Mon. Not. R. Astron. Soc. 000, 1–?? (2007) Printed 29 October 2018 (MN LATEX style file v2.2)
The Angular Separation of the Components of the Cepheid
AW Per
D. Massa
1⋆ † and N. R. Evans2 ⋆
1SGT, Inc. NASA’s GSFC, Code 665, Greenbelt, MD 20771, USA
2Center for Astrophysics, 60 Garden St., MS 4, Cambridge, MA 02138
Accepted 2007 December 15. Received 2007 December 14; in original form 2007 August 13
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, sin i, for the system if the mea-
surements are repeated at additional epochs. Since the system is potentially a double
lined spectroscopic binary, the combination of spectroscopic orbits for both compo-
nents 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.
Key words: Cepheids – stars: AW Per – binaries: visual – binaries: spectroscopic.
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 inte-
riors modelling. The use of Cepheids as primary extragalac-
tic 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 metal-
icity is still not fully understood (Romaniello et al. 2005).
Consequently, accurate distances (absolute magnitudes) to
Galactic cepheids are needed to fully understand and quan-
tify 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 cal-
culations 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
⋆ E-mail: massa@derckmassa.net; evans@head-cfa.harvard.edu
† Based on observations with the NASA/ESA Hubble Space Tele-
scope, obtained at the Space Telescope Science Institute, which
is operated by the Association of Universities for Research in As-
tronomy, Inc. under NASA contract No. NAS5-26555.
maxima or by a parameterization of the pulsation con-
stant. 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 Si-
mon, 1990, for a summary). We see, therefore, that in ad-
dition 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 spec-
troscopic 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., Groe-
newegen & Oudmaijer 2000). More recently, a large cam-
paign 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 ob-
servations (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 re-
solve in the optical. This is because of the inevitable, enor-
mous magnitude differences of the components in the opti-
cal, 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 bi-
nary, and the contrast between the primary and secondary
throughout the optical and IR is obvious.
c© 2007 RAS
http://arxiv.org/abs/0704.1051v2
2 D. Massa and N. R. Evans
Figure 1. Kurucz models for a typical Cepheid (large/red) + hot
star (small/blue) binary. The top panel shows how the secondary
is roughly 10 times fainter in the optical, making the system ex-
tremely difficult to resolve from the ground. On the other hand,
the secondary dominates the flux from the system in the UV. The
remaining 5 panels demonstrate how the wavelength dependence
of the spectrum centroid changes with orientation of the axis of
the binary relative to the dispersion for 5 different orientations,
shown to the left of each panel. Notice that in the spectral re-
gion accessible from the ground, the centroid shifts by less than
10% of the full separation. The “cross-over” point is not reached
until λ ∼ 2500Å. A color version of the figure is available in the
electronic version of the paper.
Thus, while the measurement of a Cepheid mass by di-
rectly 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, in-
direct 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 sys-
tems. Clearly, a direct measurement of the masses of both
components is more desirable.
In this paper, we describe how we used the Space Tele-
scope 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∼ 12000K (Evans
1994).
The remainder of the paper is organized as follows: §2
provides an overview the approach used to “resolve” the bi-
nary, §3 describes the observations, §4 gives the data anal-
ysis, §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 “resolv-
ing 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 ba-
sic concept of this approach is quite simple. It is based on
an idea put forth by Beckers (1982) and has been indepen-
dently 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 an-
other 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 sys-
tem. 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 dis-
placed 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 ex-
posure 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
c© 2007 RAS, MNRAS 000, 1–??
AW Per 3
would have to be obtained using different optical elements,
making alignment at the sub-pixel level effectively impossi-
ble. 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 measure-
ments 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, λ, are given by λ = λ(x). Now, consider a bi-
nary whose components have an angular separation θ and
photon fluxes per unit wavelength Np(λ) and Ns(λ) for the
primary and secondary, respectively. Further, let φ 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 sec-
ondary) and let α be a similarly measured angle between
north and a line in the dispersion direction pointing in the
direction of decreasing wavelength. Thus, α can be varied by
changing the orientation of the telescope. With these defi-
nitions, the wavelength dependence of the centroid of the
spectrum of a single observation of a binary is
y(λ) =
1 +Np(λ)/Ns(λ)
+ Const. where (1)
∆y = θ sin(φ− α) (2)
(see the appendix). Thus, if Np(λ)/Ns(λ) is known, then
measurements at two or more orientations (α’s) enables one
to determine θ and φ, the separation and position angle of
the binary. Note that if the spectral energy distributions
(SEDs) of the components are vastly different, then the po-
sition of the centroid shifts from one to the other, depend-
ing 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 dif-
ferent effective temperatures, is influenced by the relative
energy distributions of the two components and the orien-
tation 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 (∼ 3000Å for the case shown). In
order to infer spatial information from the centroids, it is de-
sirable 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 ex-
tract the angular separation. Note that if the absolute flux
calibration of the instrument is well-determined, then the
Figure 2. Relative error in the angular separation of a binary
determined from fitting a cosine curve to measurements obtained
at three orientations, {−∆α, 0,+∆α} versus ∆α (abscissa) over
the interval ∆α = 1 → 90◦. The different curves are for different
values of the orientation of the system on the sky, φ, between
φ = 1 → 90◦.
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 θ and φ in terms of the
measured quantities ∆y(n) and α(n), for n > 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 an-
gles and φ. We have examined the relative error for sampling
three orientations, α(n) = {−∆α, 0,+∆α}, for position an-
gles between 1 and 90◦. Figure 2 demonstrates how the rel-
ative accuracy of the observations changes as a function of
sampling interval, ∆α, and relative orientations, φ, for this
case. For most orientations, any sampling with ∆α & 30◦
provides comparable accuracy.
The approximations developed in this section are only
valid in the sub-Rayleigh regime. Once the sources are re-
solved at any wavelength, the entire image must be modeled
using a an accurate representation of the point spread func-
tion as well as the fluxes of the two objects.
2.2 Exposure Times and Random Errors
The counts needed to centroid to an accuracy σ[y(λ)] can be
estimated for an instrument whose spread function perpen-
dicular to the dispersion is a Gaussian with FWHM = ξ.
A single count is equivalent to one estimate of the center
of the spectrum drawn from a sample with an RMS disper-
sion σ = ξ/
8 ln 2 = 0.42ξ. Therefore, N samples (counts)
determine the centroid to an accuracy of
σ[y(λ)] =
0.42ξ
. (3)
c© 2007 RAS, MNRAS 000, 1–??
4 D. Massa and N. R. Evans
Equation (3) gives the counts needed over a wavelength
band to obtain the desired accuracy. The FWHM of the
STIS PSF varies from ∼ 0.05 − 0.07′′ (depending on wave-
length) and the minimum number of counts obtained in one
10 min observation over a spectral resolution element (2 pix-
els) was 4000, and we obtained 3 of these. Therefore, the
poorest precision we can expect based upon simple sampling
arguments is ∼ 3 × 10−4′′, and this is for a single resolu-
tion element. In all, there are 512 independent resolution
elements which will be combined to determine a single mea-
surement of ∆y through the use of eq. (1). Therefore, ran-
dom noise in the angular separation determinations should
be . 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,
3 THE OBSERVATIONS
As can be seen from the top panel of Figure 1, a broad wave-
length baseline is needed to optimize the extraction process.
Furthermore, good spectral resolution is also advantageous,
since spectral features provide additional constraints. Con-
sequently, we employed the STIS on HST to obtain high
spatial resolution, excellent wavelength coverage and good
spectral resolution. We used the STIS NUV-MAMA detec-
tor together with its G230L grating, since this combination
provided good coverage (1600 6 λ 6 3160 Å) of the ex-
pected cross-over point (see, Kim Quijano, J., et al. 2003).
Spectra were obtained at three distinct roll angles (see,
Table 1) which differ by ∼ ±20◦. Although rolls of ±60◦
would be optimal, we were limited to smaller rolls by HST
restrictions for objects at the declination of AW Per. Al-
though 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 spec-
tra of the binary. At each roll, we offset the star by ±0.1′′
and obtained additional science exposures. This procedure
allows us to characterize localized distortions in the detec-
tor. 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 ±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 x0 − y0 system. In this system, the dispersion direction
(from red to blue) makes an angle (measured in the c.c.
direction) of −1.4◦ with the x0 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 measure-
ments 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 cen-
troids, y(λ), from the raw images. We chose to analyze the
raw images (in their native “highres” 2048×2048 format) be-
cause 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 gaus-
sians. 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 sys-
tematic differences were found among all three approaches.
However, the results from the non-linearly extracted cen-
troids 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 an-
gle were rebinned to 512 elements from the 2048 elements
available in the raw images, and these were used to con-
struct mean centroids at each roll and their standard devi-
ations. Because the centroids near the edges of the detec-
tor 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 Fig-
ure 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
c© 2007 RAS, MNRAS 000, 1–??
AW Per 5
Table 1. Observation log
Obs ID Off Set Roll Obs Date Exp. Time Phasea V (B − V )
arc sec Deg. MJD - 52235 Min. Φ 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
a Phase, V and (B − V ) are derived from sources in the literature, as discussed in the text.
Figure 3. Standard deviations of the three independent spectra
of AW Per obtained at each roll angle. The standard deviations
for each roll angle are over plotted.
particular roll angle to produce a mean centroid, 〈y〉, at each
roll. These measurements contain geometric distortions and
any systematic effects introduced by the centroid extrac-
tion 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 dis-
placements from one roll to another, and the scatter that
the former exhibit (Fig. 3) demonstrates that localized ge-
ometric distortions are small. Similarly, any systematic af-
fects that result from mis-matches between the actual PSF
orthogonal to the dispersion and the gaussian used to deter-
mine the centroids will cancel out, since the same process is
used in each case.
Finally, we must account for the fact that y(λ) is not
exactly perpendicular to the dispersion. As a result, we must
divide the final displacements that we measure by cos(1.4◦).
4.2 Fluxes
STIS fluxes were extracted from the images using the CAL-
STIS IDL software package developed by Lindler (1998) for
Figure 4. Plots of the mean STIS spectrum of AW Per (solid
curve) together with the available IUE spectra (dotted), cali-
brated to the HST flux system.
the STIS Instrument Definition Team. In order to constrain
the B star flux contribution, we also incorporate the avail-
able 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 us-
ing 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
(Φ = 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 discrep-
ancy 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 intro-
duces 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 ob-
servations is 1.59 and 3.19 hours, respectively. This plot
c© 2007 RAS, MNRAS 000, 1–??
6 D. Massa and N. R. Evans
Figure 5. Plots of the ratios of mean STIS spectra of AW Per
obtained at the second and third roll angles divided by the mean
flux obtained at the first roll angle. These plots demonstrate how
the Cepheid component brightened over the 3.5 hour observing
sequence. Notice that the flux at the shortest wavelengths does
not change, since it is dominated by the B star secondary.
demonstrates two things. First, the Cepheid flux changed
significantly throughout the three HST orbits spanned by
the observations. Second, the flux ratios decrease with wave-
length, becoming unity at the shortest wavelengths. This is
contrary to what is normally seen in single Cepheids like
δ 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 mag-
nitude of the system the time of the observations. We com-
bined 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, com-
bined flux of the two binary components. Ideally, one would
fit the observed flux and centroid positions with a combi-
nation of single star spectra obtained with the same instru-
ment and which experience the same reddening. However,
because there is no such library of single star spectra avail-
able, 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 procedure1 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 ef-
fects were ignored for the first attempt. First, we use Kurucz
(1991) Atlas 9 models with updated metallicities2 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 man-
ner 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 Fitz-
patrick (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 (c3), the width of the 2175 Å (γ) and far-UV
curvature term (c4) to vary independently. In this way, we
can accommodate any observed extinction curve. As a result,
the RV parameter (the ratio of visual extinction to color ex-
cess) only affects the general slope of the UV extinction and
the shape of the near-UV curve, and the wavelength depen-
dence of the total extinction to an object can be expressed
Aλ ≡ A[RV , E(B − V ), γ, c3, c4;λ] . (4)
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 λ 6 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 ∆y(n) (displacements perpendicular to the dispersion
at each roll angle), Tseff (the effective temperature of the B
star secondary), [m/H]
(the abundance parameter for the
B star), E(B−V ) (the color excess of the system, consistent
with the fluxes), RV (which determines the slope of the UV
extinction curve), γ (the width of the 2175 Å bump), c3
(the bump strength), and c4 (the strength of the far UV
curvature) – 10 parameters in all. The V magnitude of the B
star, Vs, is fixed by the observed flux attributed to the B star
at λ = 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
1 We use the Markwardt non-linear IDL fitting procedure, avail-
able at http://astrog.physics.wisc.edu/∼craigm/idl/idl.html.
2 We used the the Kurucz “preferred models” available at
http://kurucz.harvard.edu/.
c© 2007 RAS, MNRAS 000, 1–??
http://astrog.physics.wisc.edu/~craigm/idl/idl.html
http://kurucz.harvard.edu/
AW Per 7
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:
− θ2sN(Ts, log gs, vt, [m/H];λ)
θ2sN(Ts, log gs, vt, [m/H];λ)
and the unreddened flux of the Cepheid is given by
p = [N(λ)
− θ2sN(Ts, log gs, vt, [m/H];λ)]
×10A[RV ,E(B−V ),γ,c3,c4;λ] (6)
where θs 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, espe-
cially 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 inconsis-
tency 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◦ (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).
Figure 6. Diagram showing the definitions of the different angles
and coordinate systems used in the analysis, and their relations to
one another. The position angle on the sky of the binary angle, φ,
is defined as the angle measured the c.c. from north to east, with
the primary at the origin. The x− y system is the standard STIS
coordinate system, with x parallel to the dispersion (increasing
in the direction of increasing wavelength) and y perpendicular
to it. The angle α (also measured the c.c. from north to east) is
defined as the angle between North and x for a given telescope
orientation. Thus, φ− α is the angle between the dispersion and
a line connecting the binary components and ∆y = θ sin(φ−α) is
the displacement of the two spectra of the binary perpendicular
to the dispersion. If φ− α = 0 or ±180◦, then ∆y = 0.
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 over-
whelmed 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. Conse-
quently, we fixed the surface gravity at 4.0, again typical for
main sequence B stars. We allowed the abundance parame-
ter, [m/H]
, and the effective temperature of the B star to be
optimized by the least squares routine, along with the ∆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 apply-
ing 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 prop-
erties 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 Ta-
ble 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.
c© 2007 RAS, MNRAS 000, 1–??
8 D. Massa and N. R. Evans
Table 2. Parameter Values
Parameter Value Parameter Value
∆y1 −0.010 c3 4.13
∆y2 0.279 c4 0.82
∆y3 −0.269 γ 0.9686
[6297] Vs (11.084)
15735 (B − V )s0 (−0.156)
log gp (4.00) (U −B)
0 (−0.597)
log gs [1.60] Vp (7.362)
[m/H]p [0.00] (B − V )
0 (0.494)
[m/H]s -0.20 (U −B)
0 (0.359)
E(B − V ) 0.53 ∆ logL (0.95)
R(V ) 3.11
Values in parenthesis were not involved in the fitting procedure.
Values in square brackets were determined from a fit to the
Cepheid SED derived from the initial fit.
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 RV = 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 red-
dened B star model from the observed SED of the system
and then correcting this difference for the effects of extinc-
tion. 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 as-
signed 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
eff (the effective temperature of the
primary), log gp (the surface gravity of the primary) and
[m/H]
(the abundance of the primary), to vary. We had
to restrict the surface gravity to be larger than 1.6, other-
wise the fitting routine would seek log gp values that were
unrealistically small (we expect a log gp ≃ 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 in-
ferred UV Cepheid SED by comparing it to IUE observa-
tions of the single Cepheid star δ Cep. δ Cep has a period
of 5.4 days, compared to 6.5 days for AW Per, and its mean
unreddened color is 〈(B − V )〉 = 0.57. To obtain the in-
trinsic color of AW Per, we use our derived color excess
for the system and the intrinsic colors of the B star sec-
ondary from Table 2 and the mean magnitude of the system,
〈V 〉 = 7.49 mag, to correct the observed mean color of the
system, 〈(B − V )〉 = 1.06 mag, for both extinction and the
presence of the companion. The result is 〈(B−V )p0〉 = 0.57,
identical to that of δ Cep (recall that the intrinsic color
we derive for AW Per is at Φ ≃ 0.92). Thus, the compar-
ison between these two stars is expected to be quite good.
Figure 7. Fits to the mean centroids at each roll angle for AW
Per. Each mean centroid was fit simultaneously with the corre-
sponding fluxes, optical photometry and interstellar extinction.
A Kurucz model was used to fit the B star component, and the
Cepheid flux was taken to be the difference between the reddened
B star model and the observed flux.
The bottom plot in Figure 9 compares the unreddened IUE
data (points) for δ Cep from several exposures obtained
for 0.9 6 Φ(δCep) 6 1.0 to the unreddened Cepheid STIS
spectrum (solid curve) of AW Per. Several IUE exposures
are required to produce the δ Cep spectrum since the dy-
namic range of IUE was so limited and the range of the
UV SED of δ 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 Φ = 0.92 (the mean for the STIS data) and
δ Cep at Φ = 0.95 (the mean of the IUE data).
Finally, we utilize the ∆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 ∆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
c© 2007 RAS, MNRAS 000, 1–??
AW Per 9
Figure 8. Top: Best fit B star (thin curve) compared to the IUE
(points) and STIS (thick curve) fluxes. The model includes red-
dening. We only show the far-UV region, since the fit is, by defini-
tion, exact for wavelengths longward of 1650Å. Bottom: AW Per
extinction curve determined by the simultaneous fit of the flux
and centroids (solid curve) compared to a standard RV = 3.1
curve (dotted) from Fitzpatrick (1999).
used to weight the fit. The final result of the analysis is a
separation of θ = 13.74 ± 0.26 mas and a position angle
φ = 184.16 ± 1.94 deg, for an accuracy of ∼ 2%.
6.1 Errors in the parameters
In this section, we describe the internal, random, errors af-
fecting 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 deriva-
tives of the model. These errors are listed in the second col-
umn 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 indepen-
dent 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
3 applica-
ble to the error in the mean) are listed in the third column
of Table 3. Notice that the errors in the ∆y(n) determined
from the S.D.s are nearly twice as large. To be conservative,
Figure 9. Top: Inferred dereddened Cepheid SED (points) com-
pared to the best fitting Kurucz model (solid) and the dered-
dened flux of the best fit B star (dashed). Bottom: Comparison
of the unreddened Cepheid flux (solid curve) and an unreddened
IUE spectrum (dots) of δ Cep observations for 0.90 6 Φ 6 0.95.
The δ Cep flux is scaled by the difference between V = 3.54 at
Φ = 0.925 for δ Cep and V = 7.37, the magnitude of the primary
in AW Per at Φ = 0.92 (the mean phase of the STIS observations).
As discussed in the text, the δ Cep spectrum is a combination of
several IUE spectra.
these errors were used as the error shown in Figure 10 and
in determining the errors in θ and φ.
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 ±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 Teff
s, but has little effect on
the ∆y(n), which are the object of our analysis. In fact, the
only significant change in the ∆y(n) result from our inability
to determine whether the STIS or IUE fluxes are correct,
c© 2007 RAS, MNRAS 000, 1–??
10 D. Massa and N. R. Evans
Table 3. Errors
Param. Prog. S.D. |δ log g| |δ
fSTIS
fP+fs
∆y(1) 0.004 0.015 1.4× 10−4 6.5× 10−6 5.0× 10−4
∆y(2) 0.005 0.017 0.0019 1.4× 10−4 0.015
∆y(3) 0.005 0.014 0.0021 1.3× 10−4 0.015
248 105 1205 9.1 37
[m/H]
0.057 0.025 7.5× 10−5 0.0016 0.0064
E(B − V ) 0.001 0.038 0.018 0.0026 0038
RV 0.031 0.12 0.11 0.0090 2.7× 10
γ 0.015 2.4× 10−4 0.019 8.6× 10−4 0.0025
c3 0.14 0.32 0.049 0.029 0.0055
c4 0.019 0.066 0.014 4.3× 10
−3 0.0068
Figure 10. Determination of the angular separation of AW Per.
The observational errors for ∆y were determined from individual
fits to the 3 independent offset observations at each roll angle.
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 de-
termined 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 unred-
dened SED of the single Cepheid, δ 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 δ Cep, so the derived SED is
quite reasonable. Third, the flux in δ Cep is extremely small
for wavelengths shortward of 1650Å, bolstering our assump-
tion 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 nu-
merous, ill-defined physical processes that are not fully in-
corporated into the Atlas 9 models. These include spherical
extension, which can enhance the UV flux from an atmo-
sphere (see Fig. 4 in Hauschildt et al. 1999), chromospheres
(e.g., Sasselov & Lester 1994), the amount of convective en-
ergy 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 ∼ 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 prob-
In spite of these difficulties, it is of interest to exam-
ine the physical parameters determined from the Cepheid
model. To begin, Teff of the best fit model agrees reason-
ably 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 set-
tled on an even lower value if it had been allowed to do so.
It is also interesting that the Cepheid model has a signif-
icantly 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 abun-
dance (Fitzpatrick & Massa 1999).
c© 2007 RAS, MNRAS 000, 1–??
AW Per 11
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 RV = 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 ex-
cess 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 mod-
els to describe normal B star spectra are well documented.
Notice that Teff we derive is considerably hotter than previ-
ously estimated by Evans (1994), and lies somewhat closer
to the ZAMS (see, Fig. 7 in Evans 1994). However, its prob-
able mass, ∼ 5M⊙ (based on its Teff , Andersen, 1991), re-
mains significantly less than the lower limit of ∼ 6.6M⊙
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 wave-
length 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 ∼ 2%, or ± a few ×10−4′′! We also
demonstrated that the results are extremely stable to vari-
ations in the expected systematic effects in the data and its
analysis. We also showed that one possible source of uncer-
tainty 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 pri-
mary. 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.
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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, respec-
tively, with x increasing in the direction of increasing wave-
length (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 = y0 can be expressed as
f(λ, y) = N(λ)h(y − y0) (A1)
where λ = λ(x), andN(λ) is the photon flux at λ (we assume
infinite resolution in the wavelength direction).
If the spectra of the primary and secondary components
of the binary are centered at yp, and ys, then their spectra
are separated by ∆y = ys − yp, and the image of the binary
spectrum is given by
f(λ, y) = Np(λ)h(y − yp) +Ns(λ)h(y − yp −∆y). (A2)
If ∆y is small compared to structure in h(y), this equation
can be approximated by
f(λ, y) ≃ Np(λ)h(y − yp)
+Ns(λ)
4h(y − yp) +∆y
dh(y)
y=y−yp
= [Np(λ) +Ns(λ)]×
4h(y − yp) +
Ns(λ)
Np(λ) +Ns(λ)
dh(y)
y=y−yp
≃ [Np(λ) +Ns(λ)]×
∆yNs(λ)
Np(λ) +Ns(λ)
Therefore, the wavelength dependence of the centroid of the
spectrum will vary as
y(λ) = yp +
∆yNs(λ)
Np(λ) +Ns(λ)
= yp +
1 +R(λ)
where R(λ) = Np(λ)/Ns(λ) is the flux ratio of the binary
components.
Now, the separation ∆y depends on both the separation
of the binary, θ, and the orientation of the system relative
to the dispersion direction. The position angle on the sky
of the binary, φ, is defined as the angle measured the c.c.
from north to east, with the primary at the origin. The angle
α(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 nth telescope
orientation. In this case, φ − α(n) is the angle between the
dispersion and a line connecting the binary components and
∆y(n) = θ sin(φ−α(n)) is the displacement of the two spectra
of the binary (note that when φ − α = 0, ±180◦, ∆y = 0).
Therefore, the observation obtained with the telescope in
the nth orientation can be expressed as
p + θ sin(φ− α(n))[1 +R(λ)]−1 (A5)
where y
p is the wavelength independent displacement of
the nth exposure in y.
To extract both θ and φ from the observed centroids, at
least two observations at different α’s are required. There-
fore, 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(λ)]−1 gives ∆y(n) for that ob-
servation. The y
p are constant terms related to the abso-
lute 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 ∆y(n) are determined for each orientation,
these are plotted against the known quantities, α(n). Since
= θ sin(φ− α(n)) (A6)
fitting a sine function to the ∆y(n) as a function of the α(n)
determines φ and θ, the observables of an astrometric binary
at the epoch of the observations.
c© 2007 RAS, MNRAS 000, 1–??
	INTRODUCTION
	THE APPROACH(Cross-Dispersion Imaging)
	Basic Principles
	Exposure Times and Random Errors
	THE OBSERVATIONS
	DATA REDUCTION
	Centroids
	Fluxes
	ANALYSIS
	Overview
	Model Components
	Details of the Fitting Procedures
	Determining the Separations
	Weights
	RESULTS
	Errors in the parameters
	DISCUSSION
	SUMMARY
	MATHEMATICAL DETAILS

  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.

Mon. Not. R. Astron. Soc. 000, 1–?? (2007) Printed 29 October 2018 (MN LATEX style file v2.2)
The Angular Separation of the Components of the Cepheid
AW Per
D. Massa
1⋆ † and N. R. Evans2 ⋆
1SGT, Inc. NASA’s GSFC, Code 665, Greenbelt, MD 20771, USA
2Center for Astrophysics, 60 Garden St., MS 4, Cambridge, MA 02138
Accepted 2007 December 15. Received 2007 December 14; in original form 2007 August 13
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, sin i, for the system if the mea-
surements are repeated at additional epochs. Since the system is potentially a double
lined spectroscopic binary, the combination of spectroscopic orbits for both compo-
nents 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.
Key words: Cepheids – stars: AW Per – binaries: visual – binaries: spectroscopic.
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 inte-
riors modelling. The use of Cepheids as primary extragalac-
tic 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 metal-
icity is still not fully understood (Romaniello et al. 2005).
Consequently, accurate distances (absolute magnitudes) to
Galactic cepheids are needed to fully understand and quan-
tify 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 cal-
culations 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
⋆ E-mail: massa@derckmassa.net; evans@head-cfa.harvard.edu
† Based on observations with the NASA/ESA Hubble Space Tele-
scope, obtained at the Space Telescope Science Institute, which
is operated by the Association of Universities for Research in As-
tronomy, Inc. under NASA contract No. NAS5-26555.
maxima or by a parameterization of the pulsation con-
stant. 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 Si-
mon, 1990, for a summary). We see, therefore, that in ad-
dition 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 spec-
troscopic 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., Groe-
newegen & Oudmaijer 2000). More recently, a large cam-
paign 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 ob-
servations (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 re-
solve in the optical. This is because of the inevitable, enor-
mous magnitude differences of the components in the opti-
cal, 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 bi-
nary, and the contrast between the primary and secondary
throughout the optical and IR is obvious.
c© 2007 RAS
http://arxiv.org/abs/0704.1051v2
2 D. Massa and N. R. Evans
Figure 1. Kurucz models for a typical Cepheid (large/red) + hot
star (small/blue) binary. The top panel shows how the secondary
is roughly 10 times fainter in the optical, making the system ex-
tremely difficult to resolve from the ground. On the other hand,
the secondary dominates the flux from the system in the UV. The
remaining 5 panels demonstrate how the wavelength dependence
of the spectrum centroid changes with orientation of the axis of
the binary relative to the dispersion for 5 different orientations,
shown to the left of each panel. Notice that in the spectral re-
gion accessible from the ground, the centroid shifts by less than
10% of the full separation. The “cross-over” point is not reached
until λ ∼ 2500Å. A color version of the figure is available in the
electronic version of the paper.
Thus, while the measurement of a Cepheid mass by di-
rectly 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, in-
direct 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 sys-
tems. Clearly, a direct measurement of the masses of both
components is more desirable.
In this paper, we describe how we used the Space Tele-
scope 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∼ 12000K (Evans
1994).
The remainder of the paper is organized as follows: §2
provides an overview the approach used to “resolve” the bi-
nary, §3 describes the observations, §4 gives the data anal-
ysis, §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 “resolv-
ing 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 ba-
sic concept of this approach is quite simple. It is based on
an idea put forth by Beckers (1982) and has been indepen-
dently 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 an-
other 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 sys-
tem. 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 dis-
placed 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 ex-
posure 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
c© 2007 RAS, MNRAS 000, 1–??
AW Per 3
would have to be obtained using different optical elements,
making alignment at the sub-pixel level effectively impossi-
ble. 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 measure-
ments 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, λ, are given by λ = λ(x). Now, consider a bi-
nary whose components have an angular separation θ and
photon fluxes per unit wavelength Np(λ) and Ns(λ) for the
primary and secondary, respectively. Further, let φ 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 sec-
ondary) and let α be a similarly measured angle between
north and a line in the dispersion direction pointing in the
direction of decreasing wavelength. Thus, α can be varied by
changing the orientation of the telescope. With these defi-
nitions, the wavelength dependence of the centroid of the
spectrum of a single observation of a binary is
y(λ) =
1 +Np(λ)/Ns(λ)
+ Const. where (1)
∆y = θ sin(φ− α) (2)
(see the appendix). Thus, if Np(λ)/Ns(λ) is known, then
measurements at two or more orientations (α’s) enables one
to determine θ and φ, the separation and position angle of
the binary. Note that if the spectral energy distributions
(SEDs) of the components are vastly different, then the po-
sition of the centroid shifts from one to the other, depend-
ing 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 dif-
ferent effective temperatures, is influenced by the relative
energy distributions of the two components and the orien-
tation 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 (∼ 3000Å for the case shown). In
order to infer spatial information from the centroids, it is de-
sirable 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 ex-
tract the angular separation. Note that if the absolute flux
calibration of the instrument is well-determined, then the
Figure 2. Relative error in the angular separation of a binary
determined from fitting a cosine curve to measurements obtained
at three orientations, {−∆α, 0,+∆α} versus ∆α (abscissa) over
the interval ∆α = 1 → 90◦. The different curves are for different
values of the orientation of the system on the sky, φ, between
φ = 1 → 90◦.
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 θ and φ in terms of the
measured quantities ∆y(n) and α(n), for n > 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 an-
gles and φ. We have examined the relative error for sampling
three orientations, α(n) = {−∆α, 0,+∆α}, for position an-
gles between 1 and 90◦. Figure 2 demonstrates how the rel-
ative accuracy of the observations changes as a function of
sampling interval, ∆α, and relative orientations, φ, for this
case. For most orientations, any sampling with ∆α & 30◦
provides comparable accuracy.
The approximations developed in this section are only
valid in the sub-Rayleigh regime. Once the sources are re-
solved at any wavelength, the entire image must be modeled
using a an accurate representation of the point spread func-
tion as well as the fluxes of the two objects.
2.2 Exposure Times and Random Errors
The counts needed to centroid to an accuracy σ[y(λ)] can be
estimated for an instrument whose spread function perpen-
dicular to the dispersion is a Gaussian with FWHM = ξ.
A single count is equivalent to one estimate of the center
of the spectrum drawn from a sample with an RMS disper-
sion σ = ξ/
8 ln 2 = 0.42ξ. Therefore, N samples (counts)
determine the centroid to an accuracy of
σ[y(λ)] =
0.42ξ
. (3)
c© 2007 RAS, MNRAS 000, 1–??
4 D. Massa and N. R. Evans
Equation (3) gives the counts needed over a wavelength
band to obtain the desired accuracy. The FWHM of the
STIS PSF varies from ∼ 0.05 − 0.07′′ (depending on wave-
length) and the minimum number of counts obtained in one
10 min observation over a spectral resolution element (2 pix-
els) was 4000, and we obtained 3 of these. Therefore, the
poorest precision we can expect based upon simple sampling
arguments is ∼ 3 × 10−4′′, and this is for a single resolu-
tion element. In all, there are 512 independent resolution
elements which will be combined to determine a single mea-
surement of ∆y through the use of eq. (1). Therefore, ran-
dom noise in the angular separation determinations should
be . 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,
3 THE OBSERVATIONS
As can be seen from the top panel of Figure 1, a broad wave-
length baseline is needed to optimize the extraction process.
Furthermore, good spectral resolution is also advantageous,
since spectral features provide additional constraints. Con-
sequently, we employed the STIS on HST to obtain high
spatial resolution, excellent wavelength coverage and good
spectral resolution. We used the STIS NUV-MAMA detec-
tor together with its G230L grating, since this combination
provided good coverage (1600 6 λ 6 3160 Å) of the ex-
pected cross-over point (see, Kim Quijano, J., et al. 2003).
Spectra were obtained at three distinct roll angles (see,
Table 1) which differ by ∼ ±20◦. Although rolls of ±60◦
would be optimal, we were limited to smaller rolls by HST
restrictions for objects at the declination of AW Per. Al-
though 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 spec-
tra of the binary. At each roll, we offset the star by ±0.1′′
and obtained additional science exposures. This procedure
allows us to characterize localized distortions in the detec-
tor. 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 ±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 x0 − y0 system. In this system, the dispersion direction
(from red to blue) makes an angle (measured in the c.c.
direction) of −1.4◦ with the x0 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 measure-
ments 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 cen-
troids, y(λ), from the raw images. We chose to analyze the
raw images (in their native “highres” 2048×2048 format) be-
cause 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 gaus-
sians. 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 sys-
tematic differences were found among all three approaches.
However, the results from the non-linearly extracted cen-
troids 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 an-
gle were rebinned to 512 elements from the 2048 elements
available in the raw images, and these were used to con-
struct mean centroids at each roll and their standard devi-
ations. Because the centroids near the edges of the detec-
tor 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 Fig-
ure 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
c© 2007 RAS, MNRAS 000, 1–??
AW Per 5
Table 1. Observation log
Obs ID Off Set Roll Obs Date Exp. Time Phasea V (B − V )
arc sec Deg. MJD - 52235 Min. Φ 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
a Phase, V and (B − V ) are derived from sources in the literature, as discussed in the text.
Figure 3. Standard deviations of the three independent spectra
of AW Per obtained at each roll angle. The standard deviations
for each roll angle are over plotted.
particular roll angle to produce a mean centroid, 〈y〉, at each
roll. These measurements contain geometric distortions and
any systematic effects introduced by the centroid extrac-
tion 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 dis-
placements from one roll to another, and the scatter that
the former exhibit (Fig. 3) demonstrates that localized ge-
ometric distortions are small. Similarly, any systematic af-
fects that result from mis-matches between the actual PSF
orthogonal to the dispersion and the gaussian used to deter-
mine the centroids will cancel out, since the same process is
used in each case.
Finally, we must account for the fact that y(λ) is not
exactly perpendicular to the dispersion. As a result, we must
divide the final displacements that we measure by cos(1.4◦).
4.2 Fluxes
STIS fluxes were extracted from the images using the CAL-
STIS IDL software package developed by Lindler (1998) for
Figure 4. Plots of the mean STIS spectrum of AW Per (solid
curve) together with the available IUE spectra (dotted), cali-
brated to the HST flux system.
the STIS Instrument Definition Team. In order to constrain
the B star flux contribution, we also incorporate the avail-
able 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 us-
ing 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
(Φ = 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 discrep-
ancy 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 intro-
duces 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 ob-
servations is 1.59 and 3.19 hours, respectively. This plot
c© 2007 RAS, MNRAS 000, 1–??
6 D. Massa and N. R. Evans
Figure 5. Plots of the ratios of mean STIS spectra of AW Per
obtained at the second and third roll angles divided by the mean
flux obtained at the first roll angle. These plots demonstrate how
the Cepheid component brightened over the 3.5 hour observing
sequence. Notice that the flux at the shortest wavelengths does
not change, since it is dominated by the B star secondary.
demonstrates two things. First, the Cepheid flux changed
significantly throughout the three HST orbits spanned by
the observations. Second, the flux ratios decrease with wave-
length, becoming unity at the shortest wavelengths. This is
contrary to what is normally seen in single Cepheids like
δ 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 mag-
nitude of the system the time of the observations. We com-
bined 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, com-
bined flux of the two binary components. Ideally, one would
fit the observed flux and centroid positions with a combi-
nation of single star spectra obtained with the same instru-
ment and which experience the same reddening. However,
because there is no such library of single star spectra avail-
able, 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 procedure1 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 ef-
fects were ignored for the first attempt. First, we use Kurucz
(1991) Atlas 9 models with updated metallicities2 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 man-
ner 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 Fitz-
patrick (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 (c3), the width of the 2175 Å (γ) and far-UV
curvature term (c4) to vary independently. In this way, we
can accommodate any observed extinction curve. As a result,
the RV parameter (the ratio of visual extinction to color ex-
cess) only affects the general slope of the UV extinction and
the shape of the near-UV curve, and the wavelength depen-
dence of the total extinction to an object can be expressed
Aλ ≡ A[RV , E(B − V ), γ, c3, c4;λ] . (4)
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 λ 6 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 ∆y(n) (displacements perpendicular to the dispersion
at each roll angle), Tseff (the effective temperature of the B
star secondary), [m/H]
(the abundance parameter for the
B star), E(B−V ) (the color excess of the system, consistent
with the fluxes), RV (which determines the slope of the UV
extinction curve), γ (the width of the 2175 Å bump), c3
(the bump strength), and c4 (the strength of the far UV
curvature) – 10 parameters in all. The V magnitude of the B
star, Vs, is fixed by the observed flux attributed to the B star
at λ = 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
1 We use the Markwardt non-linear IDL fitting procedure, avail-
able at http://astrog.physics.wisc.edu/∼craigm/idl/idl.html.
2 We used the the Kurucz “preferred models” available at
http://kurucz.harvard.edu/.
c© 2007 RAS, MNRAS 000, 1–??
http://astrog.physics.wisc.edu/~craigm/idl/idl.html
http://kurucz.harvard.edu/
AW Per 7
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:
− θ2sN(Ts, log gs, vt, [m/H];λ)
θ2sN(Ts, log gs, vt, [m/H];λ)
and the unreddened flux of the Cepheid is given by
p = [N(λ)
− θ2sN(Ts, log gs, vt, [m/H];λ)]
×10A[RV ,E(B−V ),γ,c3,c4;λ] (6)
where θs 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, espe-
cially 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 inconsis-
tency 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◦ (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).
Figure 6. Diagram showing the definitions of the different angles
and coordinate systems used in the analysis, and their relations to
one another. The position angle on the sky of the binary angle, φ,
is defined as the angle measured the c.c. from north to east, with
the primary at the origin. The x− y system is the standard STIS
coordinate system, with x parallel to the dispersion (increasing
in the direction of increasing wavelength) and y perpendicular
to it. The angle α (also measured the c.c. from north to east) is
defined as the angle between North and x for a given telescope
orientation. Thus, φ− α is the angle between the dispersion and
a line connecting the binary components and ∆y = θ sin(φ−α) is
the displacement of the two spectra of the binary perpendicular
to the dispersion. If φ− α = 0 or ±180◦, then ∆y = 0.
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 over-
whelmed 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. Conse-
quently, we fixed the surface gravity at 4.0, again typical for
main sequence B stars. We allowed the abundance parame-
ter, [m/H]
, and the effective temperature of the B star to be
optimized by the least squares routine, along with the ∆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 apply-
ing 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 prop-
erties 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 Ta-
ble 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.
c© 2007 RAS, MNRAS 000, 1–??
8 D. Massa and N. R. Evans
Table 2. Parameter Values
Parameter Value Parameter Value
∆y1 −0.010 c3 4.13
∆y2 0.279 c4 0.82
∆y3 −0.269 γ 0.9686
[6297] Vs (11.084)
15735 (B − V )s0 (−0.156)
log gp (4.00) (U −B)
0 (−0.597)
log gs [1.60] Vp (7.362)
[m/H]p [0.00] (B − V )
0 (0.494)
[m/H]s -0.20 (U −B)
0 (0.359)
E(B − V ) 0.53 ∆ logL (0.95)
R(V ) 3.11
Values in parenthesis were not involved in the fitting procedure.
Values in square brackets were determined from a fit to the
Cepheid SED derived from the initial fit.
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 RV = 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 red-
dened B star model from the observed SED of the system
and then correcting this difference for the effects of extinc-
tion. 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 as-
signed 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
eff (the effective temperature of the
primary), log gp (the surface gravity of the primary) and
[m/H]
(the abundance of the primary), to vary. We had
to restrict the surface gravity to be larger than 1.6, other-
wise the fitting routine would seek log gp values that were
unrealistically small (we expect a log gp ≃ 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 in-
ferred UV Cepheid SED by comparing it to IUE observa-
tions of the single Cepheid star δ Cep. δ Cep has a period
of 5.4 days, compared to 6.5 days for AW Per, and its mean
unreddened color is 〈(B − V )〉 = 0.57. To obtain the in-
trinsic color of AW Per, we use our derived color excess
for the system and the intrinsic colors of the B star sec-
ondary from Table 2 and the mean magnitude of the system,
〈V 〉 = 7.49 mag, to correct the observed mean color of the
system, 〈(B − V )〉 = 1.06 mag, for both extinction and the
presence of the companion. The result is 〈(B−V )p0〉 = 0.57,
identical to that of δ Cep (recall that the intrinsic color
we derive for AW Per is at Φ ≃ 0.92). Thus, the compar-
ison between these two stars is expected to be quite good.
Figure 7. Fits to the mean centroids at each roll angle for AW
Per. Each mean centroid was fit simultaneously with the corre-
sponding fluxes, optical photometry and interstellar extinction.
A Kurucz model was used to fit the B star component, and the
Cepheid flux was taken to be the difference between the reddened
B star model and the observed flux.
The bottom plot in Figure 9 compares the unreddened IUE
data (points) for δ Cep from several exposures obtained
for 0.9 6 Φ(δCep) 6 1.0 to the unreddened Cepheid STIS
spectrum (solid curve) of AW Per. Several IUE exposures
are required to produce the δ Cep spectrum since the dy-
namic range of IUE was so limited and the range of the
UV SED of δ 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 Φ = 0.92 (the mean for the STIS data) and
δ Cep at Φ = 0.95 (the mean of the IUE data).
Finally, we utilize the ∆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 ∆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
c© 2007 RAS, MNRAS 000, 1–??
AW Per 9
Figure 8. Top: Best fit B star (thin curve) compared to the IUE
(points) and STIS (thick curve) fluxes. The model includes red-
dening. We only show the far-UV region, since the fit is, by defini-
tion, exact for wavelengths longward of 1650Å. Bottom: AW Per
extinction curve determined by the simultaneous fit of the flux
and centroids (solid curve) compared to a standard RV = 3.1
curve (dotted) from Fitzpatrick (1999).
used to weight the fit. The final result of the analysis is a
separation of θ = 13.74 ± 0.26 mas and a position angle
φ = 184.16 ± 1.94 deg, for an accuracy of ∼ 2%.
6.1 Errors in the parameters
In this section, we describe the internal, random, errors af-
fecting 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 deriva-
tives of the model. These errors are listed in the second col-
umn 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 indepen-
dent 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
3 applica-
ble to the error in the mean) are listed in the third column
of Table 3. Notice that the errors in the ∆y(n) determined
from the S.D.s are nearly twice as large. To be conservative,
Figure 9. Top: Inferred dereddened Cepheid SED (points) com-
pared to the best fitting Kurucz model (solid) and the dered-
dened flux of the best fit B star (dashed). Bottom: Comparison
of the unreddened Cepheid flux (solid curve) and an unreddened
IUE spectrum (dots) of δ Cep observations for 0.90 6 Φ 6 0.95.
The δ Cep flux is scaled by the difference between V = 3.54 at
Φ = 0.925 for δ Cep and V = 7.37, the magnitude of the primary
in AW Per at Φ = 0.92 (the mean phase of the STIS observations).
As discussed in the text, the δ Cep spectrum is a combination of
several IUE spectra.
these errors were used as the error shown in Figure 10 and
in determining the errors in θ and φ.
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 ±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 Teff
s, but has little effect on
the ∆y(n), which are the object of our analysis. In fact, the
only significant change in the ∆y(n) result from our inability
to determine whether the STIS or IUE fluxes are correct,
c© 2007 RAS, MNRAS 000, 1–??
10 D. Massa and N. R. Evans
Table 3. Errors
Param. Prog. S.D. |δ log g| |δ
fSTIS
fP+fs
∆y(1) 0.004 0.015 1.4× 10−4 6.5× 10−6 5.0× 10−4
∆y(2) 0.005 0.017 0.0019 1.4× 10−4 0.015
∆y(3) 0.005 0.014 0.0021 1.3× 10−4 0.015
248 105 1205 9.1 37
[m/H]
0.057 0.025 7.5× 10−5 0.0016 0.0064
E(B − V ) 0.001 0.038 0.018 0.0026 0038
RV 0.031 0.12 0.11 0.0090 2.7× 10
γ 0.015 2.4× 10−4 0.019 8.6× 10−4 0.0025
c3 0.14 0.32 0.049 0.029 0.0055
c4 0.019 0.066 0.014 4.3× 10
−3 0.0068
Figure 10. Determination of the angular separation of AW Per.
The observational errors for ∆y were determined from individual
fits to the 3 independent offset observations at each roll angle.
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 de-
termined 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 unred-
dened SED of the single Cepheid, δ 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 δ Cep, so the derived SED is
quite reasonable. Third, the flux in δ Cep is extremely small
for wavelengths shortward of 1650Å, bolstering our assump-
tion 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 nu-
merous, ill-defined physical processes that are not fully in-
corporated into the Atlas 9 models. These include spherical
extension, which can enhance the UV flux from an atmo-
sphere (see Fig. 4 in Hauschildt et al. 1999), chromospheres
(e.g., Sasselov & Lester 1994), the amount of convective en-
ergy 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 ∼ 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 prob-
In spite of these difficulties, it is of interest to exam-
ine the physical parameters determined from the Cepheid
model. To begin, Teff of the best fit model agrees reason-
ably 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 set-
tled on an even lower value if it had been allowed to do so.
It is also interesting that the Cepheid model has a signif-
icantly 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 abun-
dance (Fitzpatrick & Massa 1999).
c© 2007 RAS, MNRAS 000, 1–??
AW Per 11
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 RV = 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 ex-
cess 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 mod-
els to describe normal B star spectra are well documented.
Notice that Teff we derive is considerably hotter than previ-
ously estimated by Evans (1994), and lies somewhat closer
to the ZAMS (see, Fig. 7 in Evans 1994). However, its prob-
able mass, ∼ 5M⊙ (based on its Teff , Andersen, 1991), re-
mains significantly less than the lower limit of ∼ 6.6M⊙
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 wave-
length 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 ∼ 2%, or ± a few ×10−4′′! We also
demonstrated that the results are extremely stable to vari-
ations in the expected systematic effects in the data and its
analysis. We also showed that one possible source of uncer-
tainty 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 pri-
mary. 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.
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616, 912
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, respec-
tively, with x increasing in the direction of increasing wave-
length (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 = y0 can be expressed as
f(λ, y) = N(λ)h(y − y0) (A1)
where λ = λ(x), andN(λ) is the photon flux at λ (we assume
infinite resolution in the wavelength direction).
If the spectra of the primary and secondary components
of the binary are centered at yp, and ys, then their spectra
are separated by ∆y = ys − yp, and the image of the binary
spectrum is given by
f(λ, y) = Np(λ)h(y − yp) +Ns(λ)h(y − yp −∆y). (A2)
If ∆y is small compared to structure in h(y), this equation
can be approximated by
f(λ, y) ≃ Np(λ)h(y − yp)
+Ns(λ)
4h(y − yp) +∆y
dh(y)
y=y−yp
= [Np(λ) +Ns(λ)]×
4h(y − yp) +
Ns(λ)
Np(λ) +Ns(λ)
dh(y)
y=y−yp
≃ [Np(λ) +Ns(λ)]×
∆yNs(λ)
Np(λ) +Ns(λ)
Therefore, the wavelength dependence of the centroid of the
spectrum will vary as
y(λ) = yp +
∆yNs(λ)
Np(λ) +Ns(λ)
= yp +
1 +R(λ)
where R(λ) = Np(λ)/Ns(λ) is the flux ratio of the binary
components.
Now, the separation ∆y depends on both the separation
of the binary, θ, and the orientation of the system relative
to the dispersion direction. The position angle on the sky
of the binary, φ, is defined as the angle measured the c.c.
from north to east, with the primary at the origin. The angle
α(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 nth telescope
orientation. In this case, φ − α(n) is the angle between the
dispersion and a line connecting the binary components and
∆y(n) = θ sin(φ−α(n)) is the displacement of the two spectra
of the binary (note that when φ − α = 0, ±180◦, ∆y = 0).
Therefore, the observation obtained with the telescope in
the nth orientation can be expressed as
p + θ sin(φ− α(n))[1 +R(λ)]−1 (A5)
where y
p is the wavelength independent displacement of
the nth exposure in y.
To extract both θ and φ from the observed centroids, at
least two observations at different α’s are required. There-
fore, 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(λ)]−1 gives ∆y(n) for that ob-
servation. The y
p are constant terms related to the abso-
lute 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 ∆y(n) are determined for each orientation,
these are plotted against the known quantities, α(n). Since
= θ sin(φ− α(n)) (A6)
fitting a sine function to the ∆y(n) as a function of the α(n)
determines φ and θ, the observables of an astrometric binary
at the epoch of the observations.
c© 2007 RAS, MNRAS 000, 1–??
	INTRODUCTION
	THE APPROACH(Cross-Dispersion Imaging)
	Basic Principles
	Exposure Times and Random Errors
	THE OBSERVATIONS
	DATA REDUCTION
	Centroids
	Fluxes
	ANALYSIS
	Overview
	Model Components
	Details of the Fitting Procedures
	Determining the Separations
	Weights
	RESULTS
	Errors in the parameters
	DISCUSSION
	SUMMARY
	MATHEMATICAL DETAILS
