The process of measuring a trigonometric parallax is in essence the measurement of the science star’s minute motion with respect to a background field of distant “fixed” stars
(§§5.4, 6.2). The resulting motion is a combination of the heliocentric tangential motion of
the science target arising from Galactic kinematics (the proper motion), the reflex motion arising from Earth annual heliocentric motion (the trigonometric parallax itself), and any remaining motion arising from the orbital motion of the science target if it is a multiple system. The latter is observed in the form of an astrometric perturbation to the residuals of the science target’s photocentric displacement with respect to the background stars once the proper motion and the trigonometric parallax have been calculated and subtracted. If sufficient epochs have been observed, the astrometric perturbation traces out the relative orbit of the system’s photocenter with respect to the fixed barycenter. Figure 7.1 shows two examples of what astrometric residuals look like after the parallax motion and the proper motion have been subtracted. The first case shows the astrometric residuals for LP 944- 020 for which residuals in both coordinate axes are small and do not form any pattern. The second case shows the residuals for GJ 1215ABC, where a large and clear sinusoidal perturbation indicative of orbital motion is present in both axes.
(a)
(b)
Figure 7.1 Astrometric residuals for (a) LP 944-020, and (b) GJ 1215ABC. Note the different plotting scales for the two
objects in the vertical axes. The residuals represent the motion of the target’s photocenter with respect to the background of distant stars after the components of motion due to the proper motion and the trigonometric parallax have been subtracted. Panel (a) shows small residuals randomly distributed about zero in both coordinate axes, and is typical of a good parallax solution. Panel (b) shows a clear sinusoidal perturbation indicative of the orbital motion of an unresolved multiple system’s photocenter about the system’s barycenter. Each data point and its error bars represents the mean and standard deviation of the typically five consecutive observations taken in a single night. If enough observations are taken, the astrometric perturbations can be used to map out the photocenter’s orbit, as is the case with GJ 1215ABC.
7.2.1 From Photocentric Orbits to Component Masses
In order to understand how the observed photocentric orbit arises from a physical binary configuration it is useful to first consider the extreme cases of binaries with large differences in luminosities as well as equal luminosity binaries. Consider first a system in which the two components have a small mass ratio (i.e., one component is much more massive than the other) and that the secondary’s contribution to the overall luminosity is also negligible. The system’s photocenter is therefore effectively at the same location as the primary component, which contributes almost all the light in the band through which observations were taken. Both components orbit the system’s barycenter, which is very close to the location of the primary component. The resulting motion of the photocenter then maps out the small orbital motion of the primary component. Consider also the case of a binary system where the components have equal mass and equal luminosity. In that case the system’s barycenter is located exactly half way between the two components, and the components’ motions are equal and opposite to each other. In this case both components have large displacements about the system’s barycenter, but because the light distribution is always symmetric about the barycenter, the system’s photocenter does not move, and no astrometric perturbation is detected.
Now consider the case where the components’ luminosities are slightly different. The sys- tem’s photocenter is then slightly offset from the barycenter toward the direction of the more luminous component, and a small astrometric perturbation is detected. Because unresolved observations give us no information about the components’ luminosity ratio or physical sep-
aration, obtaining the mass ratio from a photocentric orbit is a degenerate problem − the same solution may be assigned to a low luminosity ratio system where the primary com- ponent moves little, but carries with it most of the system’s light, or to a high luminosity ratio system where both components have a large motion about the barycenter, but because the difference in luminosity between the two components is small the photocenter is only slightly offset from the barycenter. Figure 7.1(a) could therefore correspond to an equal luminosity binary or to a single star. In the same manner, Figure 7.1(b) could correspond to a system in which the two components’ masses and luminosities are only slightly different, or to a very low mass and luminosity ratio system where one component clearly dominates
the photocenter1.
As outlined in e.g., McCarthy et al. (1991), the degeneracy discussed above can be broken and a unique solution for the system’s mass ratio can be obtained if the system’s luminosity ratio and projected physical separation are known at a single epoch, so long as the photo- centric orbit is also known. Suppose that in a resolved image the components’ separation, p, is measured, as well as the fluxes of the two components. The following quantities can then be defined:
β = F2
F1+F2
where F1 and F2 are the fluxes of the primary and secondary components, respectively.
1Overluminosity is often detected in a color-magnitude diagram. In the equal luminosity case, the offset
Similarly, we define the secondary’s fractional mass as
B = M2
M1+M2
and ρ as the system’s photocentric perturbation at the epoch at which p was measured.
The ratio p/ρ is then a scaling factor between the photocentric orbit and the barycentric
orbit. The system’s mass ratio B can then be related to the known luminosity ratio β by
the equation
p ρ =
1
B−β.
We can then solve for B and apply the previous equation in combination with Kepler’s third law to yield the masses of the individual components.
The fractional luminosity ratio β should in principle be measured in the same band
that was used for mapping the photocentric orbit. In practice, the trigonometric parallax
observations are performed in one of the optical bands V, R, or I, and the Gemini/NIRI
observations were performed through bandsJ,H, andKs, or their narrow band equivalents
in cases where the targets were too bright for observing through broad bands. The flux ratios measured in the Gemini/NIRI observations can be converted to optical flux ratios using the relations in Table 6.5, with care taken to propagate the uncertainties associated with each relation.