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

Learning Objectives Visual Binaries Reference

frame Total MassMass Ratio

Determining the Center of Mass

Learning Objectives Visual Binaries Reference

frame Total MassMass Ratio

Determining the Center of Mass

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?

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?

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?

Visual Binary: Reference Frame In practice, astronomers measure the orbit of one star about the other star (held at

a fixed location).

Visual Binary: Reference Frame In practice, astronomers measure the orbit of one star about the other star (held at

a fixed location).

Visual Binary: Reference Frame This is equivalent to translating a 2-body problem

(at focus of ellipse)

m1

m2

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):

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)

Learning Objectives Visual Binaries Reference

frame Total MassMass Ratio

Determining the Center of Mass

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 3rd law

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 3rd 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, m1 + m2, therefore requires knowing the distance, d, to the binary system.

dα a

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. The star held fixed will appear to be located at the center of the observed elliptical orbit.

True Orbit Projected Orbit

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 (observed orbital shape, projected location of star held fixed relative to focus of observed orbit, and relative velocity of the secondary star along its orbit).

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 nodeω: argument of periastron

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.

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.

di

Learning Objectives Visual Binaries Reference

frame Total MassMass Ratio

Determining the Center of Mass

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

m2

υ1

υ2

(at focus of ellipse)

m2

υ1

υ2

Visual Binary: Mass Ratio For a visual binary whose orbital plane is in the sky plane, r1=a1(1+e) and

r2=a2(1+e) so that the mass ratio of the two components

where α1 and α2 are the angles subtended by a1 and a2 respectively. Unlike deriving the total mass, deriving the mass ratio does not require knowing the distance to the binary system.

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 a1 and a2 respectively.

Unlike deriving the total mass, deriving the mass ratio does not require knowing the distance to the binary system.

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.

Learning Objectives Visual Binaries Reference

frame Total MassMass Ratio

Determining the Center of Mass

Visual Binary: Determining the Center of Mass How can we locate the center of mass of a binary system?

⇒

⇒?

or

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

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.

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.

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.

Visual Binary: Total and Individual Component Masses If the total mass of the system can be derived from Kepler’s 3rd 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.