Fundamental Stellar Parameters

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 B1913+16, 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


- effective temperatures- radii Square of the Visibility Amplitude of Vega measured with the CHARA Array


- effective temperatures- radii- masses

Learning Objectives Celestial Orbits Circular

orbits Elliptical orbitsParabolic orbits

Hyperbolic orbitsConic sections

Conservation of Angular MomentumOrbital trajectories

Orbital velocitiesOrbital periods

Transforming a 2-Body to an equivalent 1-Body Problem Center of massReduced 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, …):

Circular orbit, unequal masses

Celestial Orbits


Circular orbit, unequal masses

Celestial Orbits


Elliptical orbit, equal masses

Celestial Orbits


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

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







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 = m1 + m2 the total mass of the system, e the eccentricity, and

.

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


where r is the separation of the two objects, L the angular momentum of the system, M = m1 + m2 the total mass of the system, e the orbital eccentricity, and

.



.

Parabolic Orbit

p2p



.

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.

Parabolic 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.

Hyperbolic Orbit

To conserve angular momentum,

when moving in circular orbits, each object must move with a constant velocity.

Conservation of Angular Momentum

m




m




m


when moving in elliptical orbits, both objects must move at higher velocities when they are closer together.


m

Conservation of Angular Momentum To conserve angular momentum,

when moving in elliptical orbits, both objects must move at higher velocities when they are closer together.

m


when moving in parabolic orbits, both objects must move at higher velocities when they are closer together.

m


when moving in hyperbolic orbits, both objects must move at higher velocities when they are closer together.

m

Orbital Periods For a circular or elliptical orbit, the orbital period is given by (see Chap. 2 of

textbook)







2-Body Problem

m1

m2

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)

m1

m2

A 2-body problem

can be reduced to an equivalent 1-body problem of a reduced mass, μ, orbiting about the total mass, M = m1 + m2, located at the origin (see Chap 2 of textbook):

Binary stars are classified according to their specific observational characteristics.

Binary Systems and Stellar Parameters

Learning Objectives Classification of Binary Stars

Optical doubleVisual binaryAstrometric binary

Eclipsing binarySpectrum binarySpectroscopic 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 pc255 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 σ2 CrB resolved with the CHARA interferometer

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Fundamental Stellar Parameters