MODELING MULTI-WAVELENGTH STELLAR ASTROMETRY. I. SIM LITE OBSERVATIONS OF INTERACTING BINARIES

As shown in Gelino et al. (2010), the SED of QZ Vul appears to be that of a reddened K2 dwarf from the optical through the mid-infrared. The system parameters are shown in Table 1. These parameters were input into reflux to model the astrometric motions, with a nextgen model atmosphere used for the secondary star. The model SED, a 3D model of the system, and its wavelength-dependent reflex motions are shown in Figure 1. Since there is only a single visible component, QZ Vul does not exhibit any discernible wavelength dependency to its reflex motions. However, since the black hole is ∼24 times more massive than the secondary star, the K2 dwarf has a large apparent motion, and thus even at a distance of 2.29 kpc, the system's astrometric reflex motion (∼8 μas), in theory, would be detectable with microarcsecond astrometry. However, in practice, this particular system is so faint (V = 21.2), and has such a short period (0.3342 days), that, according to DAPE, SIM Lite cannot reach the needed precision without integrating for longer than the orbital period. Thus, one would need to find a much closer, and thus brighter, LMXB to observe with SIM Lite. The estimated orbital inclination for QZ Vul is 64° (Gelino 2001), and the only effect one would see with a QZ Vul type system with a different inclination is a reduction or amplification of the y-component of the astrometric motion, corresponding to an increase or decrease of the inclination, respectively.

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Cyg X-1 is a well-known HMXB dominated by an O-type supergiant orbited by a 10 M_☉ black hole with a period of 5.6 d. Cyg X-1 exhibits a wide variety of behavior, such as highly variable X-ray and radio emission, some of which is presumably due to a relativistic jet. The optical light also varies, but much more weakly (Δm ≈ 0.05 mag). Thus, in contrast to QZ Vul, the primary star will be the visible source in this system. We model this system using the parameters listed in Table 2, using a blackbody for the O star. As can be seen in Figure 2, there is no wavelength dependence to the astrometric reflex motion, and a single bandpass is sufficient to determine the orbit. Even at a distance of 2.1 kpc, the large separation of the components in Cyg X-1 produces a significant astrometric signature (∼25 μas), even though it is the more massive component that is responsible for the visible astrometric motion in this system. Given the brightness of the system (V = 8.95), and the long orbital period, this system is easy to observe with SIM Lite. According to DAPE, 10 individual measurements, each with 2.5 μas precision, could be obtained in only a total of 1 hr and 40 minutes of mission time (given 5 minutes of target integration time per visit, broken into five 1 minute chops between the target and reference star). This would provide more than sufficient high-precision measurements to obtain a good solution for this system, directly yielding the inclination of the system and the semi-major axis of the primary star's orbit, and thus indirectly the masses of the components. For systems in which one component completely dominates the visual SED, having accurate photometry for the system is unnecessary in interpreting the astrometric data.

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SS Cyg is a bright, well-known, CV. In quiescence, SS Cyg has an apparent magnitude of V = 12.2. During outbursts, it brightens to V ∼ 8.8. It has an orbital period of 6.603 hr, and the secondary star has a spectral type of K5 (Harrison et al. 2004). Dubus et al. (2004) obtained near-simultaneous multi-wavelength photometry of SS Cyg. As shown in Figure 5 of Harrison et al. (2007) the WD and accretion disk dominate the blue end of the SED, while the secondary star becomes prominent in the red and near-infrared. Harrison et al. (2007) modeled the SED of SS Cyg as a combination of the two stellar components plus a free–free accretion disk around the WD. Due to the availability of simultaneous UBVRIJHK photometry, and well-constrained stellar parameters, SS Cyg is an ideal system to investigate what happens when more than one component in the binary system is visible. As we demonstrate, SS Cyg has strongly wavelength-dependent astrometric reflex motions. We only consider the reflex motion in quiescence, since during outburst, the luminosity of the system is completely dominated by the hot (10,000 K), optically thick accretion disk.

We have generated a model system with the parameters listed in Table 3, where the temperatures and radii of the stellar components have been derived from the literature. In the case of SS Cyg, we used a nextgen model atmosphere for the secondary star, and assumed a blackbody for the WD primary. We attempted both blackbody and free–free accretion disk models, and found that the free–free model provides the best match to the photometry. The results are shown in Figure 3, where the model SED is compared to the photometry, with the contributions from each of the three components shown with separate symbols. As can be seen from these plots, the WD dominates the flux contribution in the U band, and fades thereafter, with the secondary dominating from B through K (though the accretion disk contribution becomes quite important in the infrared). A wire grid representation of the SS Cyg system at phase 0.25 is also shown in Figure 3. As shown in the bottom left panel, the amplitudes of the reflex motions smoothly increase from B (semi-major axis a = 12 μas) to R (a = 28 μas), and then decline at longer wavelengths due to the contribution of the accretion disk. The reflex motion in the U band is larger than in B, as it is almost completely dominated by the WD primary. In the B band, the two stars have similar luminosities, and thus the motions are a mixture of the individual astrometric orbits. If there was no accretion disk, and the WD was invisible, the total amplitude of the reflex motions for the secondary star in this system would be a = 34.1 μas. Thus, the true reflex motions are diluted, and one cannot determine the actual astrometric orbit without modeling the system. However, with proper modeling, one can deconvolve the motion of the WD + accretion disk and the secondary star, thus directly yielding the semi-major axis of each component's orbit, which when combined with the well-known period and distance to the system, directly yields absolute masses for each component (see Section 4).

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Careful examination of the wavelength-dependent astrometric orbits reveals that the actual center about which the motion occurs is wavelength dependent. Note that the U-band reflex motion is centered on (0,0), but at the other wavelengths, the motions are offset from this position. This subtle effect has two causes: (1) the secondary star in SS Cyg fills its Roche lobe, and therefore has a teardrop shape and (2) gravity brightening for non-degenerate stars is significant. These two effects combine to offset the center of light of the secondary star from its center of mass. This creates an offset in the astrometric motions in the bandpasses where the secondary star dominates the systemic luminosity. Since the WD is spherical, and does not have gravity brightening effects, the U-band motion remains centered at (0,0). For more detailed modeling on the wavelength dependence of the astrometric center due to gravity brightening, see Coughlin et al. (2010).

One of the important consequences of these effects is that the actual orbital inclination is harder to derive. Naively, one would assume that the orbital inclination can be calculated from the ratio of the minor and major axes of the orbital ellipse. If we do this in the R band [i = cos^-1(17.5/28.0)] we derive an orbital inclination of 514, instead of the input value of 505. Depending on the bandpass, the orbital inclinations derived for our simulation of SS Cyg range from a minimum of 47° in the U band, to a maximum of 514 in the R band. While it is likely that these differences will be lost in the astrometric noise for most IB systems, there will be a small number of IBs (and many non-IB systems) where this difference should be detectable, and proper modeling is required to accurately extract the system's inclination.

With respect to observing this system with SIM Lite, according to DAPE, even in its low state with V = 12.1, a total mission time of 6 hr and 40 minutes would provide 10 measurements with respect to an astrometric standard star, each with 1.7 μas precision and an individual integration time of 20 minutes per measurement, each composed of 20 chops between the target and reference stars, with 1 minute exposures on the target and 30 s exposures on the reference. As a minimum of seven points are needed to solve for a full astrometric orbit (given that seven parameters describe an orbit), and as the orbital period of SS Cyg is ∼6.6 hr, this would successfully measure the full astrometric orbit with limited orbital smearing (see Section 4).

SS Cyg can be used as a template for cases of CVs and LMXBs where the secondary star is clearly visible in the optical. It simply requires changing the input values of the stellar components, the relative prominence (and nature) of the accretion disk, and altering the orbital inclination. Other changes, such as orbital period, mass ratio, and distance, act to scale the amplitude of the reflex motions. However, it remains critically important to have simultaneous multi-wavelength photometry, and a reasonable handle on the system properties to derive accurate astrometric orbits for these types of objects. As one would expect, the wavelength-dependent behavior of the astrometric motions for these types of systems is more complex than for systems with only a single, visible component, but the potential scientific payoff is much higher. When varying the inclination angle for an SS Cyg type system, the y-component of the astrometric motion decreases with increasing inclination angle for all wavelengths as expected, but at large inclination angles eclipse effects produce significant deviations from simple sinusoidal reflex motions.

V592 Cas is a short-period (P_orb = 2.76 hr) CV system consisting of a Roche lobe-filling M dwarf, a very hot WD, and a disk that dominates the luminosity at optical and near-infrared wavelengths. The majority of short-period CVs closely resemble V592 Cas, and thus it can act as a prototype for disk-dominated CVs. The SED of the system has been investigated by Hoard et al. (2009), who reproduced the observed photometry using a model that included a red dwarf, WD, and a two component blackbody disk around the WD consisting of an inner flat component and an outer flared component. To model the mid-infrared excess, a circumbinary dust disk was also included. This dust disk is only important at mid-infrared wavelengths, and thus can be ignored in this present study. We model the reflex motions of V592 Cas using the parameters from Hoard et al. (2009) shown in Table 4, while using a single component blackbody disk model with a flare inclination that is the average of the angles in their two component disk model. The SED produced by reflux is compared to that of Hoard et al. (2009) in Figure 4, where the contributions from each component are also shown. We find a greater contribution of flux from the M dwarf compared to Hoard et al. (2009), by a factor of ∼1.75. This is most likely due to the fact that Hoard et al. (2009) used the spectrum of a field M5.0V dwarf, while our model, which incorporates full Roche geometry, produces a star with the same effective temperature, but a larger surface area by the same factor of ∼1.75, adopting the radius for an isolated M5.0V as 0.21 R_sun, as shown by recent measurements of low-mass stars (c.f., López-Morales 2007). Either way, the M dwarf is a few percent of the total system flux, and thus does not greatly affect the derived astrometric measurements.

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As shown in Figure 4, due to the domination by the disk, the system shows less wavelength-dependent motion than SS Cyg. However, there is still a discernible effect toward longer wavelengths as the secondary becomes a more significant source of flux. Thus, with enough measurements, one could in theory derive the individual component masses. The influence of the orbital inclination is readily apparent, and is uncontaminated by the secondary star gravity effects that were visible in our models for SS Cyg. With respect to SIM Lite observing time, this system would require a moderate amount of SIM Lite mission time, given its small amplitude and short period. According to DAPE, if we constrain individual integration times to 15 minutes (0.1 of the orbital period), to prevent orbital smearing, one could obtain 2.3 μas precision per visit (composed of 20 target-reference chops of 45 s target integration each), which is on par with the amplitude of the observed reflex motions. Thus, in comparison to SS Cyg and Cyg X-1, discussed above, one might need more than 10 measurements to build up signal to noise and really nail down the system parameters. Thirty measurements would take a total mission time of 17.5 hr, and thus we can say with reasonable confidence that this system would require less than a day of total mission time to observe so that the astrometric parameters could be obtained with reasonable precision.

We decided to use this system to investigate the possibility of using the multi-wavelength astrometric capability of SIM Lite to discern two more subtle properties of disk-dominated systems: the disk temperature gradient and disk hot spots. The disk model employed in ELC allows for an inner disk temperature to be specified, along with a power-law exponent, so that the disk temperature will vary radially according to the formula

$$\begin{equation} T(r) = T_{\rm inner}\cdot \bigg(\frac{r}{R_{\rm inner}}\bigg)^{\chi }, \end{equation} \tag{ 2 }$$

where T is the temperature at a given distance from the compact object, r, given a temperature T_inner at the innermost radius, R_inner, and a power-law exponent χ (Orosz & Hauschildt 2000). Usually, a value of χ = −0.75 is assumed for a steady-state disk (Pringle 1981), but we wanted to test if changing χ to −0.5 or −1.0, which physically reflect a centrally irradiated disk (Friedjung 1985; Vrtilek et al. 1990; Bell 1999) and its corresponding extrema, would have a discernible effect on the wavelength-dependent astrometric reflex motion. Thus, we produced two models, one with χ = −0.5 and the other with χ = −1.0, while also adjusting the inner disk temperature so that the resulting SED best matched that with χ = −0.75. Neither the SEDs nor the reflex motions differed significantly from the models that used χ = −0.75. Thus, it will not be possible to constrain the overall temperature gradient in accretion disks using astrometry.

In many CV systems, the optical light is orbitally modulated due to the presence of a hot spot on the outer edge of the accretion disk. This hot spot is where the accretion stream from the secondary impacts the disk. These spots typically have temperatures of 20,000 K, and can produce modulations on order of ±50% in visible bandpasses (Mason et al. 2002). Because this feature is on the outer edge of the disk, it has the potential to distort the reflex motions in systems where it is prominent. In addition, due to its higher temperature than the surrounding disk, the hot spot could be the source of strong line emission, especially in H i or He i lines (Skidmore et al. 2002). Thus, its detection might be isolated, and the reflex motions amplified, using very narrow bandpasses centered on the strongest emission lines. To investigate the effects of a hot spot, we used the same V592 Cas model as above, but added a hot spot (20,000 K) on the outer edge of the accretion disk that leads the secondary by 30° in phase, and that is 30% of the V-band flux. The differences between a model with a spot and one without are extremely slight, with a maximum difference in models with and without a spot of 0.15 μas. Increasing the inclination of the system does not have a significant effect. To be even remotely detectable, V592 Cas would have to be ∼3 times closer, making the spot signature ∼0.5 μas, although it would still only be ∼5% of the reflex amplitude signature. Unless an IB has a hot spot that is much more dominant, such features will remain undetectable.

Sco X-1 is the proto-type LMXB consisting of a neutron star accreting from a low-mass companion. The X-ray luminosity of Sco X-1 is very close to the Eddington limit for a 1.4 M_☉ neutron star. The secondary star in Sco X-1 has never been directly detected, though an orbital period of 0.787 days was detected through the analysis of 85 years of visual photometry (Gottlieb et al. 1975), and from radial velocity measurements (Lasala & Thorstensen 1985). Steeghs & Casares (2002) detected narrow H i emission lines that they showed are from the irradiated secondary star. From these data they estimate masses for the components in this system of M₁ = 1.4 M_☉ and M₂ = 0.42 M_☉, thus making the secondary star a significantly evolved subgiant. Since the accretion disk dominates the spectrum, we have simply assumed a Roche lobe-filling M type secondary star corresponding to the observed mass, and an invisible neutron star. The stellar and accretion disk parameters are listed in Table 5, and the accretion disk model we employ is similar to that for V592 Cas. As shown in Figure 5, the accretion disk dominates until the near-infrared when the secondary star begins to contribute. Thus, as also shown in Figure 5, the reflex motion has a weak wavelength dependence, but SIM Lite cannot observe in the near-infrared where the reflex motion reverses, and thus it will be difficult to disentangle the components with such a weak wavelength dependence in the optical. However, it might be possible to recover the contribution to the astrometric reflex motion for systems with slightly more prominent secondary stars (e.g., Cyg X-2). As shown in Figure 5, Cyg X-1 has a very small reflex motion, only ∼1.25 μas, and can get relatively faint (V = 14.1 at its faintest), and thus this system is just at the limit of SIM Lite's capabilities. Adopting the faintest magnitude of 14.1 of the system to be conservative, and limiting ourselves to observations of 110 minutes (0.1 of the orbital period), to prevent orbital smearing, we find via DAPE that we can achieve individual measurements with precisions of 2.2 μas, almost twice the semi-major axis of the system. Thus, assuming we would need at least 50 measurements to begin to derive a reasonable orbit for this system, this system would require at least 4 days of SIM Lite mission time, which may prove more expensive than the scientific payoff justifies. If we are able to observe in its high state when V = 11.1, we could achieve 30 measurements with 1.5 μas precision in only 20 hr of mission time, which is far more reasonable. When varying the inclination of a Sco X-1 type system, the contribution from the secondary star becomes more significant as more of the disk becomes self-eclipsed and its total luminosity decreases, and thus alters the individual wavelength reflex motions accordingly.

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AR UMa is a "polar," a CV with a highly magnetic WD primary and a Roche lobe-filling M5/6 dwarf (Harrison et al. 2005). As shown in Szkody et al. (1999), the V-band light curve is complex, being dominated by phase-dependent cyclotron emission. The near-IR light curve shows normal ellipsoidal variations, allowing Howell et al. (2001a) to estimate an orbital inclination of 70°. The magnetic field strength in AR UMa has been estimated to be as high as B = 240 MG, though Howell et al. (2001a) argue for a lower value of 190 MG. In these systems, at some point after material from the secondary star has passed through the inner Lagrangian point, it is captured by the magnetic field of the WD, producing a "magnetic threading region" that can be a significant source of luminosity in H_α. We investigate the possibility of astrometrically detecting this region via the following modeling process. We model the system as a WD + red dwarf system without an accretion disk, using the parameters shown in Table 6, but for only the H_α bandpass add a narrow "accretion" disk halfway between the inner Lagrangian point and the WD. This disk then has a hot spot leading the inner Lagrangian point by 45° in phase. The actual contribution from the disk has been made to be completely negligible, but the spot itself makes the system 25% brighter in H_α. In effect, it is an isolated emission spot located between the WD and secondary star, mimicking a threading region.

As can be seen in Figure 6, there is a large amount of wavelength-dependent astrometric reflex motion as expected, ranging from ∼10 to 20 μas in the optical, with the WD dominating at short wavelengths and the secondary star dominating at long wavelengths. This should easily allow the determination of both component masses with orbital coverage at a few passbands. According to DAPE, limiting individual observations to 12 minutes (0.1 of the orbital period), to prevent orbital smearing, one could obtain individual measurements of ∼6 μas precision at the system's brightest (V = 14.5) or ∼48 μas at its faintest (V = 18.0). Thus, in its high state one could obtain 30, ∼6 μas precision, measurements in as little as 9 hr of total mission time, but in its low state, assuming 100, ∼48 μas precision, measurements would yield a meaningful solution, a total mission time of 30 hr would be required, which is still a reasonable amount of time for the scientific payoff in our opinion.

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Of particular interest is that in H_α the threading region produces a detectable astrometric signature that has a phase offset of ∼15° from the broadband wavelengths, due to the original 45° offset of the hot spot diluted by the light from the rest of the system. Thus, through use of narrow-band astrometry, one could recover the location of the threading accretion regions in these types of systems. When varying the inclination angle for an AR UMa type system, the y component of the astrometric motion decreases with increasing inclination angle for all wavelengths as expected, but at large inclination angles eclipse effects produce significant deviations from simple sinusoidal reflex motions.