Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://star.arm.ac.uk/~ambn/preprint.ps Äàòà èçìåíåíèÿ: Fri May 31 18:25:44 1996 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 22:03:28 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: binary star

A&A manuscript no.
(will be inserted by hand later)
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08(02.1,02.4,11.1,18.1),03.20.7
ASTRONOMY
AND
ASTROPHYSICS
7.3.1995
Cross­correlation radial velocity measurements of
chromospherically active binaries
A. G. Gunn 1 , J. C. Hall 2 , G. W. Lockwood 2 , and J. G. Doyle 1
1 Armagh Observatory, College Hill, Armagh, BT61 9DG, N. Ireland
2 Lowell Observatory, 1400 Mars Hill Road, Flagstaff, Arizona 86001, USA
February 1995
Main Journal
Running title: Radial velocity measurements of active binaries
Abstract. We present observational radial velocity curves
for 12 chromospherically active binary systems and individ­
ual measurements for a further 5 systems. These binaries are
LX Per, V471 Tau, EI Eri, OU Gem, GK Hya, TY Pyx, Z Her,
MM Her, V772 Her, ER Vul, BD­004234, MY Cyg, AR Lac, KZ
And, RT And, SZ Psc and EZ Peg. Six of our target binaries
do not agree with published ephemerides or orbital parame­
ters. We also present a rigourous derivation of the resolution
limit for binary star cross­correlation radial velocities. Using
synthetic spectral data we investigate the errors induced by
rotational broadening, signal­to­noise ratio, spectral­type mis­
match and luminosity ratio.
Key words: stars:binaries:close­stars:binaries:spectroscopic­
techniques: radial velocities.
1. Introduction
Chromospherically active binaries have been the subject of in­
tense research over recent decades in many wave­bands. Such
objects display a wide variety of phenomena; flare activity,
cool surface star­spots, extended high­temperature coronae
and large magnetic structures. Much work has been done in
determining the orbital parameters of such systems with light­
curve analysis and spectroscopic determinations of velocity
variations (see Popper & Jeong 1994 and references therein).
For many systems these parameters are poorly known or are
still based on old photographic results. We have discovered
some discrepancies between predicted velocities and those de­
duced from high­resolution data and have concluded that some
apparently well­studied systems need a radical re­assessment
of their properties. In addition to the problem of poor solu­
tions for some of these systems, actual variations in the appar­
ent orbital periods of RS CVn and related binaries are well­
documented although these are usually very small (Kreiner
1971; Guarnieri et al. 1975; Hall & Kreiner 1980; Hall et al.
1980; Kalimeris et al. 1995).
Send offprint requests to: A. G. Gunn
Cross­correlation techniques are widespread in spec­
troscopy, particularly for radial velocity determinations (eg.
Philip & Latham 1985). Simkin (1974) discussed the tech­
nique in relation to Fourier transforms of photographic spectra.
Tonry & Davis (1979) derived a relationship between the ve­
locity dispersions in target and template galaxy spectra and
the width of the peak feature in the cross­correlation function
(hereafter referred to as the CCF). Simkin (1974) described
the use of the cross­variance function, Weiss, Jenkner & Wood
(1978) explored the squared difference function while Furen­
lid & Furenlid (1990) presented an analysis of the use of delta
functions as a representation of the template spectrum which
unfortunately requires good knowledge of stellar line positions.
Although the CCF method has in recent years become refined
and widespread due to more advanced electronics and more effi­
cient Fourier algorithms, no analytical treatment of the method
has been given for the case of binary stellar data. We present
in this paper our preliminary analysis of the CCF technique
and derive the relationship between the rotation rates in the
binary components and the template spectrum and the width
of the features in the CCF. This is used to derive an ad hoc
resolution criterion which is helpful in showing at which phases
of a binary orbit a particular template spectrum will be able to
resolve the components of a given binary. We investigate this
resolution limit and other effects on the errors in CCF radial
velocities by examining synthetic data. We plan to supplement
this analysis in the future with an analysis of a large sample
of well­studied binaries and template spectra.
In this paper we present radial velocity measurements
throughout consecutive orbital cycles for a small sample of
chromospherically active binary systems. Our motivation for
this work was to (a) determine the suitability of our medium­
resolution spectrograph for the determination of stellar radial
velocities, (b) assess both quantitatively and analytically the
basis of cross­correlation techniques for binary orbit radial ve­
locity determinations and (c) confirm or refute (within our er­
ror limits) the accepted ephemerides of a small sample of chro­
mospherically active binary stars for which we require good
knowledge of radial velocities. Below we discuss in detail our
observational data, the analysis methods used, the results ob­
tained and give a rigourous assessment of the cross­correlation
procedure. We wish to emphasize the preliminary nature of
this report for both our observational data and analytical as­
sessment.

2
Table 1. The properties of our program binaries. For each system we list the CABS catalogue number (in parentheses: Strass­
meier et al. 1993), the HD number, the visual magnitude mv , the spectral types of the hot and cool components, the orbital
period Porb , the Julian date of primary conjunction Tconj , the rotational velocities for the hot and cool components (Vh sin i
and Vc sin i respectively), the systemic velocity Vo , the orbital velocity semi­amplitudes for the hot and cool components of the
system (Kh and Kc respectively), the orbital eccentricity e and the longitude of periastron
passage\Omega\Gamma Parameters not taken
from the CABS catalogue are indicated below.
Name HD mv Spectral Types Porb JD Tconj Vh sin i V c sin i Vo Kh K c
e\Omega ffi
hot/cool (days) ­2400000 (km s \Gamma1 )
LX Per (25) ­ 8.14 G0IV/K0IV 8.0382 27033.120 9 19 +28.0 75.0 70.3 ­ ­
V471 Tau (32) ­ 9.71 WD/K2V 0.5212 41913.024 ­ ­ +40 ­ 147 ­ ­
EI Eri (35) 26337 6.96 G5IV/­ 1.9472 46091.539 50 ­ +17.6 27.4 ­ ­ ­
OU Gem (62) 45088 6.79 K3V/K5V 6.9919 40202.663 5.6 5.6 ­8.4 56.5 66.9 0.150 77.6
GK Hya (77) ­ 9.32 F8/G8IV 3.5870 14968.860 17 49 +32 100 91 ­ ­
TY Pyx (81) 77137 6.84 G5IV/G5IV 3.1986 43548.666 23 23 +63.2 96.2 97.5 ­ ­
Z Her (147) 163930 7.23 F4V­IV/K0IV 3.9928 13086.348 17 34 ­45.0 85.5 105 ­ ­
MM Her (148) 341465 9.51 G2/K0IV 7.9603 44500.667 10 18 ­50.8 70.6 72.3 ­ ­
V772 Her (149) 165590 7.02 (G0V/M1V)/G5V 0.8795 47372.568 65 11 ­22.8 94.7 ­ 0.045 104
ER Vul (179) 200391 7.27 G0V/G5V 0.6981 40182.259 85 85 ­24.6 139.5 145.8 ­ ­
BD­004234 (184) ­ 9.87 K3Ve/K7Ve 3.7569 44064.847 5 5 ­109.6 43.2 53.7 ­ ­
MY Cyg ­ 8.40 ­ 4.0052 41561.598 ­ ­ ­ ­ ­ ­ ­
AR Lac (191) 210334 6.09 G2IV/K0IV 1.9832 44977.022 46 81 ­33.7 116.1 115.6 ­ ­
KZ And (200) 218738 7.98 dK2/dK2 3.0329 42371.641 12.3 11.6 ­6.9 67.6 71.2 0.034 339
RT And (201) ­ 8.95 F8V/K0V 0.6289 47803.509 a 109 ­ +5.0 131.4 168.4 b 0.089 282.2
SZ Psc (202) 219113 7.20 F8IV/K1IV 3.9659 42308.946 9 70 +12 104 82 ­ ­
EZ Peg (203) ­ 9.53 G5V­IV/K0IV 11.659 45736.665 9 7 ­27.2 24.5 24.3 ­ ­
Notes:
(a) ephemeris from Popper & Jeong (1994), (b) velocities from Wang & Lu (1993).
2. Observations and data reduction
The spectroscopic data presented in this paper were obtained
during a 13­night observing run in October/November 1994
carried out with the Solar­Stellar Spectrograph (SSS) located
at the John Hall 1.1­meter telescope at the Lowell Observa­
tory's dark­sky site near Flagstaff, Arizona. The SSS is a dual
spectrograph consisting of an echelle covering wavelengths from
5100 š A to 9000 š A and a Littrow instrument covering the Ca II
H and K lines. The resolution of the SSS is 12000, and exposure
times for the stars in our program ranged from 15 minutes (for
6th magnitude stars) to 60 minutes (for 9th or 10th magnitude
stars). After dispersion the data are recorded on two TEK 512
\Theta 512 CCDs. Data are then stored on disk as a grafted image
containing 20 spectral orders.
In Table 1 we list the important parameters of our pro­
gram binaries. We scheduled approximately one observation
per night of the primary targets on our list and observed the
other targets when possible. Observations were timed where
possible to guarantee at least two spectra of each quadrature
and one spectrum of each conjunction, given clear weather
throughout. Over our 13 nights we obtained 11 whole nights
and 2 half nights of data, so our phase coverage at the end of
the run was complete for most of our targets. We also observed
a set of radial velocity standard stars each night; these were fl
Aql (HD 186791), ' Psc (HD 222368) and ff Tau (HD 29139).
The parameters of our RV standard stars are given in Table 2.
The data were reduced and analyzed on­line using an idl
(Interactive Data Language) echelle data reduction package
called reduce 2.0 developed in­house at Lowell and described
by Hall et al. (1994). reduce allows the user to rapidly ob­
Table 2. Details of the radial velocity standard stars used in
the cross­correlation procedure. For each system we list the
HD number, the spectral type, the visual magnitude mv , the
radial velocity (RV) and the rotational velocity (v sin i).
Name HD Spectral mv RV v sin i
Type (km s \Gamma1 ) (km s \Gamma1 )
fl Aql 186791 K3 II 2.7 ­ 2.1 \Sigma 0.2 ! 15
' Psc 222368 F7 V 4.1 + 5.3 \Sigma 0.2 ! 10
ff Tau 29139 K1 III 0.9 +54.1 \Sigma 0.1 ! 15
tain wavelength­calibrated spectra normalised to unity by the
continuum, which can then be analysed using the spectrum­
analysis tool also included with the package. The data were first
de­biased and flat­field corrected using suitable frames taken
throughout each night's observations. The SSS wavelength cal­
ibrations were obtained with a Thorium­Argon (Th­Ar) hollow
cathode, which with a 55­second exposure produces about 150
well­exposed lines over the echelle frame and about 10 across
the Ca II H and K order. Using the calibration method de­
scribed in detail by Hall et al. (1994) we obtained wavelength
solutions with RMS residuals of about 0.02 š A over the echelle
frame and less than 0.01 š A at Ca II H and K. For radial veloc­
ity measurement purposes these residuals translate to about
0.9 kms \Gamma1 at Hff.

3
Lag (pixels)
Correlation
­40 ­20 0 20 40
.883
.884
.885
.886
.887
.888
b)
Lag (pixels)
­40 ­20 0 20 40
.888
.89
.892
.894
.896
d)
Normalized
Intensity
5260 5280 5300 5320 5340
.4
.6
.8
1
1.2 TY Pyx 20 Oct 1994 12:22 UT
Phase = 0.265
a)
5260 5280 5300 5320 5340 .4
.6
.8
1
1.2
TY Pyx 01 Nov 1994 12:56 UT
Phase = 0.023
c)
Fig. 1. Example of the cross­correlation procedure for the determination of component radial velocities for the TY Pyx system, 1 (a) shows
the target (full­line) and RV standard HD 222368 spectrum (dotted­line) for SSS order number 17 for a spectrum observed at 12:22 UT
(mid­exposure) on 20 October 1994 (phase 0.265). The resulting CCF (panel (b)) shows two clearly defined peaks which are fitted by a
double gaussian function (dotted). The function is fitted between two user supplied base values and provides the best positions for the two
component lags measured in pixels. Panel (c) shows a similar plot for the same spectral order for the TY Pyx spectrum observed at 12:56
UT (mid­exposure) on 1 November 1994 (phase 0.023). In this case the binary component spectra are strongly blended resulting in a single
CCF peak (panel (d)). A single gaussian function (dotted) has been fitted to the peak of the blended function.
Lag (pixels)
Correlation
­40 ­20 0 20 40
.87
.872
.874
b)
Lag (pixels)
­40 ­20 0 20 40
.81
.812
.814
.816
d)
Normalized
Intensity
5260 5280 5300 5320 5340
0
.5
1
1.5
RT And 30 Oct 1994 2:56 UT
Phase = 0.858
a)
5260 5280 5300 5320 5340
0
.5
1
1.5
RT And 21 Oct 1994 2:40 UT
Phase = 0.530
c)
Fig. 2. Example of the cross­correlation procedure for the 9th magnitude target RT And. In this case averaged CCFs have been used to
determine the component radial velocities. Panel (a) shows the data for SSS order 17 for the RT And spectrum (full­line) observed at 2:56
UT (mid­exposure) on 30 October 1994 (phase 0.858) and the RV standard HD 186791 (dotted­line). Panel (b) shows the double­peaked
CCF formed by averaging the CCFs for five separate orders of the RT And spectrum and the fitted double­gaussian function (dotted).
Panel (c) shows a similar plot for the same spectral order for an RT And spectrum observed at 2:40 UT (mid­exposure) on 21 October
1994 (phase 0.530). Once again the CCF has been averaged over five spectral orders (panel d)) and the resulting blended CCF fitted with
a single gaussian function (dotted). The broadness of the CCF in comparison to those in Figure 1 is due to the higher rotational velocities
of the RT And components.

4
All subsequent analysis of the data was done with the re­
duce 2.0 routine spectool, which among other things can
perform the cross­correlations and function fitting necessary
to extract radial velocities from the data. We used spectool
both to obtain the radial velocities from the spectroscopic data,
as well as to perform a similar analysis of the synthetic data
used to study the CCF (see x5 and x6). This ensured consis­
tency in the way all data used in this work were treated by the
software.
3. Radial velocity determinations
Our method of determining radial velocity information from
our binary spectra is to cross­correlate the observed target
spectrum with the same spectral order of a radial velocity
standard. Simkin (1974) gives a good description of the tech­
nique of cross­correlation. Many orders of the binary spectrum
are not suitable for the CCF technique since they have low
signal­to­noise, contain activity­sensitive absorption/emission
features or are heavily affected by telluric lines. We first iden­
tified those orders which were least affected by such contami­
nation and used these as our subset of orders. Popper & Jeong
(1994) have demonstrated that the use of a weighting scheme
based on the apparent quality of different spectral orders does
not give results more consistent than the mean over selected
orders. Ideally the CCF should be evaluated over a log – scale
but over a short order of 512 pixels at our resolution the errors
induced by analysing in pixel space are negligible. The CCF of
a single order of the binary spectrum with the same order of
the radial velocity standard was performed by Fourier trans­
forming each spectrum, forming the conjugate of the standard
star FFT, multiplying these together and then transforming
back into the spectral domain. The resulting CCF shows the
degree of correlation for increasing and decreasing lags and can
be more easily visualised as the common area beneath the two
superimposed spectra as one is successively moved by a positive
or negative lag from the zero­shift position. The positions of
peak values in this function occur at velocity differences where
there is high correlation between the two spectra.
With our binary spectra it is neccesary to differentiate two
scenarios for velocity resolution; firstly that the two compo­
nent velocities are easily distinguished in the peaks of the
CCF or secondly that only one peak is visible. The first cor­
responds to the complex set of spectral absorption features
being widely separated in velocity (for example at quadrature
phases), whilst the second means the two components have
very similar velocities (for example during conjunction) and
therefore have highly blended absorption profiles. We thus dis­
tinguish our velocity measurements as blended or resolved. In
x5 we present an analytical derivation of the position of this
resolution limit.
To measure the positions of the resolved peaks in the CCF
we fit a double­gaussian function across a user­specified section
of the CCF. This procedure also requires first­guess positions
for the two peaks. For blended velocities we fit a single­gaussian
using a similar iterative technique. We have found a double­
gaussian fit for resolved features in the CCF gives more consis­
tent results than two single­gaussian functions. An example of
the procedure is shown in Figure 1 which displays the target
and standard spectra for TY Pyx observed on two separate
occasions which produce resolved and blended CCFs.
The highest priority in any radial velocity study, partic­
ularly when developing an analysis protocol for a particular
instrument, is a proper assessment of the errors involved in
the measurements. We have addressed this in several ways.
Firstly we perform cross­correlation separately for several or­
ders for each program star, and perform these with all three
of our radial velocity standards. We also check the differences
in wavelength calibration solutions for the program and tar­
get spectra to see if any calibration drift is inducing spurious
velocities. Small differences in the calibrations are removed by
supplying the appropriate corrections. So as not to place undue
importance to our most error­free measurements we first form
the standard error from each set of velocities for a particular
standard and then form an RMS from the resulting maximum
error and the RMS of the set of velocities from each standard.
The resulting velocities are then corrected for Earth motion
and the motion of the template.
For some fainter stars or spectra with particularly bad
signal­to­noise we have found it necessary to adopt a slightly
different procedure. Many orders result in CCFs with a some­
what ill­defined peak or double­peak. Since the velocity­pixel
translation in our data set is linear we can add up CCFs for
many orders to increase the signal­to­noise and emphasize the
peak positions in the function. We thus average the CCFs over
many orders and proceed to fit a double­ or single­gaussian
function (using the same spectool routines) to the average
peaks. This method has proved to be helpful in determining
the positions of binary components whose profiles are not al­
ways clearly visible from a given spectral order. However, a de­
termination of a likely error in this case is more difficult since
we are reducing the number of component measurements. In
this case we assign an error of 2oe where oe is the RMS of the ve­
locity sample. An example of data treated in this way is given
in Figure 2 for observations of the RT And system.
4. Results and discussion
We present the results of our measurements in Table 3. This ta­
ble shows for each system the derived velocities of the hot and
cool components for each observation. The epoch and period
of the binary derived using the published ephemerides (Table
1) are also shown. In Figures 3 to 14 we plot the velocities
against the phase of observation and show the respective RV
curves predicted by the ephemerides. The figures show, where
known, the positions of the systemic velocity (solid lines), semi­
amplitudes of hot and cool components (dotted lines) and the
positions of our resolution limits for each system (dashed lines)
derived with Equation 19 (see x5 and x6). Figures are presented
only for those systems for which we have four or more measure­
ments of radial velocities.
The primary aim of this program was not to improve upon
the accepted ephemerides but to assess the applicability of the
SSS instrument for radial velocity determinations. We can how­
ever make some qualitative statements concerning the orbital
parameters of these systems using our medium­resolution ob­
servations. The observations were obtained on a single observ­
ing run so our epoch coverage is poor, however we intend to
improve our analysis with further observations.

5
Table 3. Results of the radial velocity measurements for each system. The table shows for each star the epoch and phase of
each observation and the derived radial velocities of the cool and hot components (Vc and Vh respectively). An asterix in the
Vh column indicates that the Vc velocity is from a blended profile. A dash in any column indicates that the component is not
visible in the data.
Star Epoch Phase Vc (km s \Gamma1 ) V h (km s \Gamma1 ) Star Epoch Phase Vc (km s \Gamma1 ) V h (km s \Gamma1 )
LX Per 2813 0.288 100.7 \Sigma 7.3 ­40.1 \Sigma 9.6 ER Vul 13553 0.297 113.5 \Sigma 11.4 ­150.9 \Sigma 18.0
2814 0.281 96.6 \Sigma 2.5 ­44.3 \Sigma 7.6 13554 0.633 ­150.7 \Sigma 12.5 81.3 \Sigma 11.0
2814 0.654 ­28.1 \Sigma 2.5 87.8 \Sigma 1.8 13555 0.991 ­26.6 \Sigma 2.5 ?
V471 Tau 14833 0.265 168.5 \Sigma 12.4 ­ 13557 0.481 ­21.7 \Sigma 7.1 ?
14835 0.207 186.0 \Sigma 5.0 ­ 13558 0.962 ­13.4 \Sigma 6.0 ?
14837 0.116 125.9 \Sigma 9.6 ­ 13561 0.805 ­173.6 \Sigma 7.0 82.3 \Sigma 8.0
14838 0.992 37.1 \Sigma 13.8 ­ 13563 0.239 114.7 \Sigma 13.4 ­163.6 \Sigma 14.6
14840 0.913 ­27.9 \Sigma 17.4 ­ 13564 0.647 ­160.0 \Sigma 16.3 73.1 \Sigma 16.3
14846 0.755 ­120.9 \Sigma 11.0 ­ 13570 0.389 66.2 \Sigma 4.8 ­118.5 \Sigma 5.7
14854 0.342 152.4 \Sigma 1.3 ­ 13571 0.760 ­167.5 \Sigma 12.4 103.2 \Sigma 8.3
14856 0.343 153.3 \Sigma 13.6 ­ 13573 0.235 113.7 \Sigma 9.8 ­162.9 \Sigma 9.2
14858 0.262 171.1 \Sigma 11.4 ­ BD­004234 1485 0.490 ­57.6 \Sigma 2.6 ­151.1 \Sigma 2.0
14866 0.115 136.6 \Sigma 15.4 ­ 1485 0.761 ­118.0 \Sigma 8.8 ?
EI Eri 1824 0.319 ­ 28.5 \Sigma 6.9 1486 0.036 ­169.5 \Sigma 9.3 ­57.1 \Sigma 4.5
1825 0.354 ­ 45.5 \Sigma 2.8 1488 0.427 ­53.4 \Sigma 13.5 ­145.1 \Sigma 4.0
1826 0.366 ­ 30.9 \Sigma 3.6 MY Cyg 2018 0.146 47.0 \Sigma 2.4 ­131.5 \Sigma 5.7
1828 0.406 ­ 19.0 \Sigma 10.1 2021 0.145 39.0 \Sigma 8.0 ­141.0 \Sigma 8.0
1830 0.463 ­ 14.4 \Sigma 2.3 AR Lac 2353 0.198 57.4 \Sigma 9.5 ­157.0 \Sigma 13.4
OU Gem 1350 0.330 52.2 \Sigma 9.1 ­61.0 \Sigma 4.9 2353 0.692 ­132.9 \Sigma 2.4 84.1 \Sigma 3.0
1350 0.616 ­9.2 \Sigma 3.3 ? 2354 0.155 55.9 \Sigma 4.0 ­140.9 \Sigma 6.0
1350 0.904 ­46.2 \Sigma 9.4 46.3 \Sigma 8.8 2354 0.669 ­136.9 \Sigma 5.0 70.6 \Sigma 7.4
1352 0.329 48.3 \Sigma 8.2 ­63.5 \Sigma 2.1 2355 0.190 66.4 \Sigma 8.9 ­149.4 \Sigma 9.8
GK Hya 9667 0.076 61.3 \Sigma 10.7 ? 2356 0.174 56.2 \Sigma 7.2 ­151.6 \Sigma 5.5
9667 0.632 ­22.6 \Sigma 9.3 121.4 \Sigma 14.8 2356 0.680 ­140.3 \Sigma 12.2 73.5 \Sigma 8.6
9667 0.917 ­15.0 \Sigma 6.0 85.9 \Sigma 21.0 2357 0.172 57.0 \Sigma 8.6 ­149.6 \Sigma 11.0
9669 0.031 36.0 \Sigma 9.2 ? 2358 0.688 ­132.9 \Sigma 8.0 75.5 \Sigma 7.3
9669 0.871 ­34.4 \Sigma 8.1 101.4 \Sigma 13.0 2359 0.191 63.5 \Sigma 5.6 ­148.8 \Sigma 8.5
9670 0.146 97.4 \Sigma 10.0 ­49.9 \Sigma 15.6 2359 0.691 ­142.1 \Sigma 6.0 76.5 \Sigma 4.9
9670 0.424 69.5 \Sigma 10.8 ­11.6 \Sigma 9.3 2360 0.194 64.8 \Sigma 6.3 ­151.6 \Sigma 6.9
9670 0.708 ­55.3 \Sigma 12.6 132.0 \Sigma 6.8 KZ And 2398 0.423 44.8 \Sigma 4.0 ­54.6 \Sigma 4.5
TY Pyx 1905 0.640 ­10.2 \Sigma 3.1 139.2 \Sigma 7.0 2401 0.385 53.1 \Sigma 4.9 ­61.9 \Sigma 4.2
1905 0.955 48.8 \Sigma 15.4 102.6 \Sigma 17.2 2402 0.389 51.2 \Sigma 3.8 ­61.2 \Sigma 2.6
1906 0.265 159.0 \Sigma 4.2 ­33.9 \Sigma 3.5 RT And 13517 0.883 64.5 \Sigma 14.0 ?
1906 0.581 19.2 \Sigma 6.1 113.4 \Sigma 7.1 13519 0.514 16.2 \Sigma 2.2 ?
1906 0.895 4.1 \Sigma 4.5 121.3 \Sigma 5.2 13522 0.530 19.5 \Sigma 3.0 ?
1907 0.830 ­19.5 \Sigma 3.8 150.0 \Sigma 6.1 13524 0.171 146.3 \Sigma 11.9 ­118.3 \Sigma 7.4
1908 0.144 129.3 \Sigma 11.8 ­23.6 \Sigma 10.4 13529 0.109 127.8 \Sigma 2.0 ­93.7 \Sigma 11.6
1909 0.085 112.4 \Sigma 4.4 16.4 \Sigma 5.5 13535 0.278 158.0 \Sigma 5.1 ­125.9 \Sigma 6.2
1909 0.395 121.4 \Sigma 2.9 3.8 \Sigma 6.5 13536 0.858 ­136.6 \Sigma 4.4 98.6 \Sigma 6.6
1909 0.707 ­29.6 \Sigma 3.4 156.5 \Sigma 4.8 13540 0.058 98.5 \Sigma 22.2 ­60.4 \Sigma 3.8
1910 0.023 86.1 \Sigma 3.0 ? SZ Psc 1849 0.751 ­39.2 \Sigma 2.3 97.6 \Sigma 4.2
Z Her 9156 0.293 51.1 \Sigma 4.8 ­127.9 \Sigma 1.8 1852 0.775 ­43.6 \Sigma 6.5 75.0 \Sigma 6.7
9156 0.795 ­135.4 \Sigma 4.1 49.4 \Sigma 4.5 EZ Peg 335 0.093 ­29.7 \Sigma 2.0 ?
9158 0.794 ­135.5 \Sigma 6.0 47.5 \Sigma 6.2 335 0.177 ­32.2 \Sigma 4.4 ?
9159 0.296 52.4 \Sigma 5.1 ­128.2 \Sigma 3.5 335 0.264 ­56.5 \Sigma 6.4 ?
MM Her 647 0.766 ­116.2 \Sigma 11.9 18.3 \Sigma 8.7 335 0.349 ­29.5 \Sigma 9.8 ?
V772 Her 2582 0.216 ­31.1 \Sigma 8.1 ­120.5 \Sigma 11.9 335 0.438 ­33.1 \Sigma 5.0 ?
2583 0.287 ­28.0 \Sigma 2.2 ­113.4 \Sigma 7.2 335 0.694 ­21.8 \Sigma 4.4 ?
2584 0.431 ­22.1 \Sigma 1.3 ­83.7 \Sigma 2.2 335 0.778 ­22.4 \Sigma 8.1 ?
2585 0.559 ­19.7 \Sigma 3.8 ? 336 0.035 ­31.0 \Sigma 5.3 ?
2586 0.710 ­9.1 \Sigma 5.8 70.1 \Sigma 6.8 336 0.117 ­30.6 \Sigma 2.9 ?
2594 0.651 ­16.2 \Sigma 2.6 53.8 \Sigma 5.0 336 0.207 ­38.1 \Sigma 4.4 ?
2595 0.779 ­13.3 \Sigma 5.3 82.0 \Sigma 14.2 336 0.297 ­48.0 \Sigma 5.0 ?
2596 0.923 ­17.9 \Sigma 3.6 42.7 \Sigma 6.5
2598 0.066 ­24.1 \Sigma 3.3 ?

6
Phase
Radial
Velocity
(km/s)
V471 Tau SSS Radial Velocity Curve
0 .2 .4 .6 .8 1
­100
0
100
200 K2 V cool
Fig. 3. Radial velocities derived from the SSS data for the V471
Tau system.
Phase
Radial
Velocity
(km/s)
EI Eri SSS Radial Velocity Curve
0 .2 .4 .6 .8 1
­20
0
20
40
60
G5 IV hot
Fig. 4. Radial velocities derived from the SSS data for the EI Eri
system.
V471 Tau is a well­studied active binary system consisting
of a K2V secondary and a white­dwarf primary which is not
detectable in our spectra. The light curve is highly variable,
the eclipse being the only constant feature, and displays the
typical wave­distortion characteristic of the RS CVn class of
binaries. The orbital parameters given by Young (1976) give
good agreement with our observations (Figure 3) and there is
no indication of any phase­shift or period change in this system.
EI Eri is a single­lined rapidly rotating RS CVn binary
with a period of 1.9472 days. All the orbital parameters are
from Strassmeier (1990). Our data for the hot G5 IV compo­
nent are extremely confused. Without information about the
secondary component it is not possible to compute a resolu­
tion limit (see x5) for this system but in any case this cannot
account for the large dispersion of our values around phase 0.4
(Figure 4) or the large errors associated with each value. Fur­
ther observations are necessary to account for our failure to
record reasonable velocities for EI Eri but we believe the very
high rotational velocity of the hot component has caused these
large discrepancies.
Phase
Radial
Velocity
(km/s)
OU Gem SSS Radial Velocity Curve
0 .2 .4 .6 .8 1
­100
­50
0
50
K3 V hot K5 V cool blends
Fig. 5. Radial velocities derived from the SSS data for the OU Gem
system.
Phase
Radial
Velocity
(km/s)
GK Hya SSS Radial Velocity Curve
0 .2 .4 .6 .8 1
­100
­50
0
50
100
150 F8 hot G8 IV cool blends
Fig. 6. Radial velocities derived from the SSS data for the GK Hya
system.
OU Gem is an interesting binary consisting of a K3V pri­
mary and a K5V secondary. The ephemeris and period for
this system come from Griffin & Emerson (1975) and Bopp
et al. (1981) respectively. Velocity information can be found in
Tomkin (1980). We only managed to observe this system four
times (Figure 5) with one blended velocity. Our results seem
to agree within the errors quite well with published results.
GK Hya is an RS CVn binary consisting of an F8 primary
and a G8 IV secondary which undergo total eclipses. Our ob­
servations of this system (shown in Figure 6) have shown ex­
cellent agreement (within our errors) with published orbital
parameters (Popper 1990).
TY Pyx is an RS CVn system with almost unity mass ratio,
luminosity ratio and identical spectral types (G5). Photometry
has shown the presence of light­curve distortions characteris­
tic of the RS CVn systems but these are relatively small. Our
observations agree remarkably well with the ephemeris of An­
dersen et al. (1981) and the velocities of Andersen & Popper
(1975). This demonstrates, we believe, the efficiency of our
medium resolution measurements since TY Pyx has rapidly

7
Phase
Radial
Velocity
(km/s)
TY Pyx SSS Radial Velocity Curve
0 .2 .4 .6 .8 1
­50
0
50
100
150
200
G5 IV hot G5 IV cool blends
Fig. 7. Radial velocities derived from the SSS data for the TY Pyx
system.
Phase
Radial
Velocity
(km/s)
Z Her SSS Radial Velocity Curve
0 .2 .4 .6 .8 1
­100
0
100
F4 V­IV hot K0 IV cool blends
Fig. 8. Radial velocities derived from the SSS data for the Z Her
system.
rotating components and does not have a large maximum ve­
locity separation. The derived velocity curve for TY Pyx is
shown in Figure 7.
Z Her is a partially eclipsing binary with F4 and K0 com­
ponents. Popper (1988) gave velocity information on the sys­
tem and the ephemeris used is usually that of Hall & Kreiner
(1980). The orbital period is so close to four days that it is
not possible to observe at widely separated phases with a sin­
gle observing run (Figure 8). Our limited observations however
seem to agree extremely well with published results.
V772 Her is a visual binary which is also a double­lined
triple system containing a close eclipsing pair. Batten et al.
(1979) assigned spectral classifications of G0 V for the hotter
star and G5 V for the tertiary and measured the rotational,
systemic and semi­amplitude velocities. More recent observa­
tions by Strassmeier & Fekel (1990) give the rotational veloc­
ities quoted in the CABS catalogue. The spectral type of the
secondary is not known and in any case is not visible in our
medium­resolution spectra. The period of the visual pair is
somewhat over 20 years (Aitken 1923). We have detected both
Phase
Radial
Velocity
(km/s)
V772 Her SSS Radial Velocity Curve
0 .2 .4 .6 .8 1
­150
­100
­50
0
50
100 G0 V hot cool blends
Fig. 9. Radial velocities derived from the SSS data for the V772
Her system.
Phase
Radial
Velocity
(km/s)
ER Vul SSS Radial Velocity Curve
0 .2 .4 .6 .8 1
­200
­100
0
100
G0 V hot G5 V cool blends
Fig. 10. Radial velocities derived from the SSS data for the ER Vul
system.
the primary and tertiary in our observations. The primary ve­
locities agree quite well with published values (ephemeris of
Reglero et al. 1991) although there is some evidence of a phase
shift (Figure 9). The tertiary component which should have
an approximately constant velocity over our two weeks of ob­
servations shows a small periodic variation in phase with the
primary. This is due to the skewing of the gaussian fitting pro­
cedure by the CCF peak from the primary since the tertiary
component has a low amplitude.
ER Vul (Figure 10) is a double­lined spectroscopic binary
of the short­period RS CVn type. A spectroscopic orbit was de­
termined by McLean (1982) using cross­correlation velocities.
He also found evidence for the presence of extended material
around the system. The components of ER Vul (G0 V and G5
V) are nearly equal in mass and luminosity and undergo shal­
low eclipses. The ephemeris and velocity components of the
system are usually taken from Hill et al. (1990). These predic­
tions agree extremely well with our observations which show
no evidence of phase­shifts or period changes.
BD­004234 has proved a very interesting system (Figure

8
Phase
Radial
Velocity
(km/s)
BD­004234 SSS Radial Velocity Curve
0 .2 .4 .6 .8 1
­150
­100
­50
K3 Ve hot K7 Ve cool blends
Fig. 11. Radial velocities derived from the SSS data for the
BD­004234 system.
Phase
Radial
Velocity
(km/s)
AR Lac SSS Radial Velocity Curve
0 .2 .4 .6 .8 1
­100
0
100 G2 IV hot K0 IV cool
Fig. 12. Radial velocities derived from the SSS data for the AR Lac
system.
11). Our results appear to suggest a phase shift of approxi­
mately 0.25 from that predicted from the ephemeris of Peter­
son et al. (1980). We checked the numbers from CABS with
the original data and found that this shift is real and not due
to differences in the phase zero­point. BD­004234 is peculiar
in that it has a very high systemic velocity (­109.6 km s \Gamma1 )
with Ca II H & K lines and Hff in emission. Spite et al. (1984)
suggested on the basis of their spectroscopic observations that
the orbital period had changed from that given by Peterson
et al. (1980) but only by about 3 minutes. Our observations
strongly suggest that the parameters of BD­004234 have been
incorrectly derived or undergone a dramatic change.
AR Lac is an extremely well­studied short­period RS CVn
system. Small period variations for this system were reported
by Hall et al. (1976), the ephemeris is from Hall & Kreiner
(1980) and velocity information from Sanford (1951). Because
the period of this system is almost 2 days it is almost impossible
to obtain reasonable phase coverage at one site during one
observing session. Our velocities therefore cluster around two
phase positions (Figure 12) but seem to give good agreement
Phase
Radial
Velocity
(km/s)
RT And SSS Radial Velocity Curve
0 .2 .4 .6 .8 1
­200
­100
0
100
200
300
F8 V hot K0 V cool blends
Fig. 13. Radial velocities derived from the SSS data for the RT And
system.
Phase
Radial
Velocity
(km/s)
EZ Peg SSS Radial Velocity Curve
0 .2 .4 .6 .8 1
­60
­40
­20
0 blends
Fig. 14. Radial velocities derived from the SSS data for the EZ Peg
system.
with the published ephemeris and velocity components.
RT And appears to be a highly variable system. This eclips­
ing binary consists of an F8 V primary and a K0 V secondary.
We have taken the ephemeris of Popper (1994) which is based
on recent times of minimum. The original velocity data of
Payne­Gaposchkin (1946) (given also by the CABS catalogue)
are severely in error as first suggested by Gordon (1955). We
found the best semi­amplitudes and systemic velocities to be
those provided by Wang & Lu (1993). The more recent orbit
deduced by Popper (1994) does not fit our measurements as
well although it should be noted that both these orbits do not
give good agreement. Our results are limited by the magnitude
of the star but appear to be consistent. There is an indication
of a large period change for this system (Figure 13). Kristen­
son (1967) and Williamson (1974) have both given evidence
for changes in the period of RT And although these are very
small.
EZ Peg (Figure 14) is a curious system with highly variable
Hff emission. Schlesinger, Barney & Gesler (1934) first classi­
fied the star as G5 but Vyssotsky & Balz (1958) later claimed

9
that the spectrum varied from B0 to G5. In fact Alden (1958)
showed that the spectrum appeared to be that of a B­star on
one occasion in 1943. Howell & Bopp (1985) later presented ev­
idence that this erroneous spectrum was due to poor resolution
data. They also suggested that the system be classified as an
RS CVn and showed that the (U­B) colour index implied a K0
spectral type for the companion. The ephemeris and orbital
parameters were determined by Griffin (1985). Our observa­
tions have failed at all phases to resolve both components of
the system. Our resolution criterion (see x5) suggests 22.7 km
s \Gamma1 is the minimum necessary velocity difference for this sys­
tem which is less than half of the maximum separation (48.8
km s \Gamma1 ). Our results however are clearly confused over most
phases of this difficult system which, assuming the orbit has
not altered, implies that the v sin i values quoted by Griffin
(1985) are under­estimated. Assuming both components have
the same v sin i the lack of resolution in our data implies a
value of v sin i ? 44 km s \Gamma1 . Clearly the SSS instrument has
failed to give reliable results for this system with the smallest
maximum velocity separation in our sample. However the re­
sults indicate that some of the orbital parameters may be in
error. High­resolution observations are required to verify the
situation.
Other systems for which we have radial velocities are LX
Per, MM Her, MY Cyg, KZ And and SZ Psc. Two of these
systems, LX Per and MM Her, agree extremely well with pub­
lished parameters, the RMS difference between measured and
predicted velocities over all phases and for both components
are 3 km s \Gamma1 and 5 km s \Gamma1 for these systems respectively. For
KZ And and SZ Psc these values are 14 km s \Gamma1 and 25 km
s \Gamma1 suggesting a substantial discrepancy with accepted values.
Kalimeris et al. (1995) recently reviewed the period changes in
SZ Psc.
5. Analytical assessment of cross­correlations
Cross­correlation techniques for the determination of radial
velocities and velocity dispersions have been employed in a
number of applications. Most analytical discussions relate to
the measurement of galaxy red­shifts and velocity dispersions
(Simkin 1974; Sargent et al. 1977; Tonry & Davis (1979); Ore et
al. 1991). General discussions of Fourier transform techniques
and the cross­correlation function in particular can be found in
Bracewell (1965), Brault & White (1971) and Scargle (1989).
Although the technique has been widely used in the determi­
nation of stellar radial velocities, stellar rotational velocities
(eg. Soderblom et al. 1989) and binary star orbits (Popper &
Hill 1991; Popper & Jeong 1994), no analytical treatment has
been presented for the form of the cross­correlation function
for binary spectra.
The cross­correlation procedure is one based on the Fast
Fourier Transform (FFT) common in many forms of signal
processing, and as such is subject to the induction of errors
due to aliasing, leakage and convolution. In order to increase
the efficiency of the technique it is possible to apply corrections
to both the measured spectral data and the Fourier frequencies
themselves. Firstly, large scale trends in spectra will result in
low­frequency spikes in the Fourier domain and can be avoided
by careful normalisation. Secondly the use of an optimal filter
to cut off the high­frequency Fourier components beyond the
Nyquist frequency (SNY = 1
2\Deltax , where \Deltax is the sampling fre­
quency) will both reduce the broadband noise and reduce the
possibility of aliasing (the overlap or folding of high Fourier
components into the lower frequencies). To completely avoid
aliasing, a careful consideration of the sampling interval should
be made and the data re­sampled if necessary. Spectral data
should also be deconvolved with the instrumental response to
give a better estimate of the true signal. Deconvolution of the
resolution function should also be performed or an artificial
smoothing applied to the discontinuities in the data (perhaps
by adding a cosine bell or zero­valued padding). This will avoid
the mixing of Fourier components known as leakage. The above
methods will all ensure that the FFTs of spectral data are not
significantly corrupted and do not induce errors in the subse­
quent manipulation. During the analysis of the observational
data presented above and that of the synthetic data described
below, FFTs were performed with the above constraints. The
instrumental profile was not deconvolved since it equalled the
resolution measured from Th­Ar lines. Any remaining errors
are certainly less than those induced by noise and the mea­
surement process itself.
The results presented in this paper have demonstrated that
a significant source of error in RV measurements with the SSS
instrument is the gaussian fitting procedure. However, these er­
rors become secondary when the CCF peaks are blended by the
dual effects of rotation and small velocity difference. In order
to assess the applicability of the CCF method for individual
measurements we present below a derivation of the approx­
imate relation between the widths of features in the binary
and standard spectra and the CCF. This analysis allows us to
make some statements on the relative importance of various
spectral parameters and define a criterion for the resolution of
CCF velocities. Our derivation is similar to that of Tonry &
Davis (1979), who analysed galactic velocity dispersions, but
with two important differences. We have generalised the tar­
get spectrum to be a weighted combination of spectra that
represents the binary star spectrum and we have derived the
relationship rigorously without the need for minimising a dif­
ference equation. Our results reduce to those of Tonry & Davis
for the case of a single spectral component.
Let b(x) be the spectrum of a binary star and t(x) the
spectrum of a radial velocity standard star of known radial and
rotational velocity and similar spectral type as the components
of the binary. Let B(s) and T (s) be the corresponding discrete
Fourier transforms of t(x) and b(x) where
B(s) = 1
N
N
X
j=1
b(x j )e \Gamma2úx j is ; T (s) = 1
N
N
X
j=1
t(x j )e \Gamma2úx j is (1)
where s is the generalised frequency variable. If b(x) and t(x)
have RMS values oe b and oe t respectively where
oe 2
b = 1
N
X
x
b(x) 2 ; oe 2
t = 1
N
X
x
t(x) 2 (2)
then the normalised cross­correlation function c(x) can be writ­
ten
c(x) = b(x) \Theta t(x) = 1
N oe b oe t
N
X
k=1
b(k)t(k \Gamma x) (3)
where the \Theta symbol indicates the cross­correlation product.
The Fourier transform of the CCF is therefore

10
C(s) = 1
N oe b oe t
B(s)T \Lambda (s) (4)
where the asterix signifies the complex conjugate. Now let us
assume that the binary spectrum is a weighted combination of
the primary and secondary spectra (p(x) and s(x) respectively)
and that each is some multiple of t(x) which is shifted and
broadened by symmetric functions rp (x) and rs(x). Then,
p(x) ' ff pt(x) \Lambda rp (x \Gamma ffi p) (5)
s(x) ' ff s t(x) \Lambda rs(x \Gamma ffi s) (6)
where the asterix signifies the convolution, ff p and ff s are the
multiple factors and ffi p and ffi s are the respective shifts of pri­
mary and secondary components. We can therefore represent
the binary spectrum by
b(x) = ff p fi pt(x) \Lambda rp(x \Gamma ffi p) + ff s fi s t(x) \Lambda rs(x \Gamma ffi s) (7)
where fi p and fi s are the intensity weights of the primary
and secondary components respectively. Using the linearity
of Fourier transforms and the shift theorem we can write the
Fourier transform of b(x) as
B(s) =
n
ff p fi pRp(s)e
2úisffi p
N + ff s fi sRs(s)e
2úisffi s
N
o
T (s): (8)
The Fourier transform of the cross­correlation function can
thus be written
C(s) = j T (s) j 2
N oe b oe t
n
ff p fi pRp (s)e 2úisffi p
N + ff s fi sRs(s)e
2úisffi s
N
o
(9)
where j T (s) j 2 = T (s)T \Lambda (s) is the power spectrum of t(x). We
now proceed by making some assumptions about the form of
t(x), rp(x) and rs(x). Firstly let t(x) have a Fourier transform
that is gaussian such that a typical spectral feature in t(x) has
a width equal to Ü , then we can write the power spectrum of
t(x) as
j T (s) j 2 = 2úNÜ oe 2
t exp
ae
\Gamma
(2úÜs) 2
N 2
oe
: (10)
Now suppose that rp (x) and rs(x) are gaussian broadening
functions of dispersion oe p and oe s respectively where
rp = 1
p
2úoep
exp
ae
\Gammax 2
2oe 2 p
oe
; rs = 1
p
2úoes
exp
ae
\Gammax 2
2oe 2
s
oe
(11)
then we can write the Fourier transforms of rp(x) and rs(x) as
Rp(s) = exp
ae
\Gamma
(2úoeps) 2
2N 2
oe
; Rs(s) = exp
ae
\Gamma
(2úoess) 2
2N 2
oe
:(12)
The assumption that rp (x) and rs(x) are gaussian is justified
only if we are considering a maxwellian distribution of veloci­
ties which is not true for axial rotation, particularly for high ro­
tation rates. We have retained this assumption in our analysis
firstly to simplify the mathematics and because any symmet­
ric function representing the degree of broadening is adequate
providing an appropriate interpretation of the broadening pa­
rameter is made. In subsequent work we will introduce the
standard axial broadening function into the procedure.
Substituting for j T (s) j 2 , Rp(s) and Rs(s) in Equation (9)
and simplifying gives
C(s) = 2 p
úÜ oe t
oe b
ae
ff p fi p exp
ae
\Gamma
2ú 2 s 2 (2Ü 2 + oe 2
p )
N 2
oe
e
2úisffi p
N +
ff s fi s exp
ae
\Gamma
2ú 2 s 2 (2Ü 2 + oe 2
s )
N 2
oe
e 2úisffi s
N
oe
: (13)
We now assume that the CCF is the simple sum of two gaus­
sian functions centred at ffi p and ffi s with dispersions ¯p and
¯s corresponding to the primary and secondary components
respectively, then
c(x) = Ap exp
ae
\Gamma
(x \Gamma ffi p) 2
2¯ 2 p
oe
+As exp
ae
\Gamma
(x \Gamma ffi s) 2
2¯ 2 s
oe
(14)
where Ap and As are the amplitudes of the two peaks in the
cross­correlation function. We can therefore write the Fourier
transform of c(x) as
C(s) =
p
2ú¯pAp exp
ae
\Gamma
(2ú¯ps) 2
2N 2
oe
e \Gamma 2úisffi p
N +
p
2ú¯sAs exp
ae
\Gamma
(2ú¯ss) 2
2N 2
oe
e \Gamma 2úisffi s
N : (15)
Rewriting Equations (13) and (15) in Eulerian form, equat­
ing their real parts and simplifying leads to the approximate
equality
p
2ú¯pAp exp
ae
\Gamma2s 2
h ú¯p
N
i 2
oe
cos
n 2úffi ps
N
o
+
p
2ú¯sAs exp
ae
\Gamma2s 2
h ú¯s
i 2
oe
cos
n 2úffi ss
N
o
=
2 p
úÜ oe t ff p fi p
oe b
exp
(
\Gamma2s 2
''
ú
p
2Ü 2 + oe 2
p
N
# 2 )
cos
n 2úffi ps
N
o
+
2 p
úÜ oe t ff s fi s
oe b
exp
(
\Gamma2s 2
Ÿ
ú
p
2Ü 2 + oe 2 s
N
– 2
)
cos
n 2úffi ss
N
o
: (16)
Since this relationship must hold for all values of the gener­
alised frequency variable s we can write the following approx­
imate relationships
¯pAp '
p
2Ü oe t ff p fi p
oe b
; ¯sAs '
p
2Ü oe t ff s fi s
oe b
(17)
¯ 2
p ' 2Ü 2 + oe 2
p ; ¯ 2
s ' 2Ü 2 + oe 2
s : (18)
Equations (17) and (18) relate the dispersions, multiple fac­
tors, intensity weights, rms values and dispersions in the binary
components and the RV standard to the width of features in
the cross­correlation function. Equation (18) is identical to the
relation derived by Tonry & Davis (1979) using a minimisation
method for the analysis of galaxy red­shift survey data. Using
reasonable assumptions about the form of spectra and cross­
correlation functions we have shown that in the case of stellar
binary spectra the individual cross­correlation components also

11
Intensity
0
.5
1
100.0 200.0 300.0 400.0
primary
secondary
binary
standard
a)
Lag (pixels)
Correlation
­40 ­20 0 20 40
.9772
.9774
.9776
.9778
b)
binary
standard
Fig. 15. Example of the formation of synthetic binary spectral data. The top panel (a) shows synthetic data for a primary and secondary
spectrum and the weighted combination of these to produce a binary spectrum with noise added. The RV standard spectrum is also shown
for comparison. The intensity values have been scaled arbitrarily. The lower panel (b) shows the resulting CCF formed between the synthetic
binary and RV standard and clearly displays the two binary components. The parameters related by Equation (17) and (18) are also shown
schematically. The inset shows the width parameters associated with the binary and RV standard spectra.
follow this relation. In addition we have derived a relationship
between the peak values of these components and their widths
in terms of the rms of the spectra, the width of features in
the RV standard spectrum, the multiple factors and intensity
weights for the binary components. The multiple factors can
be regarded as a measurement of the mismatch in spectral type
between the binary components and the RV standard star. This
will apply only across narrow ranges of spectral type since over
such a range the absorption features become generally deeper
with advancing type but little change to the sequence of fea­
tures is found (Popper & Jeong 1994). We have regarded the
multiple factors ff p and ff s as constants representing the vari­
ation of line­depths across a narrow range of spectral types.
The significance of these factors requires further observational
work. We plan, in a later paper, to investigate the calibration
and possible uses of Equation (17).
We can define a criterion for the limit of the velocity reso­
lution for binary orbit analysis as follows. We use the criterion
that the velocity difference between primary and secondary
components (ffi s \Gamma ffi p) is greater than or equal to the sum of the
two half­widths in the cross­correlation function peaks. Hence
we can write,
ffi s \Gamma ffi p – ¯p
2 + ¯s
2 (19)
–
s ae
Ü 2 + oe 2 p + oe 2 s
4
oe
+ (2Ü 2 + oe 2
p ) 1=2 (2Ü 2 + oe 2 s ) 1=2
2
For reliable measurements of binary star radial velocities to be
made the above criterion must be valid. Equation (19) shows
how the resolution limit is dependent on the broadening in the
RV standard and binary components. Further if we use a set of
delta functions as the RV template or the template has rota­
tional broadening much less than the instrumental profile then
the resolution limit is the mean of the component rotational
velocities and the width of the CCF peaks are equal to these
velocities. Undoubtedly other criteria are possible to formulate
but we believe this relationship is adequate for our preliminary
analysis.
6. Qualitative assessment of cross­correlation tech­
niques
To investigate the effects of various parameters on the deter­
mination of CCF velocities we have synthesised data for which
all the important parameters are well known. This method in­
volves synthesising the spectrum of a binary star with a given
radial velocity difference between primary and secondary com­
ponents, line strength ratio, rotational velocities and signal­to­
noise ratio. We then perform the cross­correlation procedure
outlined in x3 using a similar synthetic RV standard star for
which we also know the spectral parameters. Variation of the
synthesis parameters then gives us a qualitative assessment of
the reliability of the CCF results for this particular data set and
allows us to make some broad remarks on the efficiency of the
technique in general and to demonstrate the use of the equa­
tions in x5. Similar analyses of faked data have been performed
by Popper & Jeong (1994) and using a series of delta­functions
as the standard spectrum by Furenlid & Furenlid (1990).
We synthesise single order spectra of 512 pixels in size.
We assume that a linear relationship exists between pixel shift

12
and velocity so that a single pixel anywhere in the order cor­
responds to the same velocity difference. This is in fact the
case for the SSS used in this study where the translation is
10.422 km s \Gamma1 per pixel. We perform cross­correlation analysis
in pixel space as is done with real data and can, if we wish,
compare the results to the SSS by using this expansion factor.
The method of constructing the synthetic spectra is di­
rectly analogous to the mathematical treatment given in x5.
Our approach has been to represent a stellar spectrum with
well known parameters rather than to simulate one and there­
fore we have not synthesized data which resembles our obser­
vations. Subsequent work will make use of real data in the
synthesis but our primary aim here is to establish the validity
of Equation (19). As a template spectrum, we have thus used
a set of 20 gaussian­shaped absorption features at arbitrary
pixel positions rather than at the positions of actual stellar
lines. Popper & Jeong (1994) used a sky spectrum as their
template and Furenlid & Furenlid (1990) used an 80 š A range
of a degraded solar spectrum. Although both these techniques
are valid the use of arbitrary line positions avoids the pos­
sibility of other forms of contamination or activity sensitive
deviations and ensures that we know all the significant details
inherent in the synthetic data. Our tests have shown that an
arbitrary choice of line positions has no significant effect on
the results of the analysis. Gaussian­shaped lines which do not
accurately represent axial broadening are used in order to re­
main consistent with our mathematical formalism. By using
some multiple factor of the template spectrum we form the
spectra of the primary and secondary components of the syn­
thetic binary and broaden and shift these by convolving with
a suitable function. The binary spectrum is then formed as a
weighted sum of the two components. An RV standard spec­
trum is also formed by convolving the template with another
broadening and shifting function. Although our observational
RV standards are not broadened for the analysis this procedure
represents the broadening in the standard beyond the instru­
mental profile.
For all spectra the data are normalised to unity continuum
and then have synthetic photon noise added. To do this we
assume that the distribution of noise is gaussian with a dis­
persion of 1
S (where S is the signal­to­noise ratio) and a mean
equal to the data value. The noisy data are then represented
by,
y 0 = yo
n
1 + r
S
o
(20)
where r is a normally­distributed pseudo­random number with
a mean of zero and a oe of 1. Every synthetic spectrum has a
pseudo­random number generation which is different so that
the added noise is never correlated between spectra.
Figure 15 shows an example of synthetic data. Panel a)
shows arbitrarily scaled spectra for the primary and secondary
components, the weighted combination forming the binary
spectrum, with noise added, and the synthetic standard spec­
trum. The resulting CCF for these data is shown in panel b)
where the positions of each component are clearly measurable.
Figure 15 also shows schematically the parameters related by
Equations 17, 18 an 19. Although several procedures employed
in the construction of our synthetic data and some of our as­
sumptions are perhaps not appropriate for real binary systems
we have assumed that systematic effects so produced are small
compared to those arising in the CCF method and from the
CCF Image
0 20 40 60 80 100
Relative Lag
0
10
20
30
Velocity
Difference
Fig. 16. A combined grey­scale and contour plot of the CCF with
increasing velocity separation between the synthetic binary compo­
nents. Constant parameters used in the synthetic data were ff p=0.9,
fi p=0.6, ff s=0.8, fi s=0.4, Ü=2.0 and ffi t =5.0 (the initial shift of the
RV standard). The signal to noise ratios were 120 and the binary
components velocity difference was successively shifted by one pixel.
The primary and secondary widths were oe p=5.0 and oe s=2.0 respec­
tively which gives a resolution limit of 4.34 pixels (solid white line).
The components were shifted so as to reveal the reflection effects in
the CCF. The amplitudes have been normalised by the mean con­
tinuum level.
CCF Image
0 20 40 60 80 100
Relative Lag
0
10
20
30
Velocity
Difference
Fig. 17. A combined grey­scale and contour plot of the CCF with
increasing velocity separation similar to Figure 16. This plot is for
a synthetic binary with widths oe p = 10, oe s = 5 and Ü = 2 giving a
resolution limit of 8.07 (solid white line). The increase in component
widths is clearly visible as is the shift in the resolution limit.
synthetic binaries themselves. We intend to further investigate
the sources of error involved in CCF techniques by extending
our preliminary analysis to real spectral data obtained for a
large sample of RV standards and well­studied binaries.
Using our synthetic data maker (an idl procedure) we have
synthesised spectra where the separation of the binary compo­
nents varies linearly from 0 to 40 pixels. Knowing the width
parameters on input we can compare the resulting CCFs to
our resolution limit. Figure 16 shows a combined grey­scale
and contour plot of the CCF image for one of our data sets.

13
Separation (pixels)
­2
0
2
0 5 10 15 20 25 30 35
d)
­2
0
2 c)
­2
0
2 b)
Error
(pixels)
­2
0
2 a)
Fig. 18. The effects of varying the component rotational broadening on the measured position errors with increasing velocity separation.
See the text for a full explanation.
The image consists of the CCF amplitude plotted for varying
separation and relative lag. In this case the component widths
were oe p = 5, oe s = 2 and Ü = 2 pixels giving a resolution limit
of 4.34 pixels. Other input parameters were ff p = 0.9, fi p = 0.6,
ff s = 0.8, fi s = 0.4 and the signal­to­noise was set at 120 in all
spectra. Note that the binary components have been shifted
significantly from the CCF centroid in order to reveal reflec­
tion effects in the CCF. The amplitude scale for each CCF has
been normalised by the mean continuum level in the CCF.
Figure 16 clearly shows both components of the binary
shifting in velocity. The broader component on the right is the
primary component which demonstrates that the brighter or
hotter component does not necessarily give the greatest peak
in the CCF since it is denuded by its width. The resolution
limit is marked on the diagram as a solid white line and oc­
curs where the two components become adequately separated.
Figure 16 also shows the important effects of reflection in the
CCF. We demonstrate below that for some data the reflection
effect is one of the most significant source of error in CCF
velocity determinations beyond the resolution limit. For com­
parison Figure 17 shows a similar plot for a synthetic binary
with widths oe p = 10, oe s = 5 and Ü = 2 pixels giving a resolu­
tion limit of 8.07 pixels. The increase in component widths is
clearly visible as is the shift in the resolution limit.
We have investigated the systematic effects in the compo­
nent positions measured from our synthetic data. Our method
for assessing the technique is to derive radial velocities using
the synthetic data and compare these to the known values used
to create the data. We can then investigate the errors induced
by, for example, low signal­to­noise or small binary separation.
This is achieved with the same fitting procedure used for our
observational data (see x3). A remaining source of error which
is not possible to model is the user­supplied positions for the
double or single gaussian fitting procedure. These errors are
inherent in the results just as with the SSS radial velocities.
Our investigations have shown however that independent de­
terminations agree well within any tolerances we wish to adopt.
However a full treatment of fitting errors would be helpful.
Figure 18 shows the results of our measurements where we
plot the error in the position (actual minus measured) against
the separation of the components. Where the two CCF com­
ponents are blended the data are plotted as an open circle; in
this case the actual position of the larger peak is used to de­
termine the error. This is invariably the slower rotator, ie. the
secondary. Points plotted as filled circles are for the secondary
component and filled squares are the primary. The four panels
a) to d) of Figure 18 show the results for resolution limits of
3.74, 4.34, 8.07 and 12.87 pixels respectively. The parameters
of each component are displayed in each panel and the reso­
lution limit is displayed as a vertical dashed line. The other
parameters used in the formation of these data were ff p = 0.9,
fi p = 0.6, ff s = 0.8, fi s = 0.4 and the signal­to­noise in all
spectra was 120.
Figure 18 shows many interesting features. Firstly our def­
inition of the resolution limit appears to be consistent. Al­
though it is sometimes possible to determine the positions of
two CCF components below this limit these are likely to be
significantly in error. The RMS position of the CCF peak at
zero separation for these parameters is 0.066 pixels but the
error rapidly increases as the peak is skewed by the shifting
of blended components. The rms error over all measurements
lying above the resolution limit in panels a) to d) are 0.08,
0.18, 0.22 and 0.54 pixels respectively. Clearly there are peri­
odic variations in the errors whose amplitude increases with the
resolution limit. These systematic variations are in anti­phase
between primary and secondary components. This important
effect is caused by the reflection peaks in the CCF moving
through the component peaks as the separation increases. Al­

14
Separation (pixels)
­5
0
5
0 5 10 15 20 25 30 35
d) S/N = 20
­5
0
5 c) S/N = 60
­5
0
5 b) S/N = 100
Error
(pixels)
­5
0
5 a) S/N = 140
Fig. 19. The effects of varying the spectral signal­to­noise on the measured position errors with increasing velocity separation. See the text
for a full explanation.
though such reflection peaks can be very small their skewing
effect on the actual peaks is sufficient to induce significant er­
rors. Large errors at large separations in panel d) are caused
by the very wide CCF peak lying close to the edge of the CCF
window used in the analysis thus giving rise to fitting errors.
Figure 18 demonstrates that one of the most significant effects
in the measurement of velocities with the CCF technique is
the confusion by reflection peaks. This effect rapidly increases
with resolution limit but may be negligible in comparison with
instrumental or other systematic effects (as with the SSS). Al­
though Popper & Jeong (1994) make no reference to this effect
we believe their synthetic data also shows periodic variations
due to reflection peaks (their Figures 5­9).
We have also investigated the effects of signal­to­noise ra­
tio, luminosity weight and spectral type mismatch on the de­
rived velocities. Figure 19 shows the position errors for binary
spectra signal­to­noise ratios of 140, 100, 60, and 20. The other
parameters used in the formation of these data were ff p = 0.9,
fi p = 0.6, ff s = 0.8, fi s = 0.4, oe p = 10.0, oe s = 5.0, Ü = 2.0 and
the signal­to­noise was set at 120 in all spectra. The periodic
effect in the errors is clearly visible but as expected the dis­
persion in the values is greater for low signal­to­noise. For this
data set the signal­to­noise becomes dominant between 60 and
100. The rms of all errors lying above the resolution limit in
panels a) to d) are 0.40, 0.35, 0.85 and 1.45 pixels respectively.
Figure 19 shows that the measured velocities are systemati­
cally slightly smaller than the true ones and this effect seems
to be insensitive to the separation. Popper & Jeong (1994) also
noticed this effect in their synthetic results and attributed this
to the CCF procedure.
Figure 20 shows position errors for different multiple fac­
tors which gives an ad hoc representation of changes in spectral
type over a narrow range by varying the initial line depths in
the binary spectrum. We synthesised data sets for multiple fac­
tors ff p = 2.0 and ff s = 1.5, ff p = 1.5 and ff s = 1.0, ff p = 1.0
and ff s = 0.5 and ff p = 0.5 and ff s = 0.2. The other parame­
ters were fi p = 0.6, fi s = 0.4, oe p = 10.0, oe s = 5.0, Ü = 2.0 and
the signal­to­noise was set at 120 in all spectra. Clearly the
errors increase with decreasing line depth. The rms of errors
beyond the resolution limit (8.07) for panels a) to d) are 0.31,
0.29, 0.35 and 0.53 pixels respectively. It appears that over a
small range of spectral types where no changes in the pattern
of lines occurs a mismatch in spectral types does not induce
large errors providing that the line depths are sufficient, when
broadened, to produce a correlation well above the continuum
in the CCF. Providing lines are deeper or of similar depth as
the RV standard the spectral type mismatch will not be the
dominating source of error.
Finally we investigated the systematic errors in the mea­
sured positions for varying luminosity ratios. We synthesised
data for luminosity weights of fi p = 0.8 and fi s = 0.2, fi p =
0.66 and fi s = 0.33, fi p = 0.5 and fi s = 0.5 and fi p = 0.33
and fi s = 0.66. These then correspond to luminosity ratios of
4/1, 2/1, 1/1, and 1/2 respectively. The other parameters used
were ff p = 0.9, ff s = 0.8, oe p = 10.0, oe s = 5.0, Ü = 2.0 and the
signal­to­noise was set at 120 in all spectra. Figure 21 shows
the results of this analysis. Errors are less serious for luminos­
ity ratios of 1 and increase with the difference in luminosity.
The rms of all errors above the resolution limit in panels a)
to d) are 0.37, 0.35, 0.36 and 0.63 respectively. Although the
combined rms values are similar for luminosity ratios 4, 2 and
1 the errors for secondary and primary components scale well
with their relative weights. The large error for luminosity ratio
0.5 is due mainly to the faint primary; the rms error for the
secondary is only 0.09 pixels for values above the resolution
limit. Panel d) shows well the effects of both a large rotational
velocity and low luminosity on the primary component and the
correspondingly small errors for the secondary.

15
Separation (pixels)
­4
0
4
0 5 10 15 20 25 30 35
d)
­4
0
4 c)
­4
0
4 b)
Error
(pixels)
­4
0
4 a)
Fig. 20. The effects of varying the multiple factor on the measured position errors with increasing velocity separation. See the text for a
full explanation.
Separation (pixels)
­4
0
4
0 5 10 15 20 25 30 35
d) r = 0.5
­4
0
4 c) r = 1
­4
0
4 b) r = 2
Error
(pixels)
­4
0
4 a) r = 4
Fig. 21. The effects of varying the luminosity ratio on the measured position errors with increasing velocity separation. See the text for a
full explanation.

16
In Figures 3 to 14 where we present our observational re­
sults we have depicted as dashed lines the positions of our
resolution limits centred on the systemic velocities. These are
calculated using Equation 19 when the necessary rotational
velocities are known and taking the maximum RV standard
rotational velocity of 15 km s \Gamma1 . It can be seen from these di­
agrams that all blended velocities lie within or slightly above
these limits. Our preliminary analytical results therefore seem
to be reliable for this limited set of data. We plan to extend
this study of CCF resolution limits, the effects of various stellar
parameters and the induction of errors by the CCF technique
itself by analysing a large sample of medium­resolution spectra
for RV standards and well­studied binary systems.
7. Conclusions
We have presented observational radial velocity results for a
small sample of chromospherically active binary systems. The
data were collected with the medium­resolution Solar Stellar
Spectrograph on the John Hall 1.1­meter telescope located
near Flagstaff, Arizona. We have employed a cross­correlation
technique for radial velocity determinations and compared our
results to published ephemerides and orbital parameters. We
have excellent agreement for TY Pyx and GK Hya, very good
agreement for LX Per, MM Her, Z Her and ER Vul, reason­
able agreement for V471 Tau, OU Gem, V772 Her and AR
Lac. V772 Her has demonstrated the induced errors due to the
primary component CCF peak skewing the fitting procedure
for the small secondary peak. The system EI Eri has given ex­
tremely confused results perhaps due to the high rotational ve­
locity of this system. Although results for RT And appear to be
consistent these do not appear to agree well with any published
values for the system. EZ Peg has also given confused results
possibly due to higher rotation rates than previously published.
The system BD­004234 appears to be approximately 0.25 out
of phase with the published ephemeris. SZ Psc and KZ And
have also given individual measurements which conflict with
predicted values.
We have also presented a rigourous derivation of the rela­
tionship between binary component rotation rates, RV stan­
dard rotation rates and the width of features in the CCF.
An ad hoc resolution limit for binary star cross­correlation ra­
dial velocities is also given. Using synthetic spectral data we
have investigated the errors induced by rotational broadening,
signal­to­noise ratio, spectral­type mismatch and luminosity
ratio and identified some sources of confusion in the technique.
In summary, although we have not formally derived an er­
ror equation for CCF analysis of binary orbits nor given a
quantitative treatment we have demonstrated the relative im­
portance of rotation, signal­to­noise, spectral type mismatch
and luminosity ratio on the determination of component ve­
locities. For data with good signal­to­noise and medium reso­
lution the primary sources of error seem to be reflection effects
in the CCF, spectral type mismatch and low luminosity ratio.
In a future paper we will examine in more detail the aspects of
CCF radial velocity methods described in the present paper.
Acknowledgements.
Research at Armagh Observatory is grant aided by the Dept.
of Education for N. Ireland. We also acknowledge the computer
support by the STARLINK project funded by the UK SERC.
AGG would like to thank Armagh Observatory for a research
scholarship and the staff of Lowell Observatory for their hos­
pitality during the observations. Acknowledgements also go to
Erika Gibb (NAU) for help in the observations and data reduc­
tion. The authors would also like to thank the referee (D. M.
Popper) for judicious comments on an earlier version of this
paper. This work has also been supported through PPARC
research grant GR/K35242.
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