1. The true orbits of visual binaries can be determined from their observed orbits as projected in the plane of the sky. Once the true orbit has been computed, determining the total mass of the two stellar components only requires knowledge of the distance to the system. Determining the ratio in masses of the two stellar components requires only measurements of their projected separation and location of their center of mass. Binary Systems and Stellar Parameters

2. Learning Objectives  Visual BinariesReference frame Total Mass Mass Ratio Determining the Center of Mass

3. Learning Objectives  Visual BinariesReference frame Total Mass Mass Ratio Determining the Center of Mass

4. Visual Binary: Reference Frame  Consider a visual binary whose orbital plane is in the plane of the sky (i.e., observed face-on as in the illustrations below).  In the absence of (fixed) background stars to serve as reference points, can you tell where the center-of-mass of a binary system is located?

5. Visual Binary: Reference Frame  Consider a visual binary whose orbital plane is in the plane of the sky (i.e., observed face-on as in the illustrations below).  In the absence of (fixed) background stars to serve as reference points, can you tell where the center-of-mass of a binary system is located?

6. Visual Binary: Reference Frame  Consider a visual binary whose orbital plane is in the plane of the sky (i.e., observed face-on as in the illustrations below).  In the absence of (fixed) background stars to serve as reference points, can you tell where the center-of-mass of a binary system is located?

7. Visual Binary: Reference Frame  In practice, astronomers measure the orbit of one star about the other star (held at a fixed location). The brighter (primary) star is usually used as the reference point.

8. Visual Binary: Reference Frame  In practice, astronomers measure the orbit of one star about the other star (held at a fixed location). The brighter (primary) star is usually used as the reference point.

9. Visual Binary: Reference Frame  This is equivalent to translating a 2-body problem  (at focus of ellipse) m1m1 m2m2 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):

10. Visual Binary: Reference Frame  This is equivalent to translating the 2-body into a 1-body problem. The observed semimajor axis of the orbit corresponds to the semimajor axis of the orbit in the reduced mass system.  (at focus of ellipse)

11. Learning Objectives  Visual BinariesReference frame Total Mass Mass Ratio Determining the Center of Mass

12. Visual Binary: Total Mass  The total mass of the system can be inferred from the orbital period, P, and the semimajor axis of the orbit, a, according to Kepler’s 3 rd law

13. Visual Binary: Total Mass  The total mass of the system can be inferred from the orbital period, P, and the semimajor axis of the orbit, a, according to Kepler’s 3 rd law Astronomers measure angles in the sky. If the angle subtended by the semimajor axis is α, the dimension of the semimajor axis a = αd, where d is the distance to the binary system Deriving the dimension of the semimajor axis, a, and hence total mass of the system, m 1 + m 2, therefore requires knowing the distance, d, to the binary system. d α a

14. Visual Binary: Total Mass  What if the orbital plane is inclined with respect to the plane of the sky?  Consider a binary system with an intrinsically circular orbit. If the orbital plane is inclined to the sky plane, the observed orbit will appear to be elliptical. How can we tell the difference between an inclined circular orbit and an actual elliptical orbit? True OrbitProjected Orbit

15. Visual Binary: Total Mass  What if the orbital plane is inclined with respect to the plane of the sky?  Consider a binary system with an intrinsically circular orbit. If the orbital plane is inclined to the sky plane, the observed orbit will appear to be elliptical. How can we tell the difference between an inclined circular orbit and an actual elliptical orbit? One star has to be at the center of the ellipse if the actual orbit is circular.  It is possible to incline an elliptical orbit such that one star lies at the center of the ellipse. How then do we tell the difference between an inclined circular orbit and an actual elliptical orbit?  (at focus of ellipse) Projected OrbitTrue Orbit

16. Visual Binary: Total Mass  What if the orbital plane is inclined with respect to the plane of the sky?  Consider a binary system with an intrinsically circular orbit. If the orbital plane is inclined to the sky plane, the observed orbit will appear to be elliptical. How can we tell the difference between an inclined circular orbit and an actual elliptical orbit? One star has to be at the center of the ellipse if the actual orbit is circular.  It is possible to incline an elliptical orbit such that one star lies at the center of the ellipse. How then do we tell the difference between an inclined circular orbit and an actual elliptical orbit? Observed radial velocity along the orbit differentiates between possible orbits.  In practice, the (projected) location of the reference star relative to the focus, (projected) orbital trajectory, and (projected) orbital velocity of the binary companion along the orbit can be used to uniquely constrain the actual orbit.

17. Visual Binary: Total Mass  Consider a binary system with an intrinsically elliptical orbit. If the orbital plane is inclined to the sky plane, the observed orbit will be an elliptical orbit with a different eccentricity. The star held fixed (projected focus) will not appear to be located at the focus of the observed elliptical orbit.  This figure corresponds to the special case in which the orbital plane intersects the sky plane along a line parallel to the minor axis. The position of the star held fixed (projected focus) lies along the major axis, but does not coincide with the focus of, the observed orbit.  The true orbit can be derived from the projected orbit (projected location of the star held fixed relative to focus of observed orbit, projected orbital trajectory, and projected relative velocity of the secondary star along its orbit).

18. Visual Binary: Total Mass  In general, the orbital plane and the sky plane can intersect along a line at any angle with respect to the minor axis. The position of the star held fixed (projected focus) does not necessarily have to lie along the major axis of the orbit. Once again, the true orbit can be derived from the projected orbit.

19. Visual Binary: Total Mass  In general, the orbital plane and the plane of the sky can intersect along a line at an angle Ω with respect to the minor axis. Ω: position angle of ascending node ω: argument of periastron

20. Visual Binary: Total Mass  Let us return to the special case in which the orbital plane intersects the sky plane along a line parallel to the minor axis. Once the inclination, i, of the orbital to sky plane has been determined, the total mass can be derived from Eq. (2.37) where is the angle subtended by the projected semimajor axis. d i

21. Learning Objectives  Visual BinariesReference frame Total Mass Mass Ratio Determining the Center of Mass

22. Visual Binary: Mass Ratio  If we can locate the center-of-mass of a binary system, we can determine the mass ratio of the two stellar components. From Eq. (2.19) we find m2m2 υ1υ1 υ2υ2  (at focus of ellipse)

23. m2m2 υ1υ1 υ2υ2 Visual Binary: Mass Ratio  For a visual binary whose orbital plane is in the sky plane, r 1 =a 1 (1+e) and r 2 =a 2 (1+e) so that the mass ratio of the two components where α 1 and α 2 are the angles subtended by a 1 and a 2 respectively. Unlike deriving the total mass, deriving the mass ratio does not require knowing the distance to the binary system.

24. Visual Binary: Mass Ratio  In the special case where the orbital plane intersects the sky plane along a line parallel to the minor axis, the orbital semimajor axes of the two stellar components are foreshortened in the same manner so that where and are the projected angles subtended by a 1 and a 2 respectively.  Unlike deriving the total mass, deriving the mass ratio does not require knowing the distance to the binary system.

25. Visual Binary: Mass Ratio  In the general case where the orbital plane does not intersect the sky plane along a line parallel to the minor axis, we first have to derive the orbital inclination before deriving the mass ratio in the same manner as described earlier.  Unlike deriving the total mass, deriving the mass ratio does not require knowing the distance to the binary system.

26. Learning Objectives  Visual BinariesReference frame Total Mass Mass Ratio Determining the Center of Mass

27. Visual Binary: Determining the Center of Mass  How can we locate the center of mass of a binary system? ⇒ ⇒ ? or

28. Visual Binary: Determining the Center of Mass  There are two ways to determine the center of mass of a binary system. One way is to measure the center of mass of the system with respect to fixed (much more distant) background stars after the annual oscillation due to parallax (if measurable) has been removed.

29. Visual Binary: Determining the Center of Mass  There are two ways to determine the center of mass of a binary system. One way is to measure the center of mass of the system with respect to fixed (much more distant) background stars after the annual oscillation due to parallax (if measurable) has been removed. center of mass

30. Visual Binary: Determining the Center of Mass  There are two ways to determine the center of mass of a binary system. One way is to measure the center of mass of the system with respect to fixed (much more distant) background stars. In the illustration below, the annual oscillation due to parallax (if measurable) has been removed.

31. Visual Binary: Determining the Center of Mass  There are two ways to determine the center of mass of a binary system. One way is to measure the center of mass of the system with respect to fixed (much more distant) background stars. In the illustration below, the annual oscillation due to parallax (if measurable) has been removed.

32. Visual Binary: Determining the Center of Mass  There are two ways to determine the center of mass of a binary system. One way is to measure the center of mass of the system with respect to fixed (much more distant) background stars after the annual oscillation due to parallax (if measurable) has been removed.  Proper motion measurements of the very young (single) star HP Tau/G2 in the Taurus star- forming region at radio wavelengths with the VLBA. The annual wobble is due to parallax, which gives a distance to this object of 161.2 ± 0.9 pc.

33. Visual Binary: Total and Individual Component Masses  If the total mass of the system can be derived from Kepler’s 3 rd law and the mass ratio of the two components derived from the measured semimajor axes of their orbits the individual component masses can be derived.