Fundamental Stellar Parameters

1. 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.

2. Fundamental Stellar Parameters • The fundamental parameters of stars are their - effective temperatures

3. 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

4. Fundamental Stellar Parameters • The fundamental parameters of stars are their - effective temperatures - radii - masses

5. Learning Objectives • Celestial OrbitsCircular 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

6. Learning Objectives • Celestial OrbitsCircular 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

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

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

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

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

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

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

13. Celestial Orbits • Possible open orbits of one celestial object about another: Hyperbolic orbit

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

15. 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.

16. 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

17. 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.

18. 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 .

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

20. 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 orbital eccentricity, and .

21. 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 orbital eccentricity, and . r 2p p Parabolic Orbit

22. 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 orbital eccentricity, and . Hyperbolic Orbit

23. 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 Parabolic Orbit

24. 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

25. Conservation of Angular Momentum • To conserve angular momentum, when moving in circular orbits, each object must move with a constant velocity. m

26. Conservation of Angular Momentum • To conserve angular momentum, when moving in circular orbits, each object must move with a constant velocity. m

27. Conservation of Angular Momentum • To conserve angular momentum, when moving in circular orbits, each object must move with a constant velocity. m

28. 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

29. 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

30. Conservation of Angular Momentum • To conserve angular momentum, when moving in parabolic orbits, both objects must move at higher velocities when they are closer together. m

31. Conservation of Angular Momentum • To conserve angular momentum, when moving in hyperbolic orbits, both objects must move at higher velocities when they are closer together. m

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

33. 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

34. 2-Body Problem • 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: m2 m1

35. 2-Body and Equivalent 1-Body Problem • A 2-body problem m2 m1 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 center-of-mass (see Chap 2 of textbook): (at focus of ellipse) 

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

37. Learning Objectives • Classification of Binary StarsOptical double Visual binary Astrometric binary Eclipsing binary Spectrum binary Spectroscopic binary

38. 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.  Cassiopeiae  1/2 Capricorni 211 pc 6 pc 33 pc 255 pc

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

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

41. 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).

42. 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.

43. 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.

44. 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).

45. 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

46. 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 σ2CrB resolved with the CHARA interferometer