The Angular Separation of the Components of the Cepheid AW Per

TL;DR

This paper measures the angular separation of the Cepheid AW Per's binary components using HST data, paving the way for direct mass and distance determination through combined spectroscopic and visual orbit analysis.

Contribution

It introduces a method to determine the angular separation of a Cepheid binary from HST data, enabling future mass and distance measurements.

Findings

Angular separation of AW Per measured from HST data

Potential to determine system inclination and masses with additional observations

Method demonstrates feasibility of visual orbit analysis for Cepheids

Abstract

The 6.4 day classical Cepheid AW Per is a spectroscopic binary with a period of 40 years. Analyzing the centroids of HST/STIS spectra obtained in November 2001, we have determined the angular separation of the binary system. Although we currently have spatially resolved data for a single epoch in the orbit, the success of our approach opens the possibility of determining the inclination, sini, for the system if the measurements are repeated at additional epochs. Since the system is potentially a double lined spectroscopic binary, the combination of spectroscopic orbits for both components and the visual orbit would give the distance to the system and the masses of its components, thereby providing a direct measurement of a Cepheid mass.

Figures

Tables

Table 1. Table 1: Observation log

Obs ID	Off Set	Roll	Obs Date	Exp. Time	Phase111Phase, $V$ and $\left(B-V\right)$ are derived from sources in the literature, as discussed in the text.	$V$	$\left(B-V\right)$
	arc sec	Deg.	MJD - 52235	Min.	$\Phi$	Mag.	Mag.
o6f104010	$+0.0$	175.526	0.34765625	10.0	0.906	7.40	1.02
o6f104020	$+0.1$	175.526	0.35546875	10.0	0.907	7.39	1.01
o6f104020	$-0.1$	175.526	0.36328125	11.4	0.909	7.38	1.01
o6f105010	$+0.0$	205.000	0.41406250	10.0	0.916	7.34	1.00
o6f105020	$+0.1$	205.000	0.42187500	10.0	0.918	7.33	0.99
o6f105030	$-0.1$	205.000	0.42968750	11.4	0.919	7.32	0.99
o6f106010	$+0.0$	146.526	0.48046875	10.0	0.927	7.27	0.97
o6f106020	$+0.1$	146.526	0.48828125	10.0	0.928	7.26	0.97
o6f106030	$-0.1$	146.526	0.49609375	11.4	0.929	7.26	0.97

Table 2. Table 2: Parameter Values

Parameter	Value	Parameter	Value
$\Delta y_1$	$-0.010$	$c_3$	4.13
$\Delta y_2$	0.279	$c_4$	0.82
$\Delta y_3$	$-0.269$	$\gamma$	0.9686
T${}_{}{}^p{}_{\mathrm{eff}}{}^{}$	[6297]	$V_s$	(11.084)
T${}_{}{}^s{}_{\mathrm{eff}}{}^{}$	15735	${\left(B-V\right)}_0^s$	($-0.156$)
$\log g_p$	(4.00)	${\left(U-B\right)}_0^s$	($-0.597$)
$\log g_s$	[1.60]	$V_p$	(7.362)
${\left[m/H\right]}_p$	[0.00]	${\left(B-V\right)}_0^p$	(0.494)
${\left[m/H\right]}_s$	-0.20	${\left(U-B\right)}_0^p$	(0.359)
$E\left(B-V\right)$	0.53	$\Delta \log L$	(0.95)
$R\left(V\right)$	3.11

Table 3. Table 3: Errors

Param.	Prog.	S.D.	$\left|\delta \log g\right|$	$\left|\delta \frac{f_{IUE}}{f_{\mathrm{STIS}}}\right|$	$\left|\delta \frac{f_s}{f_P+f_s}\right|$
$\Delta y^{\left(1\right)}$	0.004	0.015	$1.4\times {10}^{-4}$	$6.5\times {10}^{-6}$	$5.0\times {10}^{-4}$
$\Delta y^{\left(2\right)}$	0.005	0.017	$0.0019$	$1.4\times {10}^{-4}$	$0.015$
$\Delta y^{\left(3\right)}$	0.005	0.014	$0.0021$	$1.3\times {10}^{-4}$	$0.015$
T${}_{}{}^s{}_{\mathrm{eff}}{}^{}$	248	105	$1205$	9.1	$37$
${\left[m/H\right]}_s$	0.057	0.025	$7.5\times {10}^{-5}$	$0.0016$	$0.0064$
$E\left(B-V\right)$	0.001	0.038	$0.018$	$0.0026$	$0038$
$R_V$	0.031	0.12	0.11	$0.0090$	$2.7\times {10}^{-4}$
$\gamma$	0.015	$2.4\times {10}^{-4}$	$0.019$	$8.6\times {10}^{-4}$	$0.0025$
$c_3$	0.14	0.32	$0.049$	0.029	0.0055
$c_4$	0.019	0.066	0.014	$4.3\times {10}^{-3}$	0.0068

Equations

$$y(\lambda )$$

$$\Delta y$$

$$\sigma [y(\lambda )]=\frac{0.42\xi }{\sqrt{N(\lambda )}}.$$

$$A_{\lambda }\equiv A[R_V,E(B-V),\gamma ,c_3,c_4;\lambda ].$$

$$y(\lambda {)}^{(n)}=\Delta y^{(n)}{\left[1+\frac{N(\lambda {)}_{obs}^{(n)}-{\theta }_s^{2}N({\mathrm{T}}_s,\log g_s,v_t,[m/H];\lambda )}{{\theta }_s^{2}N({\mathrm{T}}_s,\log g_s,v_t,[m/H];\lambda )}\right]}^{-1}$$

$$N(\lambda {)}_p^{(n)}=$$

$$f(\lambda ,y)=N(\lambda )h(y-y_0)$$

$$f(\lambda ,y)=N_p(\lambda )h(y-y_p)+N_s(\lambda )h(y-y_p-\Delta y).$$

$$f(\lambda ,y)$$

$$y(\lambda )=y_p+\frac{\Delta y{N}_s(\lambda )}{N_p(\lambda )+N_s(\lambda )}=y_p+\frac{\Delta y}{1+R(\lambda )}$$

$$y(\lambda {)}^{(n)}=y_p^{(n)}+\theta \sin (\varphi -{\alpha }^{(n)})[1+R(\lambda ){]}^{-1}$$

$$\Delta y^{(n)}=\theta \sin (\varphi -{\alpha }^{(n)})$$

Full text

††pagerange: The Angular Separation of the Components of the Cepheid AW Per–LABEL:lastpage††pubyear: 2007

1 INTRODUCTION

Cepheids are important stars in many respects, most notably for their roles as fundamental rungs on the cosmic distance ladder and the challenges their structure pose to stellar interiors modelling. The use of Cepheids as primary extragalactic distance indicators makes a quantitative understanding of their properties extremely valuable. While the Magellanic Clouds are perhaps the best laboratory to study cepheids, the dependence of the period–luminosity relation on metalicity is still not fully understood (Romaniello et al. 2005). Consequently, accurate distances (absolute magnitudes) to Galactic cepheids are needed to fully understand and quantify this dependence and to apply cepheid scale to more metal rich spiral galaxy stars which are more commonly used in extragalactic distance determinations.

Cepheids also present important tests for interiors calculations since, as evolved stars, their structure is dictated by their evolutionary history. In addition, the models must predict the puslational properties of cepheids, making the modelling especially challenging. This complexity is codified in the term “the Cepheid mass problem”. Forty years ago, when the first hydrodynamic pulsation calculations were made, it was realized that the mass could be derived by either matching the Herzsprung progression of secondary maxima or by a parameterization of the pulsation constant. These masses were as much as a factor of two smaller than evolutionary calculations. A reconciliation was recently achieved from re-evaluation of the interior opacities (see Simon, 1990, for a summary). We see, therefore, that in addition to absolute magnitudes, obtaining accurate Cepheid masses is also important.

If we can determine the angular separations of binary systems containing a Cepheid, which are double lined spectroscopic binaries (SB2s), then the distances and masses of the Cepheids can be derived from basic physics. Because of the central roles of Cepheids in fundamental astrophysics, it is important to have such direct measurements. While several Cepheid distances have been measured directly by the Hipparcos satellite, the quality of these measurements was only sufficient for statistical considerations (e.g., Groenewegen & Oudmaijer 2000). More recently, a large campaign using the Fine Guidance Sensor on HST has begun to yield accurate distances to single Cepheids (Benedict et al. 2002). However, to date the mass of only one cepheid, Polaris, has been directly determined from fundamental observations (Evans, et al. 2007).

Although SB2s containing a Cepheid and an A or B star are common (see, Evans 1995), these stars are difficult to resolve in the optical. This is because of the inevitable, enormous magnitude differences of the components in the optical, which result from massive stars evolving toward cooler temperatures at nearly constant luminosity. The top panel Figure 1 shows a typical example of a Cepheid + B star binary, and the contrast between the primary and secondary throughout the optical and IR is obvious.

Thus, while the measurement of a Cepheid mass by directly imaging a double lined spectroscopic binary with a Cepheid primary and an A or B star secondary has been a long-sought goal, ground-based studies have not, as yet, been able to accomplish this (even though they have been able to resolve the stellar disks of some Cepheids, e.g., Kervella et al. 2004, and references therein). As a result, indirect methods have been developed to determine the masses of Cepheids. The most popular of these uses a combination of UV and optical spectroscopy to obtain radial velocity curves for both components. Then the UV spectral energy distribution (SED) of the hot secondary is used to obtain its temperature. Finally, the mass – temperature relation for main sequence A or B stars is used to infer the mass of the secondary and, thus, (since the system is an SB2) the mass of the primary. This approach has been applied to several systems (SU Cyg, S Mus and V350 Sgr), using IUE or HST spectra to determine the radial velocity curves and SEDs of the secondaries (Evans, et al., 1998). The masses obtained by this approach agree, on average, with the mass-luminosity predictions from evolutionary calculations with moderate convective overshoot (e.g. Schaller, et al.  1992). However, this approach requires an exact understanding of the evolutionary phase of the hot secondary and relies on its spectroscopic parallax to determine the distances to the systems. Clearly, a direct measurement of the masses of both components is more desirable.

In this paper, we describe how we used the Space Telescope Imaging Spectrograph (STIS) on HST to resolve a the potential SB2 Cepheid binary AW Per using an approach we call “cross-dispersion imaging”. AW Per is a 6.4 day Cepheid which is in a roughly 40 year orbit with its hot secondary (Evans et al. 2000). Evans (1989) studied the system and determined that the secondary is a main sequence B7-8 star and that the color excess of the system is $E(B-V)=0.52$. The Teff of the secondary is expected to be $\sim 12000$K (Evans 1994).

The remainder of the paper is organized as follows: §2 provides an overview the approach used to “resolve” the binary, §3 describes the observations, §4 gives the data analysis, §5 details the analysis of the observations, §6 presents the results, §7 discusses the results and their implications, and §8 summarizes the findings.

2 THE APPROACH*(Cross-Dispersion Imaging)*

2.1 Basic Principles

Massa & Endal (1987) describe how combining imaging and spectroscopy can dramatically increase the effective “resolving power” of an instrument. Specifically, they showed how the wavelength dependence of the centroid of a spectrum can determine the angular separation of an unresolved binary whose components have distinctly different spectra. The basic concept of this approach is quite simple. It is based on an idea put forth by Beckers (1982) and has been independently discovered by a number of others (see, e.g., Porter et al. 2004, and references therein).

Like all cross-dispersion imaging techniques, some sort of a model is required to interpret the observations. This model might be extremely simple, as in the case of a binary where one assumes that the system is composed of exactly two stars, and that one contributes all of the flux at one wavelength and the other contributes all of the flux at another wavelength. This crude model would be sufficient to “resolve” the binary from the properties of its spectrum.

Consider the image of a highly unresolved binary system. To first order, the image of the combined light from the system is indistinguishable from a point source. However, the position of an image at any given wavelength will be displaced toward the location of the binary component which contributes most of the light that wavelength. In principle, one could obtain images at several different wavelengths and determine how the centers of the images shift from one exposure to the next. Analysis of this set of data (along with a model for the flux ratios in each band) would then determine the separation of the two components (Becker 1982). The drawback of this direct approach is that all of the exposures would have to be obtained using different optical elements, making alignment at the sub-pixel level effectively impossible. Instead, Massa & Endal (1987) show that tracking the centroid of the spectrum of the binary has the same effect. Furthermore, because all of the position measurements (the centroid of the spectrum at each wavelength) are obtained at one time, this method is more efficient and the measurements are differential in nature, freeing them from several sources of systematic error.

To make these notions quantitative, let $x$ and $y$ be the angular coordinates on the detector which are parallel and perpendicular to the wavelength dispersion. Therefore, the wavelengths, $\lambda$, are given by $\lambda=\lambda(x)$. Now, consider a binary whose components have an angular separation $\theta$ and photon fluxes per unit wavelength $N_{p}(\lambda)$ and $N_{s}(\lambda)$ for the primary and secondary, respectively. Further, let $\phi$ be the position angle of the binary on the sky (measured c.c. from north toward east of a line from the primary to the secondary) and let $\alpha$ be a similarly measured angle between north and a line in the dispersion direction pointing in the direction of decreasing wavelength. Thus, $\alpha$ can be varied by changing the orientation of the telescope. With these definitions, the wavelength dependence of the centroid of the spectrum of a single observation of a binary is

[TABLE]

(see the appendix). Thus, if $N_{p}(\lambda)/N_{s}(\lambda)$ is known, then measurements at two or more orientations ($\alpha$’s) enables one to determine $\theta$ and $\phi$, the separation and position angle of the binary. Note that if the spectral energy distributions (SEDs) of the components are vastly different, then the position of the centroid shifts from one to the other, depending upon which star dominates the flux at each wavelength. On the other hand, if the binary components have identical SEDs, then no spatial information can be gained from the centroid positions.

Figure 1 is a cartoon depicting how the centroid of the spectrum of a binary star, whose components have very different effective temperatures, is influenced by the relative energy distributions of the two components and the orientation of the binary relative to the dispersion direction of a spectrograph. In this case, the centroid shifts from the cool component at long wavelengths to the hot component at short wavelengths. We define the cross-over wavelength as that wavelength were each binary component contributes equally to the flux. For Cepheid binaries, this wavelength is typically in the near UV ($\sim 3000$Å for the case shown). In order to infer spatial information from the centroids, it is desirable to span as large a wavelength baseline as possible, to maximize the deflections in the centroid positions. The best case would be to cover a large enough wavelength range with a single setting, so that one end of the spectrum is totally dominated by one star and the other end is dominated by the other. If this is not practical, a wavelength band centered on the cross-over wavelength and covering a baseline large enough to experience more than a 50% centroid deflection is adequate. However, in this case, one needs an estimate of the SEDs of the two binary components in order to extract the angular separation. Note that if the absolute flux calibration of the instrument is well-determined, then the flux observations can provide additional information which can be incorporated into the determination of the angular separation (see §5).

Finally, to unambiguously determine the separation and position angle of the binary, two or more observations are required in order to solve eq. (2) for $\theta$ and $\phi$ in terms of the measured quantities $\Delta y^{(n)}$ and $\alpha^{(n)}$, for $n\geq 2$.

The final error associated with the angular separation and the position angle measurements depends upon the band pass of the observation, the signal-to-noise of the data (discussed in the next section), the number of independent orientations obtained and the relation between the these angles and $\phi$. We have examined the relative error for sampling three orientations, $\alpha^{(n)}=\{-\Delta\alpha,0,+\Delta\alpha\}$, for position angles between 1 and $90^{\circ}$. Figure 2 demonstrates how the relative accuracy of the observations changes as a function of sampling interval, $\Delta\alpha$, and relative orientations, $\phi$, for this case. For most orientations, any sampling with $\Delta\alpha\gtrsim 30^{\circ}$ provides comparable accuracy.

The approximations developed in this section are only valid in the sub-Rayleigh regime. Once the sources are resolved at any wavelength, the entire image must be modeled using a an accurate representation of the point spread function as well as the fluxes of the two objects.

2.2 Exposure Times and Random Errors

The counts needed to centroid to an accuracy $\sigma[y(\lambda)]$ can be estimated for an instrument whose spread function perpendicular to the dispersion is a Gaussian with $FWHM=\xi$. A single count is equivalent to one estimate of the center of the spectrum drawn from a sample with an RMS dispersion $\sigma=\xi/\sqrt{8\ln 2}=0.42\xi$. Therefore, $N$ samples (counts) determine the centroid to an accuracy of

[TABLE]

Equation (3) gives the counts needed over a wavelength band to obtain the desired accuracy. The FWHM of the STIS PSF varies from $\sim 0.05-0.07$″ (depending on wavelength) and the minimum number of counts obtained in one 10 min observation over a spectral resolution element (2 pixels) was $4000$, and we obtained 3 of these. Therefore, the poorest precision we can expect based upon simple sampling arguments is $\sim 3\times 10^{-4}$″, and this is for a single resolution element. In all, there are 512 independent resolution elements which will be combined to determine a single measurement of $\Delta y$ through the use of eq. (1). Therefore, random noise in the angular separation determinations should be $\lesssim 10^{-4}$″, and not a limiting factor for our observations. However, as is typical for most observations, we shall see that systematic effects will dominate the error budget (see, §4).

3 THE OBSERVATIONS

As can be seen from the top panel of Figure 1, a broad wavelength baseline is needed to optimize the extraction process. Furthermore, good spectral resolution is also advantageous, since spectral features provide additional constraints. Consequently, we employed the STIS on HST to obtain high spatial resolution, excellent wavelength coverage and good spectral resolution. We used the STIS NUV-MAMA detector together with its G230L grating, since this combination provided good coverage ($1600\leq\lambda\leq 3160$ Å) of the expected cross-over point (see, Kim Quijano, J., et al. 2003).

Spectra were obtained at three distinct roll angles (see, Table 1) which differ by $\sim\pm 20^{\circ}$. Although rolls of $\pm 60^{\circ}$ would be optimal, we were limited to smaller rolls by HST restrictions for objects at the declination of AW Per. Although not optimal, Figure 2 shows that this restricted range does not sacrifice very much in theoretical accuracy. After a standard STIS target acquisition, which centers the binary within a 0.1″ aperture, we obtained the science observations through the 25MAMA aperture, which provides slitless spectra of the binary. At each roll, we offset the star by $\pm 0.1$″ and obtained additional science exposures. This procedure allows us to characterize localized distortions in the detector. It is also useful for determining the sensitivity of the observations to their position on the detector, since each spectrum is sampled differently by the pixel lattice. Since the spectrum was repositioned to within 2 pixels ($<0.05$″) after each roll, the dispersion of measurements obtained at the $\pm 0.1$″ offsets should provide a good characterization of the errors that result from all of the changes encountered in the positioning of the spectrum. The reproducibility of these observations also provides a more realistic measurements of the centroiding errors than those based on simple signal-to-noise considerations. As a result of our observing strategy, we obtained 3 observations at each of 3 rolls, for a total of 9 spectra, with exposure times of roughly 10 min each.

The orientations mentioned above are measured with respect to the STIS coordinate system, which we define as the $x_{0}-y_{0}$ system. In this system, the dispersion direction (from red to blue) makes an angle (measured in the c.c. direction) of $-1.4^{\circ}$ with the $x_{0}$ axis.

4 DATA REDUCTION

4.1 Centroids

The first step in the reduction process was to extract the centroids. This presents a problem, since the STIS detector does not oversample the HST PSF. However, since (as will be explained shortly) only relative centroids will be needed, we can accept some level of bias in the extraction process, as long as it is consistent. This is because the ultimate measurements will be differences of the centroids, which will cancel small, uniform biases introduced in the extraction process.

We used three separate approaches to extract the centroids, $y(\lambda)$, from the raw images. We chose to analyze the raw images (in their native “highres” $2048\times 2048$ format) because initial experimentation showed that the geometrically corrected images did little to improve the relative positions of the centroids over the a range of 10 pixels or less (which are the scales important to us). Thus it was felt best to avoid the inevitable smoothing which is introduced by the resampling involved in geometric corrections.

The first approach we used was a simple cross-correlation technique relative to a set of 0.025″ FWHM gaussians. The second one employed a standard cross-correlation technique using the cross dispersion profiles of a spectrum of a standard star (the wd GD71) which was observed at roughly the same position on the detector with the same grating. We used sinc interpolation in the cross-correlation to compensate for the undersampling of the PSF by the MAMA detector. Finally, we used a non-linear least squares fit to a set of gaussians whose FWHMs, central positions and amplitudes were allowed to vary at each pixel. No systematic differences were found among all three approaches. However, the results from the non-linearly extracted centroids produced the results with the lowest pixel-to-pixel scatter, and these were adopted for the following analysis.

The 3 sets of centroid measurements at each roll angle were rebinned to 512 elements from the 2048 elements available in the raw images, and these were used to construct mean centroids at each roll and their standard deviations. Because the centroids near the edges of the detector are poorly determined, of the 512 bined pixels (in the wavelength direction) only about 490 are well-behaved. The standard deviations for these 490 pixels determined for each roll angle are over plotted as a function of wavelength in Figure 3. The RMS means for each roll angle are 0.027, 0.024, and 0.027 pixels or (0.67, 0.59, and 0.67 mas). Remember, these are the single observation standard deviations for a single pixel, and there are 9 independent observations with 490 useful pixels. Notice also that this scatter is significantly larger than the one expected from the simple signal-to-noise arguments of the previous section. The reason is that the actual uncertainties are set by random differences between the photometric and geometric centroids of the pixels, and by localized geometric distortions in the detector over the range of a few pixels. Nevertheless, the repeatability of the centroids (to a few percent of a pixel) is considered quite good, and we will use this scatter to characterize the actual measurement errors in the centroid positions.

Since the centroids are extracted from the raw images, they contain large scale geometric distortions. Consequently, we will analyze the relative centroids. To construct these, we first combine the centroids determined at each offset for a particular roll angle to produce a mean centroid, $\langle y\rangle$, at each roll. These measurements contain geometric distortions and any systematic effects introduced by the centroid extraction technique. However, when we analyze the differences between each individual mean and the grand mean of all the observations, these systematic affects are removed. This is because the offsets at each roll are larger than the displacements from one roll to another, and the scatter that the former exhibit (Fig. 3) demonstrates that localized geometric distortions are small. Similarly, any systematic affects that result from mis-matches between the actual PSF orthogonal to the dispersion and the gaussian used to determine the centroids will cancel out, since the same process is used in each case.

Finally, we must account for the fact that $y(\lambda)$ is not exactly perpendicular to the dispersion. As a result, we must divide the final displacements that we measure by $\cos(1.4^{\circ})$.

4.2 Fluxes

STIS fluxes were extracted from the images using the CALSTIS IDL software package developed by Lindler (1998) for the STIS Instrument Definition Team. In order to constrain the B star flux contribution, we also incorporate the available IUE spectra (obtained when the Cepheid component was near minimum light), into the analysis given in §5. The IUE fluxes were placed upon the HST/STIS flux system using the transformations described by Massa & Fitzpatrick (2000). Figure 4 compares the IUE and STIS spectra. It is immediately clear that the IUE long wavelength spectra were obtained when the Cepheid was near minimum light ($\Phi=0.53$, Evans 1989), while the STIS observations were near maximum light (Table 1). The effects of extinction are also clearly apparent, as is the fact that the IUE fluxes are a factor of 1.146 smaller than the STIS fluxes. This discrepancy is a constant over the region of overlap, and its origin is unknown. Consequently, we cannot be certain which set of fluxes is correct. In §6 we show that this ambiguity introduces a significant uncertainty into our results.

The variability of the Cepheid is clearly detectable in the STIS spectra. Figure 5 shows STIS flux ratios for the mean spectra obtained at the second and third roll angles divided by the first. The time lapsed between the mean observations is 1.59 and 3.19 hours, respectively. This plot demonstrates two things. First, the Cepheid flux changed significantly throughout the three HST orbits spanned by the observations. Second, the flux ratios decrease with wavelength, becoming unity at the shortest wavelengths. This is contrary to what is normally seen in single Cepheids like $\delta$ Cep (Schmidt & Parsons 1982) where the flux changes typically increase with decreasing wavelength. Consequently, this figure shows that the flux at the shortest wavelengths is dominated by the B star, which does not vary.

The following analysis also requires the color and magnitude of the system the time of the observations. We combined the data from Szabados (1980), Moffett & Barnes (1984), Szabados (1991), and Kiss (1998), using the period and HJD for zero phase from Kiss (1998). The combined data were fit with a high order polynomial, and this was used to determine the $V$ and $(B-V)$ photometry at the times of the STIS observations. The resulting phases and photometry are listed in Table 1.

5 ANALYSIS

5.1 Overview

Because our spectra cover a limited band-pass, we require an estimate for the flux ratio of the binary components in order to extract the wavelength dependence of the centroids. This flux ratio is constrained, since it must also satisfy the observed flux of the system, which is the reddened, combined flux of the two binary components. Ideally, one would fit the observed flux and centroid positions with a combination of single star spectra obtained with the same instrument and which experience the same reddening. However, because there is no such library of single star spectra available, we used an approach which employs a model for the B star SED star and for the UV extinction to construct the combined flux and the centroids. We then used a non-linear least squares fitting procedure222We use the Markwardt non-linear IDL fitting procedure, available at http://astrog.physics.wisc.edu/ craigm/idl/idl.html. to fit the centroids and fluxes simultaneously. This method is described in detail in § 5.3.

5.2 Model Components

We now describe the components of the model used to fit the observations. In a few instances, refinements might increase the accuracy, but in the interest of expediency, certain effects were ignored for the first attempt. First, we use Kurucz (1991) Atlas 9 models with updated metallicities333We used the the Kurucz “preferred models” available at http://kurucz.harvard.edu/. for the B star. We use only models with a micro-turbulent velocity of 2.0 km s*-1*. The synthetic photometry for the models was calibrated as in Fitzpatrick & Massa (2005). We set $\log g=4.0$ for the B star atmosphere. The sensitivity of our results to this assumption is tested once a fit is achieved. The model atmosphere fluxes were prepared in the manner described by Fitzpatrick & Massa (2005), which is best suited to the IUE fluxes. The dust model is quite general. We use the Fitzpatrick (1999) formulation of the Fitzpatrick & Massa (1990) model since we need a representation of the near-UV extinction, and the original Fitzpatrick & Massa (1990) formulation does not provide one. Although the Fitzpatrick (1999) curve for the near UV is largely untested, it is reasonable and the best currently available. To provide additional flexibility to the Fitzpatrick model, we allow the bump strength ($c_{3}$), the width of the 2175 Å ($\gamma$) and far-UV curvature term ($c_{4}$) to vary independently. In this way, we can accommodate any observed extinction curve. As a result, the $R_{V}$ parameter (the ratio of visual extinction to color excess) only affects the general slope of the UV extinction and the shape of the near-UV curve, and the wavelength dependence of the total extinction to an object can be expressed as,

[TABLE]

5.3 Details of the Fitting Procedures

We simultaneously fit the STIS centroids at all three roll angles and the IUE flux from the B star. We constrain the reddened model for the B star by assuming that all of the flux from the system for $\lambda\leq 1650$ Å is due to the B star. The difference between the observed flux and the reddened B star model provides the Cepheid SED which is used in fitting the centroids. The free parameters of the fit are: The three $\Delta y^{(n)}$ (displacements perpendicular to the dispersion at each roll angle), T${}_{\rm eff}^{s}$ (the effective temperature of the B star secondary), ${\rm[m/H]}_{s}$ (the abundance parameter for the B star), $E(B-V)$ (the color excess of the system, consistent with the fluxes), $R_{V}$ (which determines the slope of the UV extinction curve), $\gamma$ (the width of the 2175 Å bump), $c_{3}$ (the bump strength), and $c_{4}$ (the strength of the far UV curvature) – 10 parameters in all. The $V$ magnitude of the B star, $V_{s}$, is fixed by the observed flux attributed to the B star at $\lambda=1650$ Å and the extinction at that wavelength relative to $V$. In addition to the separations, the results also yield an empirical, unreddened UV SED and photometry for the Cepheid. These can then be and compared to models or to actual Cepheids. Since the derived Cepheid flux is identical to the observed flux minus the B star flux for wavelengths longward of 1650 Å, the flux in this region is fit exactly. The equation used to fit the centroids is:

[TABLE]

and the unreddened flux of the Cepheid is given by

[TABLE]

where $\theta_{s}$ is the angular diameter of the B star (fixed by the flux at 1650Å) and $n=1,2,3$ represents the observations obtained at each roll angle, which are means of the data for the three off-set positions. We cannot use a single mean for the fluxes, since significant changes in $V$, $(B-V)$ and the UV SED occur over the course of the observations (see, Table 1, Fig. 5) and must be taken into account. However, the data were averaged at each roll, since the time between off-sets was much smaller than the time between rolls.

A major advantage of our approach is that it only relies on a Kurucz Atlas 9 model for the B star, and recent work by Fitzpatrick & Massa (1999, 2005) has demonstrated that these provide excellent representations of low resolution B star SEDs. Further, it avoids using the Atlas 9 models for the Cepheid component, which is desirable since the accuracy of Cepheid model atmospheres has not been fully tested, especially in the UV. This issue is addressed further in §6. The disadvantage of our approach is that we must have extremely well calibrated fluxes, and we have already seen an inconsistency between the poorly exposed IUE fluxes and the STIS data.

5.4 Determining the Separations

The final step in the analysis is to fit the angular separations derived at each roll angle to a sine curve whose phase and amplitude are related to the position angle and separation of the binary (eq. 2). The amplitude of the curve is the full separation of the system and the phase is the position angle of the system on the sky. The abscissa of the plot is the position angle in the $x-y$ system, which is equal to the values listed in Table 1 minus $1.41^{\circ}$ (which accounts for the rotation to align the spectra with the $y$ axis). Figure 6 shows the definitions of the different angles used in the analysis, and their relations to one another.

5.5 Weights

The non-linear least squares involves fitting an array which consists 3 sets of centroids and the IUE fluxes all at once. To perform the fit, we must provide errors for the different components of this array. The measurement errors affecting the centroids were obtained from the standard deviations of the three independent sets of measurements obtained at each offset position. For the IUE data, we used the error vector which accompanies the MXLO fluxes (see, Nicholes & Linsky 1996).

6 RESULTS

In fitting the data, we assumed a microturbulent velocity of 2.0 km s*-1*, which is typical for main sequence B stars (e.g., Fitzpatrick & Massa 2005). Because the B star is overwhelmed by the Cepheid in the optical and near-UV, we do not have access to the classical $\log g$ diagnostics for B stars, namely, the Balmer jump and Balmer lines. Consequently, we fixed the surface gravity at 4.0, again typical for main sequence B stars. We allowed the abundance parameter, ${\rm[m/H]}_{s}$, and the effective temperature of the B star to be optimized by the least squares routine, along with the $\Delta y$’s and the extinction parameters. In addition, we assumed that the IUE fluxes were correct (so the STIS fluxes were divided by 1.146 to make them agree with the IUE data). In applying our model, we also assume that all of the STIS flux in a 30 Å band centered at 1650 Å is due to the B star. We shall examine the effects of our assumptions shortly. Only the IUE fluxes between 1250 and 1700Å are incorporated into the fit of the SED, which constrains the physical properties of the B star. This extends slightly beyond the 1650Å limit used for the STIS data, but recall that the IUE data were obtained when the Cepheid was near minimum light, and nearly a factor of two fainter in the UV (see, Fig. 4).

The parameters determined from the fit are given in Table 2, where parameters that were fixed in the fit are enclosed in parentheses. Figure 7 shows our fits to the centroids. The points are the observed data and the solid curves are the fits obtained simultaneously with the fit to the fluxes. The effects of spectral features on the centroids are clearly seen. Figure 8 shows the fit to the SED below 1650Å. We do not show the fit to the binary SED longward of 1650Å since it is, by definition, exact. The extinction curve derived from the best fit is also shown in Figure 8, where it is compared to a standard $R_{V}=3.1$ curve from Fitzpatrick (1999).

We can also estimate the physical parameters of the Cepheid component of the binary by fitting its mean SED inferred from fit. This SED is found by subtracting the reddened B star model from the observed SED of the system and then correcting this difference for the effects of extinction. The unreddened SED plus its $V$, $(B-V)_{0}$ and $(U-B)_{0}$ (also inferred from the fit) were then fit to an Atlas 9 model. The $V$, $(B-V)$ and $(U-B)$ photometry were initially assigned errors of 0.02, 0.01 and 0.02 mag, respectively. In performing this fit, we fixed the micro-turbulent velocity at 2 km s*-1*, and allowed $T_{\rm eff}^{p}$ (the effective temperature of the primary), $\log g_{p}$ (the surface gravity of the primary) and ${[\rm m/H]}_{p}$ (the abundance of the primary), to vary. We had to restrict the surface gravity to be larger than 1.6, otherwise the fitting routine would seek $\log g_{p}$ values that were unrealistically small (we expect a $\log g_{p}\simeq 2.0$, e.g., Evans 1994). Furthermore, we had to increase the weight (decrease the error) of the $(B-V)$ photometry by a factor of 10 in order to obtain reasonable agreement with the photometry. Figure 9 compares the unreddened SED of the Cepheid to the best fit model. The parameters derived from the fit are also listed in Table 2 and are enclosed in square brackets, to distinguish them from the parameters derived from the initial fit to the data.

It is also possible to test the reasonableness of the inferred UV Cepheid SED by comparing it to IUE observations of the single Cepheid star $\delta$ Cep. $\delta$ Cep has a period of 5.4 days, compared to 6.5 days for AW Per, and its mean unreddened color is $\langle(B-V)\rangle=0.57$. To obtain the intrinsic color of AW Per, we use our derived color excess for the system and the intrinsic colors of the B star secondary from Table 2 and the mean magnitude of the system, $\langle V\rangle=7.49$ mag, to correct the observed mean color of the system, $\langle(B-V)\rangle=1.06$ mag, for both extinction and the presence of the companion. The result is $\langle(B-V)_{0}^{p}\rangle=0.57$, identical to that of $\delta$ Cep (recall that the intrinsic color we derive for AW Per is at $\Phi\simeq 0.92$). Thus, the comparison between these two stars is expected to be quite good. The bottom plot in Figure 9 compares the unreddened IUE data (points) for $\delta$ Cep from several exposures obtained for $0.9\leq\Phi(\delta{\rm Cep})\leq 1.0$ to the unreddened Cepheid STIS spectrum (solid curve) of AW Per. Several IUE exposures are required to produce the $\delta$ Cep spectrum since the dynamic range of IUE was so limited and the range of the UV SED of $\delta$ Cep is so large. The IUE data had the Massa & Fitzpatrick (2000) corrections applied, were dereddened by an $E(B-V)=0.09$ (Dean et al. 1987) and scaled by $10^{-0.4(7.37-3.54)}$, which corresponds to magnitude difference of AW Per at $\Phi=0.92$ (the mean for the STIS data) and $\delta$ Cep at $\Phi=0.95$ (the mean of the IUE data).

Finally, we utilize the $\Delta y^{(n)}$ which resulted from the fits to derive the separation of the system and its position angle on the sky. These are found by fitting eq. (2) to the plot of $\Delta y$ versus roll angle shown in Figure 10. The error bars at each orientation are the quadratic mean errors for that roll determined from the dispersion in the fits to the three individual sets of observations obtained at each orientation (see, next section). The inverse of the errors squared were used to weight the fit. The final result of the analysis is a separation of $\theta=13.74\pm 0.26$ mas and a position angle $\phi=184.16\pm 1.94$ deg, for an accuracy of $\sim 2$%.

6.1 Errors in the parameters

In this section, we describe the internal, random, errors affecting our parameter determinations, and also examine the influence of systematic effects upon the results.

The random errors were evaluated in two, independent ways. One is the error estimates calculated by the least squares routine, which are determined by evaluating derivatives of the model. These errors are listed in the second column of Table 3. We also obtained error estimates by fitting the sets of observations obtained at the same off-set at each roll angle, independently. These provide 3 sets of independent observations and we used the parameters determined from each set to obtain standard deviations (S.D.s) of the model parameters. These estimates (divided by $\sqrt{3}$ applicable to the error in the mean) are listed in the third column of Table 3. Notice that the errors in the $\Delta y^{(n)}$ determined from the S.D.s are nearly twice as large. To be conservative, these errors were used as the error shown in Figure 10 and in determining the errors in $\theta$ and $\phi$.

Beside the random (or measurement) errors, systematic effects will also be present. We characterize these by varying the different assumptions which enter the fitting procedure, and then examining their influence on the result. To begin, we varied the assumed value of $\log g$ used to fit the B star by $\pm 0.5$, which should encompass all plausible values. The result (the difference divided by 2) is listed in column 4 of Table 3. Next, we tested the affect of assuming that the STIS (and not the IUE) fluxes are correct and allowed for the possibility that the B star accounts for only 95%, instead of 100% of the flux at 1650Å. These results are listed in the last two columns of Table 3

As can be seen from Table 3, the varying the $\log g$ can cause a significant change in ${\rm T_{eff}}^{s}$, but has little effect on the $\Delta y^{(n)}$, which are the object of our analysis. In fact, the only significant change in the $\Delta y^{(n)}$ result from our inability to determine whether the STIS or IUE fluxes are correct, and even these errors are only of the same order of the errors determined from the repeated observations. As a result, we conclude that the angular separation determined from our analysis is very robust to variations in the assumptions or input parameters.

7 DISCUSSION

We have seen that the separation determined from the fit is quite stable. We now discuss the physical parameters determined from our fits (Table 2), their reliability and their implications.

We first consider the Cepheid SED derived from the fit. It is compared to the best fitting Atlas 9 model in top panel of Figure 9. This “best fitting” model is not a very good fit, since it lies systematically below the observed flux in far-UV flux and over it in the near-UV flux. Furthermore, the agreement with the optical photometry is not very good. The model predicts $V=7.362$, $(B-V)=0.470$ and $(U-B)=0.309$. The the agreement with the $(B-V)$ color given in Table 2 is fair, but recall that it was given a very large weight. The agreement with the inferred $(U-B)$ is not very good at all. The poor overall fit probably results from the short comings of Atlas 9 models for Cepheids discussed below.

The bottom panel of Figure 9 compares the unreddened SED of the Cepheid component of AW Per to the unreddened SED of the single Cepheid, $\delta$ Cep at approximately the same phase. This figure demonstrates three points. First, the two SEDs agree surprising well. Second, the strong far-UV flux in the derived SED relative to the models is also present (and slightly larger) in $\delta$ Cep, so the derived SED is quite reasonable. Third, the flux in $\delta$ Cep is extremely small for wavelengths shortward of 1650Å, bolstering our assumption that all of the flux in AW Per observed below 1650Å is due to the B star secondary.

So, why is the Atlas 9 model fit of the Cepheid so poor? One must remember that Cepheid UV SEDs depend on numerous, ill-defined physical processes that are not fully incorporated into the Atlas 9 models. These include spherical extension, which can enhance the UV flux from an atmosphere (see Fig. 4 in Hauschildt et al. 1999), chromospheres (e.g., Sasselov & Lester 1994), the amount of convective energy transport (Castelli, Gratton, & Kurucz 1997) and the details of the line blanketing (Prieto, Hubeny, & Lambert, 2003). In addition, there are inevitably dynamical effects that are not treated by the models.

In fact, we initially attempted to fit the data with using an approach that employed models for both the Cepheid and the B star. However, we abandoned it because it produced poor fits and the separations that were $\sim 10$% larger than those derived from the adopted technique. The origin of the systematic difference in the centroids can be traced to the gradient in the flux residuals seen in the top of Figure 9. These propagate into the fits of the centroids. Perhaps the use of more detailed Cepheid models could solve this problem.

In spite of these difficulties, it is of interest to examine the physical parameters determined from the Cepheid model. To begin, Teff of the best fit model agrees reasonably well with previous estimates for Cepheid temperatures near maximum light (Evans & Teays 1996, Fry & Carney 1999, Kovtyukh & Gorlova 2000). On the other hand, the fit selects a very low surface gravity and would have settled on an even lower value if it had been allowed to do so. It is also interesting that the Cepheid model has a significantly different metallicity than the B star. However, this may not be too strange. Instead, it may simply reflect the fact that the [m/H] parameter in cooler models responds more to spectral features produced by CNO elements, while the same parameter in the B stars responds to the Fe abundance (Fitzpatrick & Massa 1999).

Next, we consider the parameters determined for the B star. The model fit to the far-UV (Fig. 8) is quite good, and the extinction curve, while distinctly different from the canonical $R_{V}=3.1$ curve, is rather unremarkable, with parameters well within normal bounds (e.g., Fitzpatrick & Massa 1990, Valencic et al. 2004). Also, the [m/H] for the B star is well within the expected range for such stars (e.g., Fitzpatrick & Massa 1999, 2005) and the inferred color excess is quite close to previous determinations (Evans 1994). It should not be surprising that these fits are so good, since both the extinction model and the ability of the Atlas 9 models to describe normal B star spectra are well documented. Notice that Teff we derive is considerably hotter than previously estimated by Evans (1994), and lies somewhat closer to the ZAMS (see, Fig. 7 in Evans 1994). However, its probable mass, $\sim 5M_{\odot}$ (based on its Teff, Andersen, 1991), remains significantly less than the lower limit of $\sim 6.6M_{\odot}$ determined from the radial velocity orbit of the primary by Evans et al. (2000). Thus, it still appears likely that the B star component of AW Per must also be a binary.

8 SUMMARY

We have shown that the signatures of the Cepheid and B star components of AW Per are clearly present in the wavelength dependence of the centroid of its spectrum. This result demonstrates the power of our approach. A simple model was devised to extract the angular separation of the binary from the centroid measurements. The accuracy of the angular separation is $\sim 2$%, or $\pm$ a few $\times 10^{-4}$″! We also demonstrated that the results are extremely stable to variations in the expected systematic effects in the data and its analysis. We also showed that one possible source of uncertainty in the current data is the absolute level of the far-UV data. Higher quality far-UV observations to secure the B star flux level and secure its parameters would be extremely useful.

Our final results are listed in Table 2. In addition to the angular separations and position angle, these include a Cepheid temperature and systemic extinction that agree with previous estimates and a B star secondary temperature that is considerably hotter than previously thought (e.g., Evans, 1994). However, the likely mass of the secondary still appears too small to account for the minimum mass of the secondary inferred by the radial velocity of the primary. Consequently, it is likely that the B star component of AW Per is also be a binary.

Finally, the long period of AW Per’s orbit means that it will be a few years before the separation changes enough for the second independent observation needed to determine $\sin i$ can be obtained.

Acknowledgments

We would like to thank Karla Peterson and Charles Proffit of STScI, who provided valuable guidance in preparing the observations. This work was supported by NASA grants to SGT, Inc. and SAO.

Appendix A MATHEMATICAL DETAILS

This appendix provides a detailed derivation of how the wavelength dependence of the centroid of the a dispersed image can be used to determine the separation of a binary whose components have different colors.

Consider the set of angular coordinates $x$ and $y$ which are parallel and perpendicular to the dispersion, respectively, with $x$ increasing in the direction of increasing wavelength (this is the standard STIS coordinate system, Kim et al. 2003). Now, let $h(y)$ be the instrumental profile in the cross-dispersion direction, $y$. Then the spectrum of a single star located at $y=y_{0}$ can be expressed as

[TABLE]

where $\lambda=\lambda(x)$, and $N(\lambda)$ is the photon flux at $\lambda$ (we assume infinite resolution in the wavelength direction).

If the spectra of the primary and secondary components of the binary are centered at $y_{p}$, and $y_{s}$, then their spectra are separated by $\Delta y=y_{s}-y_{p}$, and the image of the binary spectrum is given by

[TABLE]

If $\Delta y$ is small compared to structure in $h(y)$, this equation can be approximated by

[TABLE]

Therefore, the wavelength dependence of the centroid of the spectrum will vary as

[TABLE]

where $R(\lambda)=N_{p}(\lambda)/N_{s}(\lambda)$ is the flux ratio of the binary components.

Now, the separation $\Delta y$ depends on both the separation of the binary, $\theta$, and the orientation of the system relative to the dispersion direction. The position angle on the sky of the binary, $\phi$, is defined as the angle measured the c.c. from north to east, with the primary at the origin. The angle $\alpha^{(n)}$ (also measured the c.c. from north to east) is defined as the angle of a line in the dispersion direction pointing in the direction of increasing wavelength for the $n^{th}$ telescope orientation. In this case, $\phi-\alpha^{(n)}$ is the angle between the dispersion and a line connecting the binary components and $\Delta y^{(n)}=\theta\sin(\phi-\alpha^{(n)})$ is the displacement of the two spectra of the binary (note that when $\phi-\alpha=0$, $\pm 180^{\circ}$, $\Delta y=0$). Therefore, the observation obtained with the telescope in the $n^{th}$ orientation can be expressed as

[TABLE]

where $y_{p}^{(n)}$ is the wavelength independent displacement of the $n^{th}$ exposure in $y$.

To extract both $\theta$ and $\phi$ from the observed centroids, at least two observations at different $\alpha$’s are required. Therefore, as long as the relative fluxes of the binary components are known, a linear regression of the wavelength dependence of the centroid against $[1+R(\lambda)]^{-1}$ gives $\Delta y^{(n)}$ for that observation. The $y_{p}^{(n)}$ are constant terms related to the absolute position of the primary star, although in practice the they cannot be reliably disentangled from the large random errors in the absolute position of the binary on the detector at each orientation.

Once the $\Delta y^{(n)}$ are determined for each orientation, these are plotted against the known quantities, $\alpha^{(n)}$. Since

[TABLE]

fitting a sine function to the $\Delta y^{(n)}$ as a function of the $\alpha^{(n)}$ determines $\phi$ and $\theta$, the observables of an astrometric binary at the epoch of the observations.