Double Star Astronomy

Part 1: A Double Star Primer


Terminology
Double Star
Star System
Binary Star
Technical Categories

The Probability of Optical Pairs

Common Visual Patterns

Hierarchical Orbits

Population Characteristics
The Multiplicity Fraction
The Scale of Binary Orbits
Distribution of Binary Orbits
Distribution of the Mass Ratio
Orbital Eccentricity

Distance Estimates

The Triple Horizon
The Brightness Horizon
The Resolution Horizon
The Time Horizon
Contour of the Triple Horizon

Double Stars & Star Formation
How Stars Form
The Natal Star Cluster
How Double Stars Form
Mechanisms That Fail


A common feature of deep sky and solar system astronomy is the emphasis on context. We understand the size, distance, Hubble type and cluster membership of galaxies; we know the distance, diameter, Trumpler type, age, aggregate mass and spectral color of open star clusters; we know at the very least the distance, size and composition of bright nebulae. We also understand how each of these features fits into a larger picture — the Galactic context.

Yet the visual astronomer's traditional approach to double star astronomy is to view each system as a detached curiosity — as a "showcase pair" remarkable for the contrasts of brightness and color, or as a challenge to resolve with a specific aperture.

This page outlines the information necessary to bring double star astronomy to the same level as other areas of visual astronomy: to see double stars in context. This means to see them as representative of hierarchical dynamic relationships, as displaying a specific orbital separation and period, as "fossils" of star formation and star cluster dissolution, and as inhabiting the solar neighborhood of the Galaxy. 

Terminology

The terminology in the double star literature has been made disorderly through the identical visual appearance of two gravitationally bound stars and the random propinquity between two unrelated stars on the illusory celestial sphere, a nonchalance in the requirement to distinguish between dynamic and optical pairs, the grammatical awkwardness of applying the label "double" to a category of multiple objects, and a vocabulary that has changed and amplified across two centuries.

As a point of departure, I adopt the definition given by Wulff Heintz in his classic Double Stars (1978):

Two or more stars held, by their mutual gravity attraction, in a long-term (usually lifelong) association are termed a double star.

This identifies the three essential criteria: (1) two or more (an unspecified number of) stellar components, (2) a gravitational (dynamic) bond, and (3) a "usually lifelong" association (billions of years for most stars). A stable periodic orbit is only implied and the visual appearance of the system is entirely ignored. 

Double Star. In the original 18th century usage, double star meant nothing more than "two stars that appear side by side on the celestial sphere." In that era it was assumed by most astronomers that these pairs were widely separated in space along the line of sight; stars were "insulated" from each other and only planets orbited stars. Galileo Galilei even proposed using an annual change in the apparent separation between the closer and farther stars as a method to measure stellar parallax.

This usage was challenged when William Herschel announced the possibility of dynamical stellar pairs in his Catalogue of 500 new Nebulae, nebulous Stars, planetary Nebulae, and Clusters of Stars; with Remarks on the Construction of the Heavens (1802) and demonstrated their existence with detailed observations of six binary systems in Account of the Changes That Have Happened, during the Last Twenty-Five Years, in the Relative Situation of Double-Stars; With an Investigation of the Cause to Which They Are Owing (1803). In descibing the "construction of the heavens" in 1802 he stated:

II. Of Binary sidereal Systems, or double Stars

The next part in the construction of the heavens, that offers itself to our consideration, is the union of two stars, that are formed together into one system, by the laws of attraction.

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 should really be situated very near each other, and at the same time so far insulated as not to be materially affected by the attractions of neighboring 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.

It is easy to prove, from the doctrine of gravitation, that two stars may be so connected together as to perform circles, or similar ellipses, round their common centre of gravity. In this case, they will always move in directions opposite and parallel to each other; and their system, if not destroyed by some foreign cause, will remain permanent.

With phrases such as the appearance of a double star and this should be called a real double star, Herschel struggles with the first cause of disorderly terminology.

Historically, any close pair of stars was conventionally a double star when first discovered or entered into a double star catalog. Any such pair that was later discovered to be too far apart in space to be gravitationally bound was termed an optical double star or optical pair — Heintz disdainfully refers to them as "so-called optical or perspectivic objects." As the name implies, these are only a form of optical illusion created by the appearance of adjacent stars on the illusory celestial sphere. This exclusionary meaning took hold in the 20th century, once it became possible to distinguish a significant number of illusory pairs by means of measured rectilinear motion, divergent proper motion or discrepant parallax estimates of distance from the Sun.

But a common remedy was simply to ignore "perspectivic objects" entirely and to wield the term double star to mean an assumed, highly probable or confirmed gravitational bond. Thus one routinely finds in the astronomical literature titles such as On the period of the double star η Coronae Borealis (Wilson, 1875), On the orbits of double stars (See, 1893), A graphical method of finding the apparent orbit of a double star (Barnes, 1909), On the masses of double stars (Doberck, 1912), An attempt to determine the true distribution of the double stars (Kreiken, 1928) or The astrometric study of unseen companions in double stars (Strand, 1944) and finally, of course, the title of the Wulff Heintz overview of physical systems. In all these cases, the term "double star" expressly excludes mere optical propinquity on the celestial sphere.

I can't help but see a regrettable dose of wishful thinking in this usage, but it also signifies the large uncertainty in the 19th century as to the physical limits that gravitation would place on double stars, or how this would limit the appearance criteria for identification. A double star could not be separated by more than some arbitrary maximum which depended on the judgment or purposes of the astronomer: < 5" (W. Herschel Class II doubles), 5" (R.G. Aitken), 10" (T.E. Espin), 32" (F.W. Struve), or > 60" (W. Herschel Class VI doubles). R.G. Aitken attempted to corral this disorder in his On the Definition of the Term Double Star (1911).

As it is grammatically confusing to give the same name to a thing and to the illusion of the thing — as if a mirage were called optical water — the received usage is profoundly defective. I adopt the following terminology:

• Optical pair (or optical group) for any stars joined by visual appearance only, without any evidence either for or against a possible gravitational bond;

• Pair asterism (or group asterism) for two or more stars assumed from available physical evidence — an extended linear relative motion, divergent proper motions or discrepant geometric or spectroscopic parallax distances — to be dynamically unrelated; and

• Double star (without the redundant multiple star) to denote the three defining criteria that Heintz assigns to it.

Binary Star. The term binary star has been used restrictively throughout the 19th century and up to the present to mean a pair of stars with a confirmed gravitational bond. This has required an orbital solution (often "calculated" using a graphical or curve fitting method) that amounted to observation of nonlinear relative motion.

This usage originates in William Herschel's (1802) declaration that he had discovered six "real binary combinations of two stars, intimately held together by the bond of mutual attraction." The subtext is that it is not possible to compute a fixed orbit among three stars. That is, "binary" meant not only a confirmed gravitational (periodic orbit) association but also that the calculated orbit excluded any third component.

Herschel's difficulty was his poor grasp (as a "practical astronomer") of complex gravitational dynamics, as his perfectly accurate diagram of binary orbits, and his wildly implausible diagram of ternary orbits, illustrates (diagram, right). The fundamental problem is that three similar orbits around a common center of mass are dynamically unstable: even a tiny inequality in mass or orbital radius will evolve to eject one of the components. The only stable orbits are fundamentally binary orbits: this is the reason that physical systems of three stars, such as ζ Cancri, are analyzed as the orbit of a close binary and the orbit of the binary pair with the distant third component.

Heintz observed that c.1970 "binary" was not an international standard synonym for "double star", but some recent research papers (e.g., Shaya & Olling, 2011) use binary to mean "any system that contains more than one star."

I feel there is good reason to retain "binary" in its limited traditional denotation: all enduring double stars consist either of a binary pair or one or more binaries in combination with single stellar components, and binaries are building blocks of all double stars, the gravitational sink or lowest energy state for the components. (See the double star diagrams below.) With the Shaya & Olling alternative we once again stumble on idiocies of grammar: the Castor sextuple system would be "a binary star composed of three binary stars."

Star System. This is the third cause of disorderly nomenclature: using a categorical label for star systems that also seems to denote the number of components.

Nineteenth century astronomers, again following William Herschel, denoted the number of components as a double, triple, quadruple, quintuple or sextuple star — although a triple star might also be a ternary star (J. Herschel) or a treble star (W. Herschel). Apparently, a triple star could not also be a double star: T.W. Webb's Celestial Objects for Common Telescopes (c.1860) refers to the third component of ζ Cancri as the "triple" star. Here the catchall category of multiple star was coined by Herschel for a combination of three or more stars.

Current usage seems divided between the phrase double or multiple star (for example, at SIMBAD) to ensure that double star is not misinterpreted to mean only two stars, and the phrase binary or multiple star to exclude visual criteria and emphasize the gravitational bond. Both are equivalent to the Heintz definition. The forthcoming GAIA data reports refer to these systems as non-single stars.

As an alternative to multiple star, the term star system is often used to signify any combination of stellar objects — a single star with a circumstellar disk, two or more stars, or one or more stars with one or more planets. These heterogeneous dynamical systems all revolve a single center of mass, the barycenter, and it is this common barycenter that determines what is included in the "star system".

Because "star system" ignores the number and relative mass of components, it most often turns up when discussing the multiplicity fraction where both single and double stars must be counted as equivalent astronomical objects. The limitation that star systems must consist of stars is still the common usage, but several recent studies have included "failed stars" (brown dwarfs and massive planets) in the definition of double or multiple systems.

Multiple stars that lack a hierarchical orbit structure are highly likely to be dynamically unstable and will eventually eject one or more members to form a hierarchical structure. Empirically, the upper limit of dynamically stable star systems seems to be about 7 components; systems that comprise 10 or more components are classified as small star clusters.

Technical Categories. After the confirmation of common proper motion pairs and the discovery of spectroscopic binaries in the late 19th century, the tradition of labeling double stars according to the number of components was augmented by the convention of classifying double stars according to the method used to detect and measure them:

• A visual double star, the original type of double star, can be resolved into separate components whose relative position can be described by angular separation and position angle; historically this meant visual measurement using a telescope and a micrometer, although nowadays astrophotography, drift timing and interferometry (including long baseline measurement in invisible IR wavelengths and computer processing of speckle images) are included. A maximum separation, if stipulated at all, is usually visually wide (for example, 120" or two arcminutes).

• A spectroscopic binary is identified by periodic Doppler shifts in the absorption lines of the star system spectrum due to changes in the radial velocity of the components as they orbit toward or away from the Earth. This can appear as a double line spectroscopic binary (SB2) if the absorption lines of both stars can be detected (the stars are approximately of equal brightness and typically of similar mass), or as a single line binary (SB1) if the absorption lines of only one spectrum are visible (the stars are of unequal brightness and mass). Note that a double star can be both a visual and spectroscopic binary, especially if the star system is in the solar neighborhood.

• A spectrum binary combines in a single spectrum the absorption lines of two recognizably different stars but without significant Doppler shifts between them.

• An eclipsing binary is identified by highly regular and periodic changes in the magnitude of a star system due to mutual eclipses by the two components.

• A photometric binary is identified as an apparently single star whose light is much greater than its apparent spectral and luminosity type would predict; alternately, it refers to an eclipsing variable star.

• An astrometric binary is discovered by a wobble in the path or a periodic change in the pace of a single star's proper motion.

• An occultation double star is detected as a stepwise rather than instantaneous extinction of the star's light when occulted by the Moon.

• A common proper motion double star or CPM double is identified by parallel motion or small divergence in the direction and speed of the two stars' proper motions across the celestial sphere.

In addition, very close binaries can display some remarkable phenomena of mass transfer, which is indicated by additional categorical labels:

• A contact binary typically has a period of less than one day; the orbital radius is equivalent to the diameter of the stars, so that both stars orbit inside a single peanut shaped photosphere.

• A semi-detached binary typically has a period ≤ 10 years (an orbital radius ≤ 6 AU in a 2 solar mass system) and contains a component that has expanded so far during the late giant or supergiant stage that the giant star pours mass onto its companion via a Roche lobe overflow.

• A common envelope binary has a much smaller orbital radius with mass transfer; the overflow of matter is so extensive that the stars revolve inside a single plasma envelope, stirred into a cloud by their orbital motion.

As late as 1935 Aitken could quibble that spectroscopic binaries, which are usually not visually resolved, were "not double stars at all in the ordinary sense of the term." This ordinary sense would also exclude common proper motion (CPM) binaries, which can be visually separated by hundreds of arcminutes or even two or three degrees. The Heintz definition covers them all, because visual appearance or periodic orbital motion are not defining criteria.

The Probability of Optical Pairs

In c.150 CE Ptolemy of Alexandria referred to the naked eye pairs μ1 and μ2 Scorpii (separated by 6 arcminutes), ν1 and ν2 Sagittarii (separated by 14 arcminutes) and ξ and ν Orionis (separated by 72 arcminutes) — visually equivalent to a separation between 4″ to 40″ viewed at 100x magnification — as διπλοὖς αστήρ (double star).

None of these pairs are gravitationally bound. Ever since, the fundamental difficulty faced by double star astronomers is the possibility of mistaking a random alignment of two unrelated stars on the celestial sphere with a gravitationally bound double star. Given the long orbital periods of most double stars, which can make them appear fixed over decades or centuries of time, how can we determine if an optical pair is a physical, gravitationally bound system?

Visual ambiguity was once a problem in planetary and deep sky astronomy. Galileo saw the rings of Saturn as two balls attached to opposite sides of the planet; throughout the 19th century, it was uncertain whether nebula, globular clusters and galaxies were composed of stars or gas. Larger apertures and the advent of astrophotography resolved those uncertainties: but neither aperture nor photography can discriminate double stars from pair asterisms.

Different types of evidence create different levels of certainty. A gravitational bond is identified with the highest confidence in the roughly ~780 pairs with high quality orbits in the 6th Orbit Catalogue, in several hundred periodic (eclipsing) variable stars, and in spectroscopic, interferometric or astrometric systems where components cause an observable change in the proper motion or spectrum of the primary star. High confidence applies to bright stars (m ≤ ~10.0) with large and common proper motions, similar radial velocities and parallax calculated true separations less than ~0.1 parsec. Moderate confidence applies to stars with common proper motion but no radial velocities. Weak confidence applies to systems with a projected spatial separation inferred from angular separation and geometric or spectroscopic distance estimates.

Lacking robust evidence of orbital interaction, common motion or spatial proximity, the astronomer is left with the probability that the visual appearance of a double star could be due to a random alignment of stars on the celestial sphere.

Various methods to estimate probabilities have been proposed since the English naturalist John Michell asserted in the late 18th century that the large number of telescopic double stars identified by Christian Mayer and William Herschel could not be produced by randomly distributed single stars. Most of these methods calculate the probability of an optical double from the number of stars (N) placed at random within an area of sky (A) that would lie within a given separation (rho, ρ) of another star. The simplest of these, proposed by Friedrich Wilhelm Struve in 1827, is:

[1] OD(m,ρ) = [ N·(N–1)·π·ρ2] / 2A.

where OD(m,ρ) is the expected number of optical doubles that will appear below a given limit magnitude and separation within that area of sky. For example, there are on average 13.3 stars down to the visual magnitude 10.5 in any random square degree of sky, which is equal to almost 13 million square arcseconds. The number of pairs that would appear at random within a 30 arcsecond separation (approximately Struve's original catalog limit) is:

OD(10.5,30") = [13.3·12.3·3.142·302] / 2·36002 = 0.018

which implies that there is only about a 1 in 50 chance that any pair of 10th magnitude or brighter stars closer than 30" in a random square degree area of sky is an optical alignment.

Other methods have been proposed, some taking into account the relative magnitude of the two stars or the resolution limits of the telescope. (Some of these methods are described by Halbwachs (1988), linked under Further Reading.)

The simplest rule to utilize stellar magnitudes was first suggested by E.C. Pickering (Aitken, 1911) and adopted by the American astronomer R.G. Aitken for his New General Catalogue of Double Stars (1932). This is really an editorial rule, used by Aitken to select from past catalogs only those pairs likely to merit further study and without making any claim that the pairs were actually physical. This rule accepts any pair separated by ρ arcseconds or less if:

[2] log(ρ) ≤ 2.8–0.2·[m+m']

where the integrated magnitude of a primary star (m) and its fainter component (m') is:

[3] m+m' = m–log(1+(1/2.512m'-m))/0.4.

Using the Aitken rule, the double star Albireo would be considered a probable physical system, given ρ=35.4", m = 3.19, m' = 4.68 and m'-m = 1.49, because:

log(35.4) = 1.55;
2.8-0.2*(m+m') = 2.8–0.2*(3.19-log(1+(1/2.5121.49))/0.4) = 2.21;
and 1.55 < 2.21.

Sinachopoulos & Mouzourakis (1991) developed a table of the maximum separation in arcseconds between two stars of given magnitude, down to magnitude 15; pairs separated by a smaller angular distance are 95% likely to be physical. Their values (table, below) illustrate the effect on the probabilities of the primary and secondary magnitudes.


maximum separation (ρ") between physical binary stars

for an all sky 95% probability that a pair of magnitudes m1,m2 is not optical


primary magnitude (m1)
≤ 456789101112

secondary
magnitude
(m2)
≤ 4193.8
5141.4103.2
684.061.336.4
750.236.721.813.0
830.322.113.17.84.7
918.513.58.04.82.91.7
1011.48.34.92.91.81.10.6
117.15.23.11.81.10.70.40.2
124.53.31.91.20.70.40.30.20.1
132.92.11.20.70.40.30.20.10.1
141.91.40.80.50.30.20.10.1< 0.1
151.30.90.50.30.20.10.1< 0.1< 0.1

Thus, Sinachopoulos & Mouzourakis (S&M) claim a 95% probability that a matched pair of 4th magnitude or brighter stars separated by less than 194" that we accept as a physical system will actually be physical. The required separation for similar confidence decreases as the pair becomes fainter: 36" at 6th magnitude, 5" at 8th magnitude, and a barely resolvable 0.6" at 10th magnitude.

The chief problem with the S&M method is that the angular limits decrease with magnitude faster than the angular width of a constant orbit decreases with distance. Adopting the Sun's absolute magnitude of 4.83 to calculate the distance from the Sun implied by the apparent magnitudes in the table, the angular limits imply a projected separation of 624 AU at the 36" of a matched 6th magnitude binary but only of 65 AU at the 0.6" of a matched 10th magnitude binary. A related problem is that the projected separation depends on the mass (absolute magnitude) of the stars: for a bright A5 binary the orbits would be 2193 AU and 228 AU.

In contrast, the Pickering/Aitken formula essentially projects a constant orbital radius to a distance determined by the apparent magnitude of the star (it's a form of distance modulus). This method also depends on the brightness of the primary star rather than its mass. For example, again assuming a matched pair of stars and the constant 2.8 in formula [2], the absolute magnitude of the Sun yields a log separation equivalent to a projected orbital radius of roughly 780 AU (78" at 10 parsecs):

log(ρ) = 2.8 – 0.2·(4.83+4.83) = 2.8 – 0.2·(4.58) = 1.89" (log of 77")
R = D*10log(ρ) = 10*101.89 = 776 AU

and we obtain the same radius if we remove the Sun to 400 parsecs where its binary magnitude would be 12.53 and the apparent separation 2.0". But the absolute magnitude of Vega (type A0, M = 0.58) yields a projected radius of 6830 AU, and the overbright, blue supergiant Rigel (type B8 Ia, M = –7.92) gets a radius of 342,400 AU (1.7 parsecs)! The mass of Vega is only 2 times and Rigel 18 times that of the Sun, yet they are allowed orbital radii that are 7 times and 355 times larger. Overall, the Pickering/Aitken rule is biased to accept binaries as gravitationally bound when they are bright, early (O, B and A) spectral types, and grossly biased to include very luminous giant or supergiant binaries, such as Albireo.

It is practical to compensate for these problems by changing the value of the constant to match the star's spectral and luminosity types to produce a constant orbital radius or period across all similar stars, but this recourse is only useful if we lack a parallax estimate of the system distance and can compute the projected separation directly.

Both the Struve and Pickering/Aitken tests are vulnerable to the statistical problem of multiple comparisons, which occurs when a probability that is accurate when applied once to a small area of sky or to a single binary system becomes entirely too optimistic when applied dozens or thousands of times. Sinachopoulos & Mouzourakis avoid this issue by calculating probabilities for the entire sky. But because they average the variations in stellar densities across galactic latitudes, their method is relatively lax for systems located within the Milky Way band (where there are on average 150 stars ≤ v.mag. 10 per square degree) and too strict for systems near the Galactic poles (on average 25 stars per square degree) — where only one or two stars may be visible in a wide, low power field of view.

Where does this leave the visual astronomer? With a process of inference and elimination using available physical evidence. Binary orbital or projected separations beyond ~50,000 AU are suspect as very weakly bound, though they can be gravitationally robust in systems with a massive or binary primary. The WDS appends codes indicating observational evidence that a system is physical or optical based on common proper motion and/or similiar parallax, although the CPM flag often disagrees with a calculation based on the proper motions of both components.

This type of analysis produces sobering results. In the January 2015 WDS, I find that about two thirds of all pairs either show physical evidence of being pair asterisms or lack any evidence of a gravitational bond (are optical pairs); in more than 2,000 pairs there is conflicting physical evidence. Only about 11% of all pairs in WDS are "high probability" double stars within reach of amateur equipment.

Clearly probabilities are a drastic and imperfect solution to the problem of optical pairs. But they motivate us to look for observational evidence of a physical system, and temper our cheerful asterism habit of identifying dynamical connections in the random visual scatter of stars.

Common Visual Patterns

Double stars appear in a wide variety of combinations, but some of these create patterns that are especially striking or memorable due to a unique arrangement, close separation, vivid brightness and/or color contrasts, or the location of the system in a rich field of stars or even inside a star cluster.

The diagram (below) illustrates some of the more frequently encountered visual patterns. These by no means exhaust the combinations that an active observer will encounter, nor are the labels standard in the double star literature.

• Matched – (gam VIR, mu DRA, gam ARI). Two visually close stars of similar or identical magnitude (to a magnitude difference of about 0.3), and therefore very often of equivalent mass and identical spectral/luminosity type. These are the most likely of all visual doubles to be a physical system. They serve as standard targets for a test of the resolving power of a telescope and observer (or of the seeing), and are charming to discover while wandering through a rich stellar field (my favorite is LAL 53).

• Giant type – (alp HER, alp SCO). A binary composed of a late spectral type (K or M) giant or supergiant primary and an early spectral type (O, B or A) main sequence companion. These combinations are relatively frequent because most binaries have unequal mass and both stars are massive and visible across enormous distances. The pair typically creates a yellow vs. blue (Y/B) or orange vs. green (O/G) color contrast that the opponent processing of the eye enhances to striking effect.

• Unequal mass – (bet ORI, del CYG). Two stars of dissimilar magnitude that are close but easily resolved. I find this pattern especially attractive when the primary star is 6th to 9th magnitude with a companion 2 to 3 magnitudes fainter and within a separation of 4 to 6 arcseconds; in this configuration the pair seems visibly (and often is) very far away. Magnitude contrast with a wide (>20 to >30 arcseconds) separation is less impressive as the component can resemble a field star.

• Double single (2+1 or 1+2) – (mu HER, zet CNC). A close primary binary with a distant fainter companion, the components in a ratio of the separations greater than 7:1 (about 42% of triple systems); the reverse single double (1+2) configuration, a bright primary with a fainter, close binary companion, is less common (about 10% of all triples). When parallax or proper motion are available, the single star is sometimes found to be unrelated to the binary. Computer simulations suggest this configuration will evolve naturally from a triple system of protostars, and should not be rare.

• Double double (2+2) – (eps 6,7 LYR, STF 1964). Two binaries orbiting as a single system, typically with a ratio in the separation between and within binaries of more than 10:1. The term is often applied to two unrelated binary systems visible in the same low power field (e.g., STF 2470 and STF 2474 in Lyra), or as two unrelated binaries in optical conjunction and catalogued as a single multiple system (e.g., STF 2538 and BU 438).

• CPM – (nu1,2 CRB, xi SCO). Two stars that are visually and spatially very distant from each other, sometimes at distances greater than 50,000 AU; visually most convincing when both stars are of roughly equal magnitude or spectral color (twin K or M giants are especially fine and can be observed at great distances). Parallax or CPM evidence is necessary to confirm that a widely separated pair is physical, which is feasible only for systems within a few hundred parsecs of the Sun.

These few patterns are the basic themes in an astonishing variety in double star configurations, which extend into complex multiple systems and small star clusters. One of the unanswered questions in double star astronomy is how these patterns may be related through the orbital and astrophysical evolution of stellar systems. This requires us to understand their dynamical structure.

Hierarchical Orbits

The visual patterns exemplify a general feature of double star systems: a process of dynamical segregation that produces hierarchical orbits in multiple star systems.

All evidence indicates that multiple stars cannot form "solar system" arrangements of nested, concentric orbits. This is because the stars in multiple systems are almost always similar in mass, rarely differing by a mass ratio of less than 0.2 (or 5:1) — in contrast to the 0.0013 (746:1) mass ratio between the Sun and all its planets.

Because close binary stars exert a strong gravitational attraction on each other and a disruptive influence on other stars at a similar orbital distance, concentric orbits would be gravitationally torn apart in a relatively short time. So the dynamical segregation of orbits appears to develop early, either among the multiple protostars within their enclosing cloud core or as a result of interactions between the multiple system and other stars in the natal star cluster:

• When three stars within a multiple protostar happen to approach periastron at about the same time, the two more massive protostars transfer momentum to the orbit of the third protostar, drawing the two closer or more massive components into a closer binary orbit and propelling the third, more distant or less massive component into a higher energy, larger and more elliptical orbit. This process can be repeated many times within a few million years, creating a close or "hard" binary with a wide companion (a 2+1 double star). In roughly 92% of the triple protostar systems in computer simulations by Reipurth & Mikkola (2013), the third star is eventually ejected from the system.

• In addition, most double stars are influenced by interactions with other stars in the star cluster where they originally formed, and these near encounters most often widen already wide or "soft" orbits or strip low mass companions from a multiple system.

The result of both these processes — internal orbital instabilities and external near encounters — is summarized as the Heggie-Hills Law of dynamical evolution: "over time, soft binaries soften and hard binaries harden."

As a result of these different forms of dynamical processing, orbits of one or more close binary systems and companion single or binary stars are nearly always hierarchical: a binary joins with a third star or a second binary pair at a much larger orbital radius.

For this to be a stable arrangement of the orbits, the orbital radii must differ by a very large proportion. In the Raghavan et al. study of "solar type" stars within 25 parsecs of the Sun (cited below), the average ratio between the orbital radii of components at different levels of triple systems was on the order of about 1:1000, with a ratio in the periods of about 1:20,000 or more. This means the gravitational attraction between the close binary pair is about one million times stronger than from any wider component, ensuring the system as a whole is dynamically stable.

The diagram (below) shows these hierarchical arrangements as "mobile diagrams" for a sample of the multiple systems catalogued by Raghavan et al. (2010) survey. Spectral type is listed below each component, and angular distance in arcseconds (or period in hours, days or years) is listed under each horizontal bar.

In the HD 137763 system (bottom row, yellow), the Aa,Ab binary is effectively a single body insofar as it affects the B component, and the Aa,Ab,B triple is a single body insofar as it affects the C component.

(Incidentally, this system illustrates the standard procedure for labeling components in modern double star catalogs. Thus, HD 137763 consists of three visual components — an AB binary with a C companion. The AB pair comprises spectral type G9 and K2 stars and is identified by common proper motion (CPM) at a separation of 52 arcseconds. The C type M4.5 star is joined to them by CPM at a separation of 1200 arcseconds (20'). In addition, the A primary has a type K9 companion forming a double lined spectroscopic binary (SB2) with a period of 2.4 years. This recently discovered pair is designated with lowercase letters appended to the previously identified star letter name as Aa and Ab. If one of these stars had an even closer companion, the new pair would be designated by numbers, e.g. Ab1 and Ab2.)

These considerations not only rule out a concentric or "solar system" arrangement of orbits in double stars: they indicate that all stable systems will eventually develop a hierarchical arrangement of close binaries and/or single stars, making the binary and the single star the two basic units of enduring double star dynamics.

They also indicate that "trapezium" arrangements of four or more systems at roughly equal distances will be relatively short lived and only appear in recently formed star systems. A remarkable example is the well known "quadruple" system θ1 Orionis (diagram, right), which actually comprises at least 20 components, possibly including a black hole. It is so young that it is just emerging from (and illuminating with the radiation of its massive type O and B components) the natal cloud of hydrogen gas around it, known as the H II region M 42 — the Great Orion Nebula. The four brightest components are each close binary systems, although the orbital radii (A: 1 AU, B: 0.13 AU, C: 18 AU and D: 8 AU) are too close to show in the diagram or discern visually. These systems are minimally separated from each other by 3300 to 13,000 AU and are moving at high velocities, evidence that the system is gravitationally bound and will produce dynamical disruptions among its components. Over the next several tens of million years, the cluster will likely eject some of the widely separated and lower mass components and dissolve into separate binary or hierarchically segregated multiple systems.

The term "cluster" indicates there is continuity between large multiple star systems and small star clusters, which appears because most or nearly all multiple stars form within a star cluster (see below) and most natal star clusters dynamically "evaporate" into separate single, binary or multiple star systems in less than 100 million years.

The 2013 WDS contains almost 84,000 catalogued pairs, but also 6200 triple and 1500 quadruple targets, and over 100 targets with 10 or more "components" (chart, left), although in most cases these higher order components are really field stars. On the other hand, the catalog of Open Clusters and Galactic Structure by Dias, Alessi, Moitinho & Lépine lists about 210 "clusters" with 10 or fewer members at the core, and almost 30 with 10 or fewer members in total.

Origin and continuity indicate that mutiple stars and star clusters are the same type of astrophysical object: a gravitationally bound star system. But in star clusters all the cluster members orbit around a single barycenter created by the total cluster mass, the orbits randomly are deflected into new paths each time the stars pass through the cluster, and clusters "evaporate" or eject members over time. Double stars are enduring, which makes hierarchical segregation among components and both the physical separation and relative peculiar motions consistent with a gravitational bond the three criterion features for their identification.

Population Characteristics

Dynamical segregation explains how all double stars are similar. But how are they different from each other? What are the common forms of double stars, or the rare ones? Those are questions about the population distribution of double star attributes. The dynamical attributes of binary systems are primarily the radius, period and eccentricity of their mutual orbit and the mass ratio between the two stars. But most fundamental of all is the multiplicity fraction — the proportion of double stars among all star systems. Because double stars are linked by gravitational attraction and gravity depends on mass, the multiplicity fraction varies with the spectral type (mass) of the primary star.

The Multiplicity Fraction. Two recent and meticulous surveys by Duquennoy & Mayor (1991) and by Raghavan, McAlister et alia (2010) converge on very similar estimates of the multiplicity fraction in main sequence solar type star systems within 25 parsecs of the Sun.

Setting aside for the moment very massive (Types O, B or A) or low mass (smaller type K and all type M) stars and brown dwarfs, the multiplicity fraction is easily remembered as a 60% rule: roughly 60% of all star systems are single stars, but roughly 60% of all stars are components of binary or multiple star systems (table, below).


the solar type multiplicity fraction
Duquennoy & Mayor
(1991, p.509)
Raghavan et alia
(2010, p.29)

Star
Systems

Stars Within
Systems1

Star
Systems

Stars Within
Systems1


Sample radius from Sun

22 parsecs

25 parsecs

Spectral range
of primary stars

F7 to G9

[13 categories]

.F6 to K3

[18 categories]

.

Mass range (M)
of primary stars

1.2 to 0.8.1.3 to 0.6.

Total (N)2

164246454728

Multiplicity
of System

1

93 (57%)93 (38%)254 (56%)254 (35%)

2

62 (38%)124 (50%)144 (33%)288 (40%)

3

7 (4%)21 (9%)42 (8%)126 (17%)

4+

2 (1%)8 (3%)14 (3%)60 (8%)

Total
Double Systems

71 (43%)153 (62%)200 (44%)474 (65%)

1Includes white dwarfs (Type D) and brown dwarfs (Types L or T), but not planets. 2For both samples,
the observed rather than incompleteness corrected results are shown.

These results emend the frequently cited figure by Wulff Heintz (1969), which relied heavily on projected counts and corrections for observational biases, that roughly 85% of all systems are double (binary or multiple) systems.

Duquennoy & Mayor relied on the Gliese catalog to identify their sample of stars; Raghavan et al. used the more accurate Hipparcos survey. As a result the samples are in fact quite different: the second study includes a wider range of spectral types, and 72 of the 164 stars in the first study were actually outside the sample spectral or distance limits while another 56 stars that met the sample criteria were left out. In addition, the second study includes a much larger sample of stars, and benefited from two decades of improvements in astrometry, interferometry and spectroscopy, which have reduced the longstanding difficulty in identifying systems separated by ~1 to 10 AU (which are often too far from Earth to be resolved by visual observation, but too widely separated to have orbital velocities that spectroscopy can detect as Doppler shifted spectral lines). So it is reassuring that the two studies converge on very similar results.

The mass of the primary star, usually estimated from the spectral type or color index, has a very large effect on the multiplicity fraction. The results of several research papers that have attempted to control for observational or selection biases are summarized in the graph (left). The gray squares show the uncertainty bounds and gray dots the central estimates in 10 separate studies, most of them focused on subsolar masses (K and M stars, and brown dwarfs). The red line is a computer simulated multiplicity distribution based on the "two step" model of Sterzik & Durisen (2004), which explains mass and multiplicity frequencies as the outcome of separate fragmentation processes in the collapsing interstellar cloud and in condensing protostar disks.

The multiplicity fraction increases with the mass of the primary star, consistent with the expectation that more massive stars are more likely to have companions, are more likely to have massive companions, and are more likely to be in multiple systems, a characteristic of binary formation known as mass biasing. Type K and M stars are apparently more likely to be stripped of companions, and Type L or T bodies (brown dwarfs) are apparently more likely to be ejected from double star systems in which they form, which accounts for their lower multiplicity fraction, although most generalizations about these bodies are limited by the detection challenge resulting from their low surface temperatures and infrared luminosity.

The Sun is currently voyaging through a relatively sparse corner of the Galaxy, remote from molecular clouds or young star clusters and far from the massive and brilliant spectral type O and B stars; several nearby Type A or early Type F stars (Altair, Sirius, Canopus and Vega among them) were excluded by the spectral limits of the Duquennoy & Mayor and Ragahvan & alia samples. These higher mass stars have a higher multiplicity fraction but are also rare. At the other extreme, low mass stars and brown dwarfs are excluded, yet these are so numerous and have such a low multiplicity fraction (below ~20% in most studies) that one astronomer (Lada, 2006) has estimated the single star fraction is actually around 66% to 69%. Overall, solar type stars comprise the largest proportion of double star systems and remain the most useful frame of reference for understanding their origin and abundance.

The Scale of Binary Orbits. The common practice in the astronomical literature is to describe binary systems by the order of magnitude period (in days) of the components. If we use the solar system units of years (y) for time, astronomical units (AU) for distance and solar masses (M) for mass, the calculations based on Kepler's Third Law reduce to the simple form:

[1] r = ((M1+M2P2)1/3

[2] P = (r3/(M1+M2))1/2

We can use these formulas to calculate the orbital radius and period of an "average" binary system comprising 2M (two solar mass) stars (table, below).


the scale of binary orbits1
Log Period

Nominal [Actual]

Period2

days/years

Orbital
Radius2

Solar Radii/AUs

Mean
Eccen-
tricity
Percent of
6th Orbital
 Category Label (Note)

–1 [–0.63]0.23/0.00062.0/0.0090.00. CONTACT (smallest orbit before stars merge)
0 [0.01]1.02/0.00285.4/0.0250.050.006
1 [0.91]8.2/0.02222/0.100.230.014INTERACTING
2 [1.96]91/0.25108/0.500.3312.7(Venus R = 0.72 AU, limit of corotation)
3 [3.01]1021/2.82.500.4119.7CLOSE (asteroid belt R = 2.8 AU)
4 [3.91]22100.5043.7(Saturn R = 9.6 AU)
5 [4.96]250500.5220.4MEDIAN (Kuiper Belt R = ~50 AU)
6 [6.01]28002500.620.012(Heliosphere R = ~120 AU)
7 [6.91]22,00010000.650.002WIDE3 (longest period of solved orbits)
8 [7.96]250,0005000..(all identified as CPM pairs)
9 [9.01]2,800,00025,000..FRAGILE4 (widest confirmed = ~75,000 AU)
11 [10.7]150,000,000350,000..tidal radius (comoving but not bound)

1For a binary of two solar mass stars, M1+M2 = 2M. 2Period and orbital radius rounded for simplicity; actual log period shows the period of the rounded radius values. 3Approximately the largest orbital radius of a permanently bound system. 4An orbit likely to be disrupted by passing stars or giant molecular clouds within the lifetime of the components.

The period/radius category labels are not standard in the double star literature but correspond to frequently used dynamical categories. Orbital radii and periods are rounded to make them memorable: log 4 to 6 orbits are approximately 10, 50 and 250 AU (multiplied by either 100 or 0.01 to produce larger or smaller orbits); periods are approximately 22, 250 and 2800 years (multiplied by either 1000 or 0.001 to produce the matching larger or smaller periods).

The diagram (below) shows the scale of the orbital range in the 2 solar mass binary orbits, from "contact" binaries at a log –1 orbit to log 8 systems at 5000 AU. The log period is shown for each binary orbital radius, and the star disks at the bottom are to scale with the separation. The log 9 or "fragile" systems at 25,000 AU are not shown, but they correspond to the radius of the outer Oort cloud, estimated to be 20,000 to 50,000 AU. Spectroscopic binaries include some systems with orbits >250 AU and are therefore also visual double stars, but about 60% of spectroscopic pairs identified in the Pourbaix (2014) catalog have an orbital period in the range shown (100 days or less).

The physical scale of orbits in the table varies with the total binary mass. Given the same orbital period, more massive systems create a greater gravitational potential and this increase in binding energy must be balanced by a larger orbital circumference traveled at a higher orbital velocity. A useful benchmark is the Jacobi radius or tidal radius, which is the estimated semimajor axis between two stars at which the gravitational bond between the stars becomes weaker than the gravitational pull from passing stars and giant molecular clouds:

[3] rJ = 1.7*(M1+M2/2M)1/3 parsecs (masses in Solar mass units)

Two solar mass stars would have a tidal radius of 1.7 parsecs; two type M stars, with a median mass of about 0.5M, would have a tidal radius of about 1.1 parsecs; a type B binary (11.8M) would have a tidal radius of 3.1 parsecs. (These calculations apply to the mass density of the Galaxy at the solar circle; in star clusters or the galactic nucleus the radius would be smaller.)

The diagram (below) compares the orbital scale of the matched solar type binary with the orbital radius for twin Type B (11.8M) or Type M (0.5M) binary systems with orbital periods from log 3 days (~2.8 years) to log 7 days (~22,000 years). The median log 5 period of 250 years has a radius that varies from 85 AU (Type B) to 29 AU (Type K) and an orbital velocity from 10 to 3.5 km/s.

Some sources suggest that 1000 AU to 5000 AU is the largest orbital radius in a solar type binary system that will survive disruption by random encounters with other stars or clouds of interstellar matter; these systems are bound by less than 0.01% the gravitational force of an equal mass "median" system. Yet Fomalhaut forms a ~2.6M binary with TW Piscis Austrini at an orbital radius of 57,000 AU (0.28 parsecs, or nearly one light year). Binaries are more massive than single stars of the same spectral type, and the combined system mass increases the gravitational attraction between components, allowing systems to remain bound at wider distances. For example, Mamajek & alia (2013) suggest that 79/80 Ursae Majoris (Mizar and Alcor), which comprise three spectroscopic binaries with Type A primaries, form an ~11M sextuple system linked across an estimated 75,000 AU orbital distance.

Many "fragile" or log 9 systems are likely not older than a single galactic orbit or ~108 years, and may be more common in the Galactic halo or at the Galactic corotation radius where encounters with the galactic disk or spiral arms are less frequent. A study by Bahcall & Soneira (1981) that used spectroscopic parallax to determine the spatial separation among faint field stars near the North Galactic Pole found that about 15% of them, including many Population II stars, were binaries with separations out to about 0.1 parsec. Many of these systems turned out on closer examination to consist of one or two spectroscopic pairs.

These limiting arguments distract from the enormous potential range of binary orbital periods and distances. At one extreme are contact binaries, so close that they share tidal streams of plasma or revolve under a single, peanut shaped photosphere. These can have orbital periods as short as a few hours and include some of the most exotic star systems — binaries containing a white dwarf star that occasionally blazes into view as a Type Ia supernova. Even binaries that are detached or separated by distances of several million kilometers (~0.1 AU or ~10 days) have evolved into mutually circular orbits with entangled magnetic fields that can produce enormous "star spots" on the surfaces of both stars and corotating orbital periods formed through mutual tidal friction.

At the other extreme, wide binary systems with separations of more than 1000 AU and fragile systems at distances of more than 25,000 AU and periods greater than two million years test the orbital limits of enduring 2M binary systems. The diagram (right) shows the orbital separation of high and low probability binaries (mostly solar type) from Shaya & Olling (2011). The low probability (<95%) objects (orange dots) consist of unbound but comoving companions — typically lower mass bodies (K and M type stars and brown dwarfs) that were formerly bound to the primary — mixed with randomly passing stars from the galactic background flow. The high probability (≥95%) companions for a solar type primary star (yellow dots) become sparse in proportion to the lower probability components at around the predicted tidal radius of 350,000 AU (1.7 parsecs), but the diagram also shows that high probability orbits begin to mix with the flow much closer than the tidal radius, at a "fragile" orbit of around 25,000 AU (about 0.1 parsec). Clearly there is no firm outer limit to the orbital radius of a star system, only a decreasing probability with increasing orbital radius that the system will survive for the lifetime of both components.

By looking at computer simulations of the dynamic evolution of binaries inside star clusters, Parker et alia (2009) interpret binary orbital separations as products of the natal environment (blue dashed lines). Binary orbits >10,000 AU are "always soft" (weakly bound); they likely form as neighbor protostellar objects or as random pairs of stars ejected in parallel from dissolving low density natal star clusters. Binaries at separations >1000 AU are "sometimes soft" and can survive only within low density natal clusters, while binary orbits ≤1000 AU are "sometimes hard" and can resist disruption even in moderately dense clusters. Binary orbits ≤50 AU are "always hard" (strongly bound) and can emerge intact even from high density natal clusters. Thus, most of the spectroscopic or visual pairs in the orbital catalogs have probably come from high density star forming regions or as triple stars that have hardened the binary orbit through the ejection of the third member. Wide pairs seem most likely to come from dispersed star forming regions and dissolving low density star clusters.

Once binaries have formed in the original star formation process, the Heggie-Hills law of binary evolution states that soft binaries soften and hard binaries harden over time due to random encounters with other stars in the natal star cluster and, once the natal cluster has dissolved, with stars or molecular clouds in the galactic orbit. In addition, wide binary surveys and computer simulations indicate that bound components and unbound but comoving companions tend to move away from the tidal limit in opposite directions: bound components perturbed by passing stars tend to move closer to the primary star; "escaped" components drift away from the former primary star but can continue to shadow it at a distance of up to 100 to 300 parsecs.

Distribution of Binary Orbits. We use the log period (in days) to describe the distribution of binary dynamical attributes — orbital period and radius, eccentricity, mass ratio and multiplicity fraction — including binary components of multiple star systems in the population.

In the most reliable multiplicity surveys the orbital periods form a "bell shaped" distribution on log period, and this log normal distribution (first proposed by Gerard Kuiper in 1935) is currently believed to best represent the distribution of binary orbits. The survey by Raghavan et alia (2010) cited above found a median period in binary systems of about 260 years or log 4.98 days (chart, left); Duquennoy & Mayor found a slightly shorter median period of about 180 years or log 4.82 days. Both are at the high end of log 4 orbits, so a log 5 (100,000 days) orbit can be taken as the median value, as reported in the table (above).

The chart shows that the log 4.0 to 4.9 systems comprise the largest number of double stars that can be resolved by visual observation (green bar areas), which matches the proportion of log 4 solved orbits in the 6th Catalog of Orbits of Visual Binary Stars (table, above). Note that systems of Types 1 to 3, which have orbits smaller than the orbit of Saturn, are predominantly identified by spectroscopy (mixed spectra or Doppler shifted spectral lines), interferometric imaging (considered a "visual" technique) or by astrometric detection of the periodic orbital wobbles or changes in the radial velocity of a star's true motion. In contrast, systems of log 6 and higher (orbits of ~250 AU or more) are almost entirely identified by common proper motion or a projected spatial separation calculated using geometric or spectroscopic parallax estimates of distance (described below). This is because the log 6 period is roughly 3000 years, which means only an ambiguously small part of the visual orbit will have been observed in the 200 years of double star astronomy.

Superimposed on these results is the distribution of orbital radii calculated from Hipparcos or spectroscopic parallaxes in the Washington Double Star Catalog. These peak in log 6 and 7 orbits and include very few orbits below log 3. This illustrates that most visual double stars are widely separated, on the order of 100 AU or more.

Contrasting the survey and catalog distributions, we can infer that something like 2 out of 3 log 4 systems, and 9 out of 10 log 3 systems, are undetected among the very many faint stars observable down to magnitude 15; at the same time, as many as 3 in 4 log 9 systems are optical. (As a plausibility check, if we only consider the one fourth of log 9 systems in WDS with the smallest orbital radius, we find the widest orbital separation in this subset is about 30,000 AU.)

Distribution of the Mass Ratio. Raghavan et. al. also calculated the mass ratio of all binary systems, defined as q = M2/M1 (where M2 < M1); values of q are bounded as 0 < q ≤ 1.0. The mass ratio can be estimated in two ways: from the period and relative orbital velocities of a binary (especially close binaries measured with spectroscopy), or from the ratio of the visual magnitudes, calculated as

[4] q = 10–[(m2-m1)/10]

and assuming both stars are on the main sequence. As a rule of thumb, a 1.0 magnitude difference indicates a mass ratio of 0.8, a 2.0 magnitude difference indicates a mass ratio of 0.6, and a 3.0 magnitude difference a mass ratio of 0.5.

These results suggest three features: (1) a spike in the mass ratio near 1.0 (two equal mass stars), especially in binary stars that are components of multiple systems; (2) a broad plateau of distributions that may have a secondary peak around q = ~0.4 to 0.5 (more apparent in isolated binary systems); and (3) a fall off in mass ratios below 0.2. However, different surveys produce different q distributions, and theoretical computer simulations of star formation predict there should be a large number of systems below 0.2 and few that actually reach 1.0. (Another factor may be the observational bias caused because low mass K or M stars — and the brown dwarf "failed stars" with masses < 0.07M — are faint and difficult to detect.)

Orbital Eccentricity. The remaining dynamical attribute is orbital eccentricity, calculated as the ratio between the semimajor (a) and semiminor (b) axis of the orbit:

[5] e = √ 1–(b2/a2)

Circular orbits (a = b) have an eccentricity of 0.0 and bind the largest angular momentum and orbital velocity for the given system mass and radius; elliptical orbits have an eccentricity 0 < e < 1.0 and become less energetic as the eccentricity approaches 1.0, entirely because the semimajor axis of the orbit is elongated. (Eccentricity has no effect on the orbital period, which is entirely determined by the system mass and the semimajor axis.) Unbound (parabolic or hyperbolic) trajectories have an eccentricity ≥ 1.0, and the geometry of the orbit means that such components are not bound to the primary star. (Proxima Centauri has an eccentricity that is near or slightly above 1.0, so it is unclear whether it is actually bound to the primary star, Rigel Kent.)

Eccentricity divides double stars into two distinct populations. Orbits with periods around 100 days or less (log 2, r < ~ 0.5 AU, circular) tend to have roughly circular orbits (e < 0.3), and at periods below 10 days the circularization is complete (e = ~0.0). In contrast, the orbits of log 4 and higher systems are almost randomly distributed, with a median eccentricity e = 0.5 to 0.6. It is likely that "fragile" (lower energy) systems have more eccentric orbits, although the increasing eccentricity shown in the table above (summarizing the 6th Orbital Catalog) is in part a selection bias. Raghavan et al. conclude that high eccentricity and multiple systems are relatively young and more likely to be disrupted by chance encounters with other massive bodies along their Galactic path.

Distance Estimates

The double star astronomer Paul Couteau, writing in 1978, opened his chapter "A Voyage to the Country of Double Stars" with this observation:

Looking at a system of stars through a telescope transports the mind on a distant voyage. If the system is at a known distance and is composed of objects that the astronomer can compare with our Sun, it is easy to travel there in thought and to imagine the sights seen by the inhabitants of that place. Often, the system will not give up its secret, neither the distance nor the orbit is known. Almost all the double stars that have been classified are of this kind." (Observing Visual Double Stars, p. 174)

A poetic sentiment, but today increasingly incorrect. In 1993 the Hipparcos satellite completed an astrometric mission that measured the geometric parallax of 118,000 stars (the Hipparcos Catalog) and the celestial coordinates and basic spectral types of an additional 2.4 million stars down to magnitude 11 (the Tycho 2 Catalog). The precision of the parallax measurements, with median measurement errors of 1 milliarcsecond inclusive of stars up to magnitude 8, allowed unprecedentedly accurate estimates of stellar distance out to about 1000 parsecs. The GAIA satellite launched in 2013 is scheduled to produce a three dimensional map of roughly one billion stars; the data, planned for publication sometime around 2020, will include the distances of about 200 million stars down to v.mag 15 measured to an accuracy of ±10% out to a distance of 8000 parsecs.

We therefore have a substantially improved understanding of the distances and spatial relationships among hundreds of double stars. And until the GAIA data become available, there is an alternative method to infer the distance and orbital scale of a double star system — the spectroscopic parallax. This utilizes the stellar attribute that amateur astronomers usually consider to be an esthetic ornament: the star color.

Stellar mass and radius almost entirely determine a star's spectral type and luminosity class, and these almost entirely determine the star's absolute magnitude, which forms the basis of the Hertzsprung-Russell temperature/luminosity diagram. This relationship between spectral type and brightness allows us to predict a star's absolute magnitude (M) in cases where the spectral class is known but the astrometric parallax is unavailable.

The most direct method to find the absolute magnitude of a specific spectral type and luminosity class is to read the value from the average luminosity curves for the corresponding spectral type and luminosity class (chart, below). The luminosity classes are: Ia/Ib (supergiant), II (bright giant), III (giant), IV (subgiant), V ("dwarf stars" or main sequence) and D (white dwarf). (If the luminosity class is not indicated, use the Class V curve.) Alternately, you can use this table.

Morgan-Keenan calibration absolute magnitudes, from Appendix B of Stellar Spectral Classification by Gray & Corbally (2009)

With an estimate of absolute magnitude (M) and the visual magnitude (m) of the star — and assuming that interstellar extinction (discussed below) does not affect the star's apparent brightness — the star's distance (in parsecs) can be calculated using the distance modulus. This distance can in turn be used to estimate the projected spatial separation between the stars (in astronomical units AU) from the observed angular separation (ρ, in arcseconds):

[1] Distance (Parsecs) D = 10[1+((m - M)/5)] (distance modulus)

[2] Projected separation (AUs) R = D·ρ

[3] Projected separation (AUs) R = D·10[log(ρ)+0.13] (with Couteau correction for foreshortening)

The term projected separation means we calculate the spatial separation of the stars as projected onto the celestial sphere — in other words, assuming that the inclination of the orbit is 90°. This is almost never true, so R is actually the minimum of the actual distance. We also cannot know from this estimate the actual semimajor axis of the orbit.

Statistical analyses by Paul Couteau suggested that on average we observe a binary orbital radius reduced or foreshortened by roughly one fourth. He suggested adding the factor 0.13 to the exponent of the orbital radius to compensate for the fact that we almost never see a binary orbit with the stars at apastron and the semimajor axis perpendicular to our line of sight.

Data for m and ρ are provided in WDS and most other double star catalogs. For example, the primary star of the visual binary zet UMA (Mizar) has a catalog visual magnitude of 2.23, a separation ρ = 14.5 arcseconds and a spectral type A2V. The chart (above) indicates an average absolute magnitude of 1.9 for a type A2V star, so we can calculate

Distance D = 10[((2.23–1.9)/5)+1] = 101.07 = 11.7 parsecs (38 light years)
Projected Separation R = 11.7·10[log(14.5)+0.13] = 11.7·101.29 = 228 AU

The Hipparcos parallax values for zeta UMa are 26 parsecs and the semimajor axis of the orbital solution is 537 AU, suggesting poor accuracy in the spectroscopic method. However, on average across all systems, the overall accuracy is better, for example by comparing the spectroscopic parallax estimates of distance in a sample of 14,000 WDS systems for which Hipparcos astrometric distance estimates are also available. The sample is truncated at 1000 parsecs as larger distance estimates using either method become unreliable; a log/log plot is used to reduce the visual scatter.

This validity test shows very good agreement between the two methods (at least, as astronomical distance estimates go), with a scatter of about ±20% across the sample range — without making adjustments for extinction or reddening. However there is a detached group of stars at the top center of the chart for which the spectroscopic method underestimates the parallax distance by roughly a factor of 5 — the absolute magnitude of the stars is too large (too faint) by about 3.5 magnitudes. Inspection of the data shows these are of all spectral types from B to M, but all lack any luminosity code: if a luminosity classification were made, these would plausibly be giant or supergiant rather than "dwarf" or mainstream stars. This underscores the importance of using both spectral and luminosity data to determine the spectroscopic absolute magnitude. The smaller number of stragglers below the diagonal line occur because we "underestimate" the visual magnitude of the stars due to extinction.

Note that this procedure is useful to identify likely optical doubles, even when these have not been identified as such in WDS or other catalogs, when the estimated minimum semimajor axis (R) is greater than some arbitrarily large value — such as 100,000 AU.

The Triple Horizon

With a method available to estimate the distance to double star systems, we can evaluate three obstacles to double star observation — brightness, orbital radius and relative motion. If a system is (1) too faint to be detected, (2) orbiting too closely to be resolved, or (3) moving too slowly to be confirmed as a physical pair, by either measurable orbital trajectory or common proper and radial motion, the double star will be undetectable, unresolved, or indistinguishable from a pair asterism.

Olling (2005) illustrated the problem (diagram, left) by first plotting the distribution of spatial separations (orbital radii) and system distances from the Sun in the Hipparcos catalog (white dots), then adding systems identified in the 4th Catalog of Interferometric Measurements of Binary Stars (2004, yellow crosses) and systems identified in the 9th Catalog of Spectroscopic Binaries (2004), the California/Carnegie Planet Search Compilation (2003), or the Geneva-Copenhagen and Tycho faint star catalogs (blue dots). The exclusion at large distances of close orbiting systems and wide, faint companions (including brown dwarfs and planets) is a striking illustration of the multiplicity sampling bias in the Hipparcos (visual) catalog and, in fact, all catalogs bounded by brightness, resolution or distance limits.

There is also a fourth uncertainty — our definition of what constitutes a double star. This appears historically as the visual criterion: optical pairs that were wider than some arbitrary limit, at most a few arcminutes (see discussion above) were excluded from visual double star catalogs. Although bound pairs are now confirmed at angular (visual) separations of several degrees (hundreds of arcminutes), and pairs wider than 17 arcminutes are included in the current WDS, a second issue has been the outer gravitational limit for bound pairs, which lacks a clarifying empirical basis and is ambiguously located by computer simulations or astrophysical theory. At times, brown dwarfs and planets are included as companions of a "double star". I exclude these issue from the following discussion.

The limitations of detection, resolution and dynamic measurement are the three horizons in double star astronomy and they correspond equally to physical facts (double star absolute brightness, distance from Earth, orbital radius, and orbital motion or shared galactic motion) and to limitations in our observing technology (light grasp, resolution, limits of detectable orbital, proper or radial motion).

We can see the combined effect of these three horizons in the parallax measured or spectroscopically estimated distance of star systems in the 2014 WDS (graph, right), where 50% of the catalogued systems are within 175 parsecs of the Sun and 90% are within 600 parsecs.

In order to trace the triple horizon I assume the observing instrument is a "typical" amateur aperture of 250mm (10"), which provides a limiting visual magnitude limit of about 14.6 and a limiting resolution (using the λ/D criterion) of about 0.5 arcseconds. To measure dynamic change in orbit or proper motion, I assume a 200 year observational period, equivalent to the current record of double star astronomy.

The Brightness Horizon. The brightness horizon is determined by the intrinsic luminosity of a star and its distance from Earth. The distance modulus allows us to use the estimated absolute magnitude of the star (M) and its visual magnitude (m) to calculate a maximun stellar distance D (completely unobscured by interstellar dust).

If we limit ourselves to main sequence (luminosity class V) stars, what is a reasonable apparent magnitude limit? Disregarding the limits of "lucky imaging" astrophotography, we cannot visually resolve or measure close double stars by averted vision, so we must use the limit magnitude determined by the foveal (direct vision) brightness threshold. For naked eye stars at a dark sky site, given a dark adapted pupil aperture of 6mm, this is comfortably about magnitude 4, which in a 250mm aperture is the equivalent brightness of a 12th magnitude star:

ΔA = (250/6)2 = 1736; m'–m = 2.512·log(1736) = 8.1; 4.0+8.1 = 12.1 magnitude.

However Paul Couteau (1978, p.181) has suggested a limit of magnitude 10.5 for double star observations regardless of aperture ("Experience shows that, whatever the aperture, magnitude 10 is a barrier") and 11th magnitude is approximately the limit for inclusion in most 19th century double star catalogs. So let's adopt magnitude 10.5 as a reasonable limit magnitude for the brightness horizon; in the 2015 WDS, this comprises about 47,600 unique double star systems.

How far away is a star observed at that magnitude? That depends on its intrinsic brightness. Excluding the obscuring effect of interstellar matter, the massive "Orion type" O8 V stars (M = –4.4) will present an apparent magnitude of 10.5 at a distance of roughly 9600 parsecs:

DO8 = 10[(10.5 – -4.4)/5 + 1) = 103.98 = 9550 parsecs

and a Type B5 V star (M = –1.1) at magnitude 10.5 will be roughly 2100 parsecs away. The diagram (below) shows these distances superimposed on a rendering of the Galaxy spiral arm structure by Robert Hunt.

Both type O and B stars, and even more luminous giant or supergiant (Ia, Ib or II) A to M stars, are relatively rare — comprising 0.2%, 10% and 1% respectively of stars brighter than magnitude 10 in the Hipparcos-Yale-Gliese catalog. Most double stars are far less luminous, and therefore much closer to the Sun. A main sequence Type A5 V star (about 2.3 solar masses, M = 2.1) is visible at magnitude 10.5 out to about 480 parsecs (circle A5, diagram above).

The diagram (below) zooms in on this 480 parsec radius to show the location in the Galactic plane of naked eye stars (down to v.mag. 6.5) using Hipparcos data for distance and spectral type. Over 90% of all naked eye stars and almost 80% of all stars down to 10th magnitude are within 400 parsecs — roughly the distance to the Great Orion Nebula. More remarkable is the dense concentration of stars within 100 parsecs of the Sun, roughly twice the distance to the Hyades and two thirds the distance to the Pleiades — which comprises half (49%) of all naked eye stars.

Beyond 200 parsecs most naked eye stars are Type O or B dwarfs mixed with giants of all spectral types. Type F and G stars are especially concentrated near the Sun because their giant forms are relatively rare. 

The extinction of stellar light caused by scattering or absorption in the interstellar medium substantially contracts brightness horizon, especially for early type (O and B) stars that radiate primarily in the "blue" visual and ultraviolet wavelengths. Extinction in visual wavelengths by a uniformly distributed interstellar medium is estimated to cause between a 0.7 to 1.0 magnitude increase for every 1000 parsecs of distance. However this average disguises a very wide range: extinction is strongest along the galactic plane and very weak toward the galactic poles, and is highly variable with galactic longitude due to the presence of discrete, dense clouds of gas or dust. Finally, extinction varies approximately as 1/λ (the reciprocal of wavelength): for a unit extinction at 550 nm, extinction is 1.56 magnitudes per kiloparsec at 365 nm (ultraviolet) but only 0.28 mag/kpc at 1250 nm (infrared). (This is the reason Galaxy spiral arms have been traced at large distances by means of giant K and M stars, which are visible through the interstellar medium to more than 10 kiloparsecs.)

Extinction can be measured as the color excess [E(B–V)], which estimates the amount of the magnitude extinction as AV = 3.2*E(B–V). Then the distance formula with correction for extinction becomes:

[3] Distance (Parsecs) D = 100.2*[m–M+5–AV] (extinction corrected distance modulus)

Thus, an O8 star at 9550 parsecs is fainter than magnitude 19 when average extinction is taken into account, and must be on average only 2900 parsecs distant to appear at magnitude 10.5; a B5 star at 2090 parsecs is fainter than magnitude 12 and must be less than 1300 parsecs away to reach the criterion magnitude. (These estimates do not take into account the enhanced extinction in OB stars, which would reduce the extinction adjusted distances by at least half!) The density of stars is slightly greater on the left half of the plot because significant concentrations of gas and dust within the galactic disk are more distant from us in this direction.

Overall, the brightness horizon constrains our view so much that the study of double stars hardly seems to qualify as a branch of "deep sky" astronomy. We find that visually detectable double stars are generally systems within a few hundred parsecs of the Sun.

The Resolution Horizon. Distance also affects our ability to resolve the angular separation of a given orbital radius. The happenstance aspect or orientation of the orbit to our line of sight will also affect whether a binary can be resolved, so I assume the most favorable configuration — a circular orbit that we observe perpendicular to the orbital plane.

The table (below) illustrates the angular width of six orbital radii: the close orbit of 2.5 AU, the approximately Jovian orbit of 10 AU, the median orbit of 50 AU, the gravitationally robust wide limit of 1000 AU, the CPM limit of 5000 AU, and the fragile limit of 25,000 AU, calculated for a 2 solar mass system. The illustrative distances chosen are approximately: to the nearest stars (4 pc), the absolute magnitude benchmark (10 pc), the solar neighborhood (~25 pc), the nearest star forming regions in Ophiuchus (~100 pc), the Gould Belt radius and approximate 10th magnitude brightness horizon (~400 pc), the near point of the Sagittarius spiral arm (~1000 pc), and the limit of our "downstream" view along the Local arm (~2500 pc).


angular width of standard binary orbits at seven distances

separations measured in arcminutes are shown in bold type

Distance (parsecs) / Apparent Magnitude (Sun)

Orbit Radius4
2.8
10
4.8
25
6.8
100
9.8
400
12.8
1000
14.8
2500
16.8

[close] 2.5 AU0.63"0.25"0.10"0.025"0.0063"0.0025"0.001"
[.] 10 AU2.5"1.0"0.40"0.10"0.025"0.010"0.004"
[median] 50 AU12.5"5.0"2.0"0.5"0.125"0.05"0.02"
[wide] 1000 AU4.2'100"40"10"2.5"1.0"0.4"
[.] 5000 AU20.8'8.3'200"50"12.5"5.0"2.0"
[fragile] 25000 AU104.2'41.7'16.7'250"62.5"25"10"

The resolution horizon clearly biases our detection toward the widest orbits and/or the systems nearest to us. Assuming the 0.5 arcsecond resolution limit of a 250mm (10 inch) aperture, even a "median" 50 AU system can be resolved only within 100 parsecs.

A 1000mm (1 meter) aperture, with a magnitude limit of 17.6 and a resolution limit of 0.11", might resolve median systems out to 400 parsecs, although Couteau's guidance (above) implies the increase in resolution will not be visually useful (probably due to magnification of the effects of thermal turbulence in the atmosphere). For that reason, the area of the table printed in gray is primarily the domain of interferometric and spectroscopic binaires.

Searching the 2014 Washington Double Star Catalog for binary or triple systems (which often contain a binary) that have Hipparcos or spectroscopic distances and where both components are above magnitude 10.5 at a separation of 0.5" or more, there are only 217 systems with a projected separation of 50 AU or less. The farthest of these systems is 75 parsecs away, which corresponds to the table column of 100 parsecs. This is the distance horizon for "median" binary systems observed with a 250mm telescope.

Given that binary components are only rarely observed in the most favorable aspect, the table overestimates the effective distance limits on our ability to resolve binary orbits. This underscores the fact that visual double stars are widely orbiting or nearby systems, even when we observe them with meter class instruments.

The Time Horizon. The third horizon in double star astronomy is the detection of positional evidence that a pair is gravitationally bound. This means the two stars must show a positional change larger than the resolution limit of the telescope within a given observation interval. The positional change can be either proper motion or orbital motion.

A star's apparent proper motion is the change in its relative position against the background of fixed stars. For a binary pair to be identified as physically related, the divergence in proper motions of the two stars must be a small proportion of their mutual proper motion. If there is little or no divergence between them, then the stars appear to be moving in the same direction at the same speed, which implies they are gravitationally bound. If we consider the proper motion data in the 2015 WDS, 95% of the 47,600 systems in the magnitude limited sample with useable proper motion measures are within 750 parsecs; the median distance is about 165 parsecs.

If we follow Paul Couteau (1981, p.180-84) and require measurable orbital motion that is sufficient to calculate a binary orbit over the 200 years of astrometric double star astronomy, then we are limited to systems that have by now been added to the 6th Orbit Catalog or notated as having an orbital solution in WDS. These must be orbits no larger than about 100 AU or the period will be too long, and this orbital radius sets a limit on the distance at which orbits can be resolved. Again avoiding assumptions by relying on WDS, 95% of systems identified as having an orbital solution are within 260 parsecs.

Orbital solutions depend on the combined mass of the binary components: the orbital radius that yields the same orbital period decreases with mass, and therefore the system must be closer to be resolved. Assuming that our orbital resolution is no less than 0.5" arcsecond and our time horizon is 200 years for the completion of a half orbit, a distance of about 235 parsecs or less is necessary in order to resolve an orbit for a matched Type B5 system, 150 parsecs for an F5 binary, 135 parsecs for a G5 binary, and 80 parsecs or less for a matched M5 binary.

Overall we assess the lower and upper limits of the time horizon, combining proper motion and orbital evidence, to be between 80 to 750 parsecs from the Sun. Given that most mass (spectral type) based orbital calculations suggest values less than 300 parsecs, we can adopt a distance of about 400 parsecs as a reasonable estimate of the time limited horizon of double star observations.

Contour of the Triple Horizon. The complex interplay of double star mass, brightness, distance, orbital separation and orbital period or common proper motion, combined with telescopic light grasp, resolution and the measurement of orbital change or common proper motion over time, produces a highly selective observational sample for double star astronomy. In general, as we move from least to most massive main sequence spectral types (from M to O), the distance determined observational limits are set first by absolute magnitude, then by orbital radius, and finally by our ability to measure dynamic change across time. These overlapping limits form the triple horizon of double star astronomy.

The table and graphic (below) illustrate the contour of this triple horizon for visual astronomers. The table columns present the four orbital periods standardized on orbital radii calculated for a 2M binary system — the "close" orbital period of 2.8 years, the "median" period of 250 years, the "wide" period of about 22,000 years, and the "fragile" period of about 2.8 million years. (Periods shorter than ~3 years are typically spectroscopic and not visual.)

These four standard orbital periods (P2) are used to calculate three binary parameters:

[1] aAU = (2M·P2)1/3 – the orbital radius aAU of a matched binary system across the nominal main sequence masses for spectral types O8 to M5, and a giant type G5III and supergiant type K5Ia;

[2] d = aAU/Ro – the distance d at which this orbital radius will subtend an angular width equal to the resolution limit of a 250 mm aperture (Ro = 0.5"); and

[3] m = 5·(log(d)–1)+M – the sum apparent magnitude 2m of the system at that resolution determined distance across the nominal absolute magnitudes for each spectral type.

Then the table values are derived in the following steps:

• Time Horizon. If the combination of bright absolute magnitude and wide orbital radius place the observational limit at a distance greater than 400 parsecs, the system is assumed (from the discussion above) to be time limited in either detectable orbital or proper motion change. For these systems, the system distance is fixed at 400 parsecs and shown in magenta type, and the apparent magnitude (m) and minimum angular separation of the orbit (ρ) at that distance are calculated as:

[4] m = 5·(log(400)–1)+M

[5] ρ = aAU/400.

The system cannot be farther away than the indicated distance or the system will appear fixed across time and we will lack dynamic evidence that it is a double star rather than a pair asterism.

• Brightness Horizon. Among the remaining systems, the calculated apparent magnitude at the resolution limit may be greater than 10.5, and the system is brightness limited. For these systems, the magnitude is fixed at 10.5 and shown in cyan type, and the maximum distance of the system at that apparent magnitude and the corresponding apparent orbital radius are calculated as:

[6] d = 10(1+((10.5-M)/5))

[7] ρ = aAU/d.

The system cannot be farther away than the indicated distance or the system will be too faint to measure with the standard aperture.

• Resolution Horizon. By definition, the remaining systems are resolution limited by the angular resolution criterion (0.5"). For these systems, the minimum resolution is given as 0.5" and shown in yellow type, with the maximum distance and apparent magnitude given by formulas [2] and [3].

These systems cannot be farther away than the indicated limit distance or the average orbital separation will be too small to be resolved by the aperture, assuming an optimal inclination of the orbit and relative position of the components.

The results (table, below) demonstrate the effects of the triple horizon. The benchmark orbital radii for solar mass systems are shown in red type. The large effects of system mass are apparent in the range of values for the "median" orbital period of 250 years, which produces orbital separations from 29 to 146 astronomical units.


contour of the "triple horizon" in double star astronomy

assuming a 0.5" aperture resolution limit, a 10.5 foveal magnitude limit, and a 400 parsec dynamic limit


Spectral Type
Absolute Magnitude
System Mass (M)
Avg. Orbital Velocity

Illustrative Orbital Scales

close

(P = 2.8
years)

median

(P = 250
years)

wide

(P = 22,360
years)

fragile

(P = 2,795,000
years)


O8V
-4.7
46M
17.4 km/s
Radius (AU)
Apparent Mag.
Angular Separation
Max. Distance (pc)
7.1
-3.9
0.5"
14
142
2.6
0.5"
284
2844
3.3
7.1"
400
71,100
3.3
3.0'
400
B5V
-1.2
11.8M
10.2 km/s
Radius (AU)
Apparent Mag.
Angular Separation
Max. Distance (pc)
4.5
-1.4
0.5"
9
90
5.1
0.5"
181
1810
6.8
4.5"
400
45,170
6.8
113"
400
A5V
1.95
4.0M
8.1 km/s
Radius (AU)
Apparent Mag.
Angular Separation
Max. Distance (pc)
3.2
0.9
0.5"
6
63
7.5
0.5"
126
1260
10.0
3.1"
400
31,500
10.0
79"
400
F5V
3.5
2.8M
6.4 km/s
Radius (AU)
Apparent Mag.
Angular Separation
Max. Distance (pc)
2.8
2.2
0.5"
6
56
8.7
0.5"
112
1120
10.5
4.5"
251
27,970
10.5
111"
251
G5V
5.1
1.84M
6.0 km/s
Radius (AU)
Apparent Mag.
Angular Separation
Max. Distance (pc)
2.4
3.5
0.5"
5
50
10.2
0.5"
100
1000
10.5
8.7"
115
25,000
10.5
3.4'
120
K5V
7.4
1.34M
5.3 km/s
Radius (AU)
Apparent Mag.
Angular Separation
Max. Distance (pc)
2.2
5.6
0.5"
4
44
10.5
1.0"
43
875
10.5
20.5"
43
21,880
10.5
8.5'
43
M5V
12.3
0.5M
3.5 km/s
Radius (AU)
Apparent Mag.
Angular Separation
Max. Distance (pc)
1.6
10.5
0.4"
4
31
10.5
7.1"
4
620
10.5
2.4'
4
15,540
10.5
1.0°
4
G5III
0.9
2.2M
6.0 km/s
Radius (AU)
Apparent Mag.
Angular Separation
Max. Distance (pc)
2.6
-0.5
0.5"
5
52
6.0
0.5"
103
1030
8.9
2.6"
400
25,800
8.9
65"
400
K5Ia/b
-5.9
26M
6.0 km/s
Radius (AU)
Apparent Mag.
Angular Separation
Max. Distance (pc)
5.9
-5.4
0.5"
12
118
1.1
0.5"
235
2350
2.2
5.9"
400
58,780
2.2
147"
400

The log/log plot of system distance as a function of system mass (below) shows how the three horizons limit double star astronomy.

The principal restriction arises from the resolution limit (yellow line), which is sloping because the orbital radius for a constant orbital period increases with mass, and this larger radius can be resolved at greater distances. These 250 year orbits can be resolved at a distance of about 100 parsecs for a type G5 binary and about 290 parsecs for an O8 binary.

We can extend the reach of visual observation by either increasing the orbital radius that we observe — the "wide" 22,000 year orbit shows how the horizon shifts in that case — or by increasing the aperture to shrink the resolution limit. We can press this advantage until the domain of visual double stars is entirely bounded by the brightness and dynamic limits. The systems we gain in this way will be more massive "solar type" systems from about G5 to F0. At that point we need a resolution limit of 0.18" (which requires a 630mm or 25 inch aperture). The minimum orbital period at this point is about 1400 years: 95% of the systems in the 6th Catalog of Orbits of Visual Binary Stars have estimated periods no greater than 1210 years.

Increasing the resolution would increase our ability to detect small changes in position or proper motion, which ought to push back the time horizon. In fact, the time horizon is difficult to overcome with larger aperture or space based astrometry because these are instruments of recent construction; the new orbits they can detect will have changed very little over a few decades. The only alternative with apparently "fixed" (unchanging) systems beyond the time horizon is the dubious and inconclusive coin toss of a probability test. Note in particular that this dynamic horizon severely censors our ability to learn the distribution of wide or fragile orbits among high mass (especially O, B, giant and supergiant) stars, which are rare but exceedingly bright and therefore typically far away.

The brightness horizon (cyan line) is uneven due to the complex effects on brightness of stellar temperature, opacity and radius. However it clearly censors our view of the "late" K and M spectral types, because of their intrinsic faintness, to within a few dozen parsecs of the Sun. At those distances the largest orbits can appear enormous — as wide as 3.2° in a "fragile" M5 system. These separations will not be visually recognizable as systems: the orbital radius between components must be triangulated from astrometric or spectroscopic parallaxes, then confirmed with comoving proper motions.

Pushing back the magnitude limit with larger aperture gives us visual access to less massive K and M systems and to planetary scale orbits such as the "close" orbit of the table. The association of smaller physical orbits with less massive stars is evident in the 240 log 3 or log 4 systems (out of the 262 systems described above) with spectral classifications in WDS: none are Type O, 2 are Type B, 8 are Type A, 54 are Type F, 88 are Type G, 67 are Type K, and 22 are Type M.

In typical amateur telescope apertures of 350 mm or less, the meat of double star astronomy is with local, "solar type" F, G and larger K type stars. The twin aperture constraints of light grasp and angular resolution limit the double star astronomer to widely spaced double star systems within about 400 parsecs of the Sun. If we also require an orbital solution from two centuries of visual measurements, then we are limited to the brightest stars within 250 parsecs, or less massive solar type systems within 150 parsecs, and in a 250mm telescope "median" orbits can be resolved at no more than half that distance. Double star astronomy is to a great extent the study of only the nearest and brightest corner of the solar neighborhood.

Double Stars & Star Formation

William Herschel was perhaps the first to surmise that the luminous matter of H II regions and planetary nebulae was "fit to produce a star by its condensation." The large proportion of binary systems and the wide range of binary orbital characteristics seem to require the twin conclusions that most individual stars originally form as binary or multiple star systems and that most or all star systems form as members of a star cluster. Thus a basic understanding of star formation processes and the role of star clusters in their early history is essential to place double star systems within the Galactic context.

How Stars Form. In outline, star formation occurs across three distinct stages, which are completed after roughly 10 to 30 million years.

1. Spiral arm shock waves and supersonic turbulence fragment and compress cold, giant molecular clouds (GMCs) into sheets and filaments, which contract gravitationally into dense cloud cores.

About 4% to 8% of the Galaxy mass (equivalent to 8 to 16 billion solar masses) is in the form of an interstellar medium (ISM) consisting of atomic hydrogen and helium, molecular hydrogen and "dust" (grains of ice, graphite and silicates). The ISM is confined to the Galaxy disk by lines of magnetic flux and internal friction, and revolves at different speeds than the Galaxy spiral arm density waves. As a result, the gas clouds encounter spiral waves sweep across the ISM and compress it into dark, cold (10 K) giant molecular clouds (GMCs), whose turbulent and fragmented texture is beautifully captured in early Milky Way photographs made by E.E. Barnard (image, left). A single cloud can be comparable in mass (~104-106 M) and diameter (50-150 pc) to a globular cluster.

Clouds are normally supported against their own gravity and held within the Galaxy disk by lines of magnetic flux. But turbulence from the passage of spiral arm shock waves and from nearby supernova explosions can compress the filaments further into cloud cores with a mass of around 102-103 M and a radius of 0.5 to 1 parsec (more than 105 AUs). These cores are the seeds of star formation.

2. At a critical density, cloud cores collapse under their own gravity to form protostellar objects (PSOs); these burn deuterium, gather mass and dissipate angular momentum inside a circumstellar accretion disk and a cloud "cocoon" of hot dust.

Each cloud core is a concentration of mass within a certain spatial radius; some of these cloud cores are visible as discrete Bok gobules found in star forming regions (image, right top). When this mass concentration exceeds a limit known as the Bonnor-Ebert density, gravity overwhelms the thermal pressure and magnetic support within the cloud core and an isothermal free fall collapse occurs, forming one or more disklike, rotating protostellar objects (PSOs), each with a radius of ~1-5 AU and a mass of around 10–3 M.

The collapse removes thermal support underneath the surrounding gas, which falls toward the PSO. This "late" infalling gas forms a rotating accretion disk around the PSO with a radius up to ~100 AU. Computer simulations suggest that accretion disks with masses near and above ~1M form spiral shock waves and asymmetric mass concentrations which create binary/multiple protostellar objects through disk fragmentation. In other words, binary and multiple star systems begin to form well before star formation is completed.

When gravity contracts the PSO far enough to raise the central temperature above 2000 K, molecular hydrogen (H2) is ionized and the protostar becomes opaque, creating a radiant surface or "photosphere". A second core collapse to around 0.3 AU occurs, deuterium fusion begins and raises core temperature to 106 K, and the protostar luminosity rises from 5 L to over 1000 L, almost entirely in the infrared. The clouds of dust and gas become almost transparent in the infrared, and at their high luminosities these protostars become visible deep inside the surrounding dust clouds and while still within the "brown dwarf" mass range. This has made star formation regions crucial sampling areas for an unbiased specification of the stellar initial mass function (IMF).

Mass continues to accrete from the surrounding cloud into the circumstellar disk, and from the disk into the YSO. This process is still poorly understood, but it seems likely that the accelerating angular momentum in the collapsing matter is braked by torque and stellar winds from the accretion disk, while matter that cannot adhere to a rapidly spinning protostar is blown away in bipolar jets along the rotational axis, at velocities up to ~100 km/s, which create the energetic mass outflows observed as Herbig-Haro jets. These jets create shock waves in the surrounding cloud core that can initiate new protostar formation.

3. After roughly one million years of accretion, the PSO attains sufficient mass and density to initiate hydrogen fusion. Planets may form from the remnant disk, otherwise radiation from the new star evaporates the gas/dust "cocoon" and the star takes its place on the main sequence.

The protostar gains more than 90% of its mass through irregular accretion events. At the same time, it dissipates roughly 99% of the angular momentum it inherited from the GMC via disk gravitational/magnetic torque, binary orbital energy, and dynamic interactions with other protostars.

The protostar gradually gains mass and contracts under its own gravity until the core reaches fusion temperatures (the hydrogen burning limit at about 107 K). The resulting enormous increase in surface temperature and UV radiation pushes back and photoevaporates the accretion disk and surrounding cloud "cocoon", ending mass accretion. At this point the object appears as a young stellar object (YSO) or T Tauri star (image above, bottom). The new star descends to its "mature" position on the main sequence, where it remains for most of its existence.

The Natal Star Cluster. A giant molecular cloud may form hundreds or thousands of protostars or young stellar objects (YSOs) within a few hundred cubic parsecs of space, which becomes the natal star cluster for the emerging stars. This cluster is the environment that shapes the mass distribution of stars and the final distribution of dynamical elements of double star systems, and can even stimulate further star formation within nearby regions of the cloud.

As an example (panel A, left), star formation may already be under way in a GMC roughly 50 parsecs long that is compressed or shocked by a spiral arm density wave, supernova shock waves or turbulence. This cloud compresses local regions of the cloud to the point where cloud cores collapse and protostars begin to accrete, often in close proximity to one another.

The most massive Type O and B stars, formed in the densest concentrations of gas and accreting mass most rapidly, descend to the main sequence first (panel B). Heated to 104 K and ionized by the UV radiation from young Type O and B stars, the surrounding GMC lights up as hydrogen emission nebulae or H II regions, such as the Messier nebulae M 8, M 20 and M 42.

These massive stars have a lifetime of a few hundred thousand to a few million years. Shock waves from the supernova demise of the largest Type O stars disperse some of the cloud mass and create "runaway" (high proper motion) single stars (panel C). But the shock fronts and radiation from new stars also compress the surrounding gas within the extended GMC, accelerating star formation. This "cascade" effect may explain why many star forming regions and new star clusters show evidence of increased star formation near the end of the star formation process.

The final stage is the dissolution of the natal star cluster (panel D). As new stars form, cold mass outflows and OB radiation disperse ~95% of the GMC mass within ~3x107 years, leaving a naked star cluster. This loss of mass drastically reduces the gravitational potential of (and escape velocity from) the cluster. The cluster unbinds into field stars and expanding OB stellar associations. About 90% of star clusters disperse within a few 107 years — very few star clusters last longer than 108 years, and those that do (such as M 11 or the Hyades) are massive fossils of massive cloud collapse.

All these stages are exemplified in an area about 1680 parsecs wide from the high resolution Hubble Legacy image of NGC 5194 (M 51) a grand design spiral galaxy in Ursa Major. (This illustration is adapted from Elmegreen, 2007.) The orbital flow of stars and molecular clouds is approximately from upper left to lower right; these are compressed along the spiral arm shock wave outlined by a nearly continuous concentration of gas and dust. This contains many collapsing giant molecular clouds and protostars (1), visible as small dots of red within the spiral arm. By the time these star forming regions have passed through the shock wave they have matured into giant H II regions (2) that have begun to disperse the natal gas. This process is completed as the star clusters revolve farther from the shock wave, resulting in emerging Galactic star clusters and secondary star forming regions (3) as well as expanding OB associations (4). Note that many cloud fragments survive the dispersion turbulence, and secondary star forming regions appear within the sheared dust and gas filaments visible in the lower left quadrant of the image.

A spectacular example of an emerging "starburst" cluster is the dense central concentration of massive stars (R136) at the center of the 200 parsec wide star forming region NGC 2070 (30 DOR or Tarantula Nebula, image right) in the Large Magellanic Cloud. Here the infalling molecular gas has been compressed by ram pressure against the central concentration of LMC interstellar medium, producing an H II region with an estimated total mass of 5x105M. This detail image shows the natal gas and dust being pushed back by radiation from the dense concentration of high mass Type O and B stars, a vivid example of mass segregation at the center of a star forming region.

The surprising outcome is that only a small fraction (<5%) of the total GMC mass has been transformed into gravitationally bound bodies: the rest has been dispersed back into the interstellar medium and enriched by the material ejected in supernovae explosions, where it will be mixed and compressed by future turbulence. In addition, roughly one in four protostars never reach the hydrogen burning limit and instead become brown dwarfs or "failed stars".

Many Galactic clusters display a variety of binary systems, especially near the cluster center. Look for mass segregation (bright and/or red giant stars and binary systems near cluster center, e.g. M 47); also look for close, "matched" binary systems.

How Double Stars Form. Several decades of research into the formation of binary and multiple stars has concluded that a single formation process or dynamic interaction cannot explain large number of double stars and the very large variety in their mass ratios and orbital elements. Instead, double stars seem to emerge at several points in the star formation process. Six mechanisms have been proposed to influence double star formation, and in three of these the development and influence of the protostar cluster is a critical factor.

• Molecular Cloud Fragmentation (r > 100 AU): Binaries form "in place" during the turbulent collapse of a massive, dense cloud core, which creates a clustered central distribution of stars that become gravitationally bound in the last stages of contraction and mass accretion.

• Protostar Disk fragmentation (r < 100 AU): Computer simulations strongly suggest that the disk of accreting material around a contracting protostar is highly susceptible to density variations that cause fragmentation into a companion star or, in systems with a high content of heavier elements (high metallicity), planetary systems. These companion protostars modulate the angular momentum, accretion rates and mass ratio of the evolving protobinary, and are strongly associated with the appearance of Herbig-Haro jets. Infrared surveys suggest that >80% of mass outflows and polar jets occur in binary protostellar systems.

An early smoothed particle hydrodynamic (SPH) computer simulation of protostar disk fragmentation (Bonnell & Bate, 1994b)

• Competitive Accretion & Mass Segregation: A protostar grows more massive by accreting gas and dust from its surroundings, but at masses characteristic of a Type A star (~10M) radiation from the emerging star becomes powerful enough to push back surrounding dust and prevent further accretion. How then do the most massive (O and B) stars attain masses as high as 150M? Forced accretion of gas and dust and the occasional collision or merging of protostars, within the gravitational well of a dense cluster core, appear to be the only plausible mechanisms, and mimic on a small scale the formation of massive black holes at the center of many galaxies. This is consistent with the observation of the most massive stars and binary systems near the center of many protostar and star clusters. The same environment that produces forced accretion and protostar collisions would produce sufficiently frequent encounters to explain the high multiplicity fractions in high mass stars.

• Dynamical interactions (r < 10 AU, and "brown dwarf desert"): The orbits of double systems within a recently formed protostar cluster will bring binary and multiple systems into gravitational interaction, with complex results. The main effect will be to strip low mass and brown dwarf components from multiple systems, sending them out of the protostar cluster as high velocity or "runaway" stars. Disruptive gravitational encounters can also "harden" (increase the orbital energy and reduce the orbital radius) already formed binary orbits. These interactions have been proposed to explain the many binary systems with orbits < 1 AU, which are too close to emerge from protostellar disk fragmentation. Simulations by Reipurth & Mikkola (2013) suggest that an originally triple system can evolve by dynamical interactions into a wide "pair" consisting of a close binary with a distant component that is later stripped from the binary by disruptive encounters with passing stars or clouds of interstellar matter.

• Gas induced orbital decay (r < 1 AU): Orbital periods of already close protobinary systems may be shortened further by friction between the protostars and the surrounding protocluster gas cloud, during the interval after the stars have accreted most of their mass but before the stars begin nuclear fusion and the cloud is dispersed by stellar radiation. This would produce the contact and interacting binary systems observed in the Galactic population.

• Parallel Dispersion (r > 1000 AU): "Soft" or "fragile" binaries are too widely separated to survive even one or two orbits through a protostar cluster. Instead, they may bind when released in parallel trajectories (Galactic orbits) as their natal star cluster becomes gradually unbound, or when unstable triple or higher order systems break apart.

The consensus opinion of double star astronomers is that the very different orbital period, distance and eccentricity profiles of binary systems requires the contribution of very different star formation mechanisms, acting in a very different sequence or with different relative effects, due to differences in the mass, structure and turbulence of the natal giant molecular cloud, the presence and size of a protostar cluster, and the spacing and dynamics of protostar systems and accretion disks.

Mechanisms That Fail. Two processes formerly proposed to explain binary star formation — gravitational capture of one star by another, or the centrifugal splitting of a contracting and rapidly rotating protostar — appear inadequate to explain the observed large number of binary systems and the range of their orbital elements.

Gravitational capture requires the temporary influence of a third star to carry away enough angular momentum to slow the remaining components into an orbital trajectory. But this is as likely to disrupt an already formed binary system as to create a new one. In fact, gravitational disruption is most likely to eject the lowest mass components of a system, or "harden" already stable binary orbits.

The "fission" hypothesis recognized that as a protostar contracts it must spin more rapidly, just as an ice skater spins more rapidly by pulling arms close to the body, and that this spinning might tear the protostar in two, forming a contact or interacting binary pair. But three decades of computer simulations have shown that rapidly rotating, "pseudo liquid" protostars shed matter from the tips of their elongated volume, which slows the rotation enough to prevent fission. The accretion disk around a protostar does seem very susceptible to fragmentation, especially when infalling matter is concentrated on one side, but this process forms binaries at much larger distances than the orbits of the closest binary pairs.

Further Reading

The literature on double stars is enormous, and overlaps with research on star formation, star clusters, galactic processes and stellar evolution. The following articles provide an entry and include useful references to further work; many can be downloaded by entering author last names and date of publication in the search function at the SAO/NASA Astrophysics Data System.

"The Definition of the Term Double Star" (1911) by R.G. Aitken – one of the earliest proposals to define separation limits for physical double stars, with interesting comments from a large number of early 20th century astronomers.

Double Stars (1978) by Wulff Heintz – a classic in the field, compact and informative, although dense and in many places technical.

Observing Visual Double Stars (1981) by Paul Couteau – reader friendlier than Heintz, and with many topics of interest to the double star observer.

Small Clusters or Double Stars? (1987) by L. Lodén – There is no well defined border between small open clusters and wide multiple systems.

"Statistical Studies of Wide Pairs" (1988) by J.-L. Halbwachs – a review of statistical methods to identify optical double stars, and as applied to wide systems where optical pairs are most likely.

"A Statistical Approach for the Recognition of Physical Visual Double Stars" (1991) by D. Sinachopoulos & P. Mouzourakis – develops limits for the maximum angular separation for any pair of stars of given magnitude, calculated for the whole sky.

"Multiplicity among solar-type stars in the solar neighbourhood: II. Distribution of the orbital elements in an unbiased sample" (1991) by A. Duquennoy & M. Mayor – for two decades the standard analysis of main sequence stellar multiplicity.

"Multiple stellar systems: From binaries to clusters" (1999) by Ian Bonnell — A concise overview of the state of research in double star formation within newly forming star clusters.

"Embedded Clusters in Molecular Clouds" (2003) by C. Lada & E. Lada - an excellent overview of advances in infrared astronomy, advanced surveys of star forming regions, and the insights these yield about star formation.

"The Physics of Star Formation" (2003) by Richard Larson – a comprehensive, technical and critical review of star and binary formation processes by a major theorist in the field.

"Are Binary Separations Related to their System Mass?" (2004) by M. Sterzik & R. Durisen – a Monte Carlo study exploring the role of system mass in the multiplicity fraction and binary orbital elements.

"Star Formation in the Galaxy, An Observational Overview" (2005) by C. Lada – a basic overview of star formation processes with emphasis on solar type stars.

"Astrometric Binaries in the Age of the Next Generation of Large (Space) Telescopes" (2005) by R. Olling – a demonstration of selection effects that vex current double star catalogs.

"On the rapid collapse and evolution of molecular clouds" (2007) by B. Elmegreen – a remarkable review paper of galactic star formation processes by one of the leading advocates of "prompt" star formation.

"Do binaries in clusters form in the same way as in the field?" (2009) by R. Parker, S.P. Goodwin, P. Kroupa & M.B.N. Kouwenhoven – summary of the dynamical processing that binaries receive from their natal star cluster, by leading theorists in this field.

"On the evolution of a star cluster and its multiple stellar systems following gas dispersal" (2010) by N. Moeckel & M. Bate – a recent paper using computer simulation to trace the evolution of double stars within a natal cloud cluster.

"A Survey of Stellar Families: Multiplicity of Solar Type Stars" (2010) by D. Raghavan, H. McAlister, T. Henry, D. Latham, G. Marcy, B. Mason, D. Gies, R. White & T. ten Brummelaar – the latest, most precise and intensive analysis of solar type multiplicity.

"Formation of the Widest Binaries from Dynamical 'Unfolding' of Triple Systems" (2013) by B. Reipurth & S. Rikkola – Computer simulations that show how both wide binaries and very close binaries can evolve from the dynamic decay of triple systems.

"Stellar Multiplicity" (2013) by Gaspard Duchene & Adam Kraus – a recent review paper on the relationships between multiplicity and stellar characteristics such as mass, age and clustering.

"Multiplicity in Early Stellar Evolution" (2014) by Bo Reipurth, Cathie Clarke, Alan Boss, Simon Goodwin, Luis Rodriguez, Keivan Stassun, Andrei Tokovinin & Hans Zinnecker – an overview of the current state of research into the multiplicity observed in star forming regions and YSOs.

The Exoplanet Catalog - An innovative online database of all 747 confirmed exoplanet systems.

 

Last revised 07/03/16 • ©2016 Bruce MacEvoy