of the secondary causes a magnitude decrease of 0.68. Unlike normal eclipsing binaries, the contact nature makes it impossible to precisely tell when an eclipse of one component by the other starts or ends."
"The two stars in W Ursae Majoris are so close together that their outer envelopes are in direct contact. Hence they have the same stellar classification of F8Vp, which matches the spectrum of a main sequence star that is generating energy through the nuclear fusion of hydrogen. However, the primary component has a larger mass and radius than the secondary, with 1.19 times the Sun's mass and 1.08 times the Sun's radius. The secondary has 0.57 solar masses and 0.78 solar radii."
"The orbital period of the system has changed since 1903, which may be the result of mass transfer or the braking effects of magnetic fields. Star spots have been observed on the surface of the stars and strong X-ray emissions have been detected, indicating a high level of magnetic activity that is common to W Uma variables. This magnetic activity may play a role in regulating the timing and magnitude of mass transfer occurs."

## V1010 Ophiuchi

Def. binaries "in which the stellar components are close enough that proximity effects are important but far enough apart that a large temperature differential may be maintained between the two stars" are called near-contact binaries.
"V1010 Oph ... is one of the brightest and best studied of the near-contact (P=0.66$^d$) binaries." "The period of the binary is known to be decreasing ..., which can be understood in terms of conservative mass transfer (Shaw 1990)." "If this star is truly an evolved system, it may have been in contact previously. ... [I]t is not now in contact".
The temperatures of the two stars have been estimated spectrally as 8200 K and 5671 ± 30 K.
The first image at the right contains two light curves for V1010 Ophiuchi: the top is in yellow at 550.0 nm and the bottom is in blue at 435.0 nm.
The "eclipses are complete and the primary minimum is a transit [...] The system is in contact [shown in the second diagram at the right], with a surface potential near that of the inner contact surface [...] The temperature difference (2529 K) between the primary and secondary is quite large. This suggests that the temperature gradient at the interface must be very steep."
"The appreciable departure between the theoretical light curves and the observations [in the first figure at the right] at the ascending branch of the secondary minimum is due to [a large asymmetry from absorbing gaseous matter]."
"There are two likely ways to form contact systems:
* through star fission with critical angular momentum, i.e. the angular momentum is just right for the star to divide, but not large enough for it to be detached (zero-age contact);
* through mass exchange in which one or both components expand to fill the common envelope during the course of stellar evolution."
"V1010 Oph has essentially a ZAMS radius. Thus this system is likely to be essentially a zero-age contact system [formed through star fission]."

## BH Centauri

"The eclipsing binary system BH Cen is a close (contact) binary in the extremely young galactic cluster Córdoba XXVI (NGC 2944)."
In 1928-30, "from ten light minima ... a (half) period P = 0.395790 7 [days is derived]." This half period estimate becomes the period P = 0.791 581 4 d, for the observations in 1928-30.
From more recent observations around 1977, P = 0.791 616 d. And, from 1979, P = 0.791 592 10 ± 0.000 14.

## Beta Lyrae

"Beta Lyrae is a semidetached binary system made up of a stellar class B7II primary star and a secondary that is probably also a B-type star. The brighter, less massive star (B7II) in the system was once the more massive member of the pair, which caused it to evolve away from the main sequence first and become a giant star. Because the pair are in a close orbit, as this star expanded into a giant it filled its Roche lobe and transferred most of its mass over to its companion. The secondary, now more massive star is surrounded by an accretion disk from this mass transfer, with bipolar, jet-like features projecting perpendicular to the disk. This accretion disk blocks our view of the secondary star, lowering its apparent luminosity and making it difficult for astronomers to pinpoint what its stellar type is. The amount of mass being transferred between the two stars is about 2 × 10$^{–5}$ solar masses per year, or the equivalent of the Sun's mass every 50,000 years, which results in a increase in orbital period of about 19 seconds each year."
"The orbital plane of this system is nearly aligned with the line of sight from the Earth, so the two stars periodically eclipse each other. This causes Beta Lyrae to regularly change its apparent magnitude from +3.4 to +4.6 over an orbital period of 12.9414 days. The two components are so close together that they cannot be resolved with optical telescopes, forming a spectroscopic binary. In 2008, the primary star and the accretion disk of the secondary star were resolved and imaged using the CHARA Array interferometer and the Michigan InfraRed Combiner (MIRC) in the near infrared H band (see video [at the right], allowing the orbital elements to be computed for the first time."

## Plaskett's star

"Plaskett's Star (HR 2422) is a spectroscopic binary at a distance of [5,245] light-years (1,608 pc). It is one of the most massive binary stars known, with [components: A = 54 M$_ʘ$ and B = 56 M$_ʘ$ for] a total mass of around [110 M$_ʘ$]. ... The pair have a combined visual magnitude of [6.06], and is located in the constellation of Monoceros. The orbital period for the pair is 14.39625 ± 0.00095 days. The secondary is a rapid rotator with a projected rotational velocity of 300 km sec$^{–1}$, giving it a pronounced equatorial bulge." Plaskett’s star is an example of fission.

## Solar-like binaries

For a solar-like binary system, the primary has T$_{eff}$ ≲ 6000 K.
"TYC 4410-01037-1 [has] a mass of 1.07 ± 0.08 M$_⊙$ and radius of 0.99 ± 0.18 R$_⊙$. ... T$_{eff}$ = 5879 ± 29 K [and] [Fe/H] = -0.01 ± 0.05".
At right is a Keck adaptive optics image of TYC 4110-01037-1 in K′. A faint candidate tertiary companion (indicated by the arrow) with red colors is separated by 986 ± 4 mas from the primary star. If it is physically associated with the primary, it is most likely a dM3-dM4 star. The companion is designated MARVELS-3B.
Its "low-mass stellar companion [has a] small mass ratio [ q ≥ 0.087 ± 0.003] and short orbital period [78.994 ± 0.012 days, which] are atypical amongst solar-like ... binary systems. [The orbit has] an eccentricity of 0.1095 ± 0.0023, and a semi-amplitude of 4199 ± 11 m s$^{-1}$. ... the minimum companion mass (if sin i = 1) [is] 97.7 ± 5.8 M$_{Jup}$."
"One possible way to create such a system would be if a triple-component stellar multiple broke up into a short period, low q binary during the cluster dispersal phase of its lifetime. A candidate tertiary body has been identified in the system via single-epoch, high contrast imagery. If this object is confirmed to be co-moving, ... it [may] be a dM4 star."