Binary stellar systems are interesting to study for many reasons. For example, most stars are members of binary systems, and so studies of binary systems in the process of formation tell us about how most stars form. Studies of the binary system PSR B , comprising two pulsars (neutron stars), provide the only (indirect) evidence thus far for gravitational waves, a prediction of Einstein’s general theory of relativity. Binary stellar systems provide the only way to directly determine stellar masses.

Fundamental Stellar Parameters  The fundamental parameters of stars are their -effective temperatures

Fundamental Stellar Parameters  The fundamental parameters of stars are their -effective temperatures -radii Square of the Visibility Amplitude of Vega measured with the CHARA Array

Fundamental Stellar Parameters  The fundamental parameters of stars are their -effective temperatures -radii -masses

Learning Objectives  Celestial Orbits Circular orbits Elliptical orbits Parabolic orbits Hyperbolic orbits Conic sections  Conservation of Angular Momentum Orbital trajectories Orbital velocities Orbital periods  Transforming a 2-Body to an equivalent 1-Body Problem Center of mass Reduced mass system

Learning Objectives  Celestial Orbits Circular orbits Elliptical orbits Parabolic orbits Hyperbolic orbits Conic sections  Conservation of Angular Momentum Orbital trajectories Orbital velocities Orbital periods  Transforming a 2-Body to an equivalent 1-Body Problem Center of mass Reduced mass system

Celestial Orbits  Possible orbits of one star (or planet, asteroid, …) about another star (or planet, asteroid, …): Circular orbit, equal masses

 Possible orbits of one star (or planet, asteroid, …) about another star (or planet, asteroid, …): Celestial Orbits Circular orbit, unequal masses

 Possible orbits of one star (or planet, asteroid, …) about another star (or planet, asteroid, …): Celestial Orbits Circular orbit, unequal masses

 Possible orbits of one star (or planet, asteroid, …) about another star (or planet, asteroid, …): Elliptical orbit, equal masses Celestial Orbits

 Possible orbits of one star (or planet, asteroid, …) about another star (or planet, asteroid, …): Elliptical orbit, unequal masses Celestial Orbits

 Possible open orbits of one celestial object about another: Parabolic orbit (minimum energy open orbit) Celestial Orbits

 Possible open orbits of one celestial object about another: Hyperbolic orbit (open orbit) Celestial Orbits

 Possible orbital trajectories are conic sections, generated by passing a plane through a cone. What physical principle do such orbits satisfy? Conic Sections

 Possible orbital trajectories are conic sections, generated by passing a plane through a cone. What physical principle do such orbits satisfy? Conic Sections

 Possible orbital trajectories are conic sections, generated by passing a plane through a cone. What physical principle do such orbits satisfy? Conic Sections

 Possible orbital trajectories are conic sections, generated by passing a plane through a cone. What physical principle do such orbits satisfy? Conservation of angular momentum. Conic Sections

Learning Objectives  Celestial Orbits Circular orbits Elliptical orbits Parabolic orbits Hyperbolic orbits Conic sections  Conservation of Angular Momentum Orbital trajectories Orbital velocities Orbital periods  Transforming a 2-Body to an equivalent 1-Body Problem Center of mass Reduced mass system

Conservation of Angular Momentum  For a system under a central force* such as the force of gravity, it can be shown (see Chap. 2 of textbook) that the angular momentum of the system is a constant (i.e., conserved) m * A central force is a force whose magnitude only depends on the distance r of the object from the origin and is directed along the line joining them.

Orbital Trajectories  In such a case, the only possible orbital trajectories are (see Chap. 2 of textbook) where r is the separation of the two objects, L the angular momentum of the system, M = m 1 + m 2 the total mass of the system, e the eccentricity, and.  E.g., if e = 0, r = constant (i.e., circular orbit).  What if e ≠ 0?

Conic Sections  Compare Eq. (2.29) with the equations for conic sections: where p is the distance of closest approach to the parabola’s one focus. Closed orbits Just open orbit Open orbit

Conic Sections  Compare Eq. (2.29) with the equations for conic sections: where p is the distance of closest approach to the parabola’s one focus. Closed orbits Just open orbit Open orbit

Orbital Trajectories  In such a case, the only possible orbital trajectories are (see Chap. 2 of textbook) where r is the separation of the two objects, L the angular momentum of the system, M = m 1 + m 2 the total mass of the system, e the eccentricity, and.  E.g., if e ≠ 0, r = equation for ellipse.

Orbital Trajectories  Note that As e , b/a   Change of perspective: imagine you are on one star of a binary system, located at one of the focal points of the ellipse. The other star would then seem to perform an elliptical orbit around you.

Orbital Trajectories  Note that As e , b/a   Change of perspective: imagine you are on one star of a binary system, located at one of the focal points of the ellipse. The other star would then seem to perform an elliptical orbit around you.

Orbital Trajectories Parabolic Orbit p 2p2p r  Change of perspective: imagine you are on the Sun, located at the focus of the parabola. The comet would then seem to perform a parabolic orbit around you.

Orbital Trajectories  Change of perspective: imagine you are on the Sun, located at the focus of the parabola. The comet would then seem to perform a parabolic orbit around you.

Orbital Trajectories  Change of perspective: imagine you are on the Sun, located at the focus of the hyperbola. The comet would then seem to perform a hyperbolic orbit around you. Hyperbolic Orbit

Orbital Trajectories  Change of perspective: imagine you are on the Sun, located at the focus of the hyperbola. The comet would then seem to perform a hyperbolic orbit around you.

Orbital Velocities  Along the possible orbital trajectories, the velocity of one object relative to the other is given by (see Chap. 2 of textbook) where a is the semimajor axis of the orbit. Parabolic Orbit r

Orbital Velocities  Along the possible orbital trajectories, the velocity of one object relative to the other is given by (see Chap. 2 of textbook) where a is the semimajor axis of the orbit. r

Orbital Velocities  Along the possible orbital trajectories, the velocity of one object relative to the other is given by (see Chap. 2 of textbook) where a is the semimajor axis of the orbit. Hyperbolic Orbit

Orbital Velocities  Along the possible orbital trajectories, the velocity of one object relative to the other is given by (see Chap. 2 of textbook) where a is the semimajor axis of the orbit.

Orbital Velocities  Along the possible orbital trajectories, the velocity of one object relative to the other is given by (see Chap. 2 of textbook) where a is the semimajor axis of the orbit. With respect to the center of mass, the orbital velocity of the more massive object is lower.

Orbital Velocities  Along the possible orbital trajectories, the velocity of one object relative to the other is given by (see Chap. 2 of textbook) where a is the semimajor axis of the orbit. With respect to the center of mass, the orbital velocity of the more massive object is lower.

Orbital Velocities  Along the possible orbital trajectories, the velocity of one object relative to the other is given by (see Chap. 2 of textbook) where a is the semimajor axis of the orbit.

Orbital Velocities  Along the possible orbital trajectories, the velocity of one object relative to the other is given by (see Chap. 2 of textbook) where a is the semimajor axis of the orbit. With respect to the center of mass, the orbital velocity of the more massive object is lower.

Orbital Periods  For a circular or elliptical orbit, the time that one object takes to make a complete orbit about the other (orbital period) is given by (see Chap. 2 of textbook)

Learning Objectives  Celestial Orbits Circular orbits Elliptical orbits Parabolic orbits Hyperbolic orbits Conic sections  Conservation of Angular Momentum Orbital trajectories Orbital velocities Orbital periods  Transforming a 2-Body to an equivalent 1-Body Problem Center of mass Reduced mass system

2-Body Problem m1m1 m2m2  Computing the orbits of a binary system is a 2-body problem. Problems involving 2 or more bodies are more easy to analyze in an inertial reference frame that does not move with respect to the system; i.e., a reference frame coinciding with the system’s center of mass.  It can be shown (see Chap 2 of textbook) that the center of mass is located at:

2-Body and Equivalent 1-Body Problem  (at focus of ellipse) m1m1 m2m2  A 2-body problem can be reduced to an equivalent 1-body problem of a reduced mass, μ, orbiting about the total mass, M = m 1 + m 2, located at the center-of-mass (see Chap 2 of textbook):

2-Body and Equivalent 1-Body Problem m1m1 m2m2  Recall that the separation of the two objects where L the angular momentum of the system, M = m 1 + m 2 the total mass of the system, e the eccentricity, and μ = m 1 m 2 /(m 1 + m 2 ).

2-Body and Equivalent 1-Body Problem  Recall that the separation of the two objects where L the angular momentum of the system, M = m 1 + m 2 the total mass of the system, e the eccentricity, and μ = m 1 m 2 /(m 1 + m 2 ). Recall that that Eq. (2.29) also describes an ellipse:

2-Body and Equivalent 1-Body Problem  Recall that the separation of the two objects where L the angular momentum of the system, M = m 1 + m 2 the total mass of the system, e the eccentricity, and μ = m 1 m 2 /(m 1 + m 2 ). Eq. (2.29) is mathematically equivalent to a reference frame where M is located at one of the focuses of the ellipse, and is orbited by μ (Chap. 2 of textbook).  (at focus of ellipse)

2-Body and Equivalent 1-Body Problem  The velocity of μ about M is given by  The orbital period of μ about M is given by  (at focus of ellipse)

2-Body and Equivalent 1-Body Problem  In the reduced mass system, orbital separation  What is the orbital separation of m 1 and m 2 from the center of mass?  (at focus of ellipse) m1m1 m2m2

2-Body and Equivalent 1-Body Problem  In the reduced mass system, orbital separation  What is the orbital separation of m 1 and m 2 from the center of mass? Use the relationship for the center of mass m1m1 m2m2

2-Body and Equivalent 1-Body Problem  (at focus of ellipse) m1m1 m2m2  Thus, if the orbit of μ about M is an ellipse with a certain eccentricity e then the orbit of m 1 and m 2 about the center of mass also is an ellipse with the same eccentricity e.

2-Body and Equivalent 1-Body Problem  In the reduced mass system, orbital velocity  What is the orbital velocity of m 1 and m 2 about the center of mass?  (at focus of ellipse) m1m1 m2m2

2-Body and Equivalent 1-Body Problem  In the reduced mass system, orbital velocity  What is the orbital velocity of m 1 and m 2 about the center of mass? Differentiate the relationship for the center of mass m1m1 m2m2

2-Body and Equivalent 1-Body Problem  In the reduced mass system, orbital period  What is the orbital period of m 1 and m 2 about the center of mass?  (at focus of ellipse) m1m1 m2m2

2-Body and Equivalent 1-Body Problem  In the reduced mass system, orbital period  What is the orbital period of m 1 and m 2 about the center of mass? Same.  (at focus of ellipse) m1m1 m2m2

Binary stars are classified according to their specific observational characteristics. Binary Systems and Stellar Parameters

Learning Objectives  Classification of Binary Stars Optical double Visual binary Astrometric binary Eclipsing binary Spectrum binary Spectroscopic binary

Optical Double  Stars that just happen to lie nearly along the same line of sight, but are far apart in physical space and not gravitationally bound.  1/2 Capricorni 33 pc211 pc  Cassiopeiae 6 pc 255 pc

Visual Binary  True binary systems where individual components can be visually (with eyes or telescopes) separated. 23.4´ 0.17˝

Astrometric Binary  Only one component visible, presence of companion inferred from oscillatory motion of visible component.

Astrometric Binary  Sirius was discovered as an astrometric binary in 1844 by the German astronomer Friedrich Wilhelm Bessel. With modern telescopes, Sirius is a visual binary (separation ranging from 3″ to 11″ depending on orbital phase).

Eclipsing Binary  Two stars not separated. Binarity inferred when one star passes it in front and then behind the other star causing periodic variations in the observed (total) light.

Eclipsing Binary  Two stars not separated. Binarity inferred when one star passes it in front and then behind the other star causing periodic variations in the observed (total) light.

Spectrum Binary  Two stars not separated. Binarity inferred from two superimposed, independent, discernible spectra. If orbital period sufficiently short, both spectra exhibit periodic and oppositely-directed Doppler shifts (hence also spectroscopic binary).

Spectroscopic Binary  Two stars not separated. Binarity inferred from periodic and oppositely-directed Doppler shifts in spectra of one (single-lined spectroscopic binary) or both (double-lined spectroscopic binary) detectable components. observer

Binary Systems  These classes of binary systems are not mutually exclusive. For example, with ever increasing angular resolutions provided by modern telescopes, some spectroscopic binaries have now been resolved into visual binaries. Spectroscopic binaries may also be eclipsing systems. Spectroscopic Binary σ 2 CrB resolved with the CHARA interferometer