U Geminorum: a test case for orbital parameters determination
U Geminorum: a test case for orbital parameters determination
Juan Echevarŕıa1, Eduardo de la Fuente2 and Rafael Costero3
Instituto de Astronomı́a, Universidad Nacional Autónoma de México,
Apartado Postal 70-264, México, D.F., México
ABSTRACT
High-resolution spectroscopy of U Gem was obtained during quiescence. We
did not find a hot spot or gas stream around the outer boundaries of the accretion
disk. Instead, we detected a strong narrow emission near the location of the
secondary star. We measured the radial velocity curve from the wings of the
double-peaked Hα emission line, and obtained a semi-amplitude value that is
in excellent agreement with the obtained from observations in the ultraviolet
spectral region by Sion et al. (1998). We present also a new method to obtain
K2, which enhances the detection of absorption or emission features arising in
the late-type companion. Our results are compared with published values derived
from the near-infrared NaI line doublet. From a comparison of the TiO band with
those of late type M stars, we find that a best fit is obtained for a M6V star,
contributing 5 percent of the total light at that spectral region. Assuming that
the radial velocity semi-amplitudes reflect accurately the motion of the binary
components, then from our results: Kem = 107 ± 2 km s
−1; Kabs = 310 ± 5
km s−1, and using the inclination angle given by Zhang & Robinson (1987); i =
69.7◦ ± 0.7, the system parameters become: MWD = 1.20 ± 0.05M⊙; MRD =
0.42 ± 0.04M⊙; and a = 1.55± 0.02R⊙. Based on the separation of the double
emission peaks, we calculate an outer disk radius of Rout/a ∼ 0.61, close to the
distance of the inner Lagrangian point L1/a ∼ 0.63. Therefore we suggest that,
at the time of observations, the accretion disk was filling the Roche-Lobe of the
primary, and that the matter leaving the L1 point was colliding with the disc
directly, producing the hot spot at this location.
Subject headings: binaries: close — novae, cataclysmic variables — stars: indi-
vidual (U Geminorum)
1email: jer@astroscu.unam.mx
2present address: Departamento de F́ısica, CUCEI, Universidad de Guadalajara. Av. Revolución 1500
S/R Guadalajara, Jalisco, Mexico. email: edfuente@astro.iam.udg.mx
3email: costero@astroscu.unam.mx
http://arxiv.org/abs/0704.1641v1
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1. Introduction
Discovered by Hind (1856), U Geminorum is the prototype of a subclass of dwarf novae,
a descriptive term suggested by Payne-Gaposchkin & Gaposchkin (1938) due to the small
scale similarity of the outbursts in these objects to those of Novae.
After the work by Kraft (1962), who found U Gem to be a single-lined spectroscopic
binary with an orbital period around 4.25 hr, and from the studies by Kreminski (1965),
who establish the eclipsing nature of this binary, Warner & Nather (1971) and Smak (1971),
established the classical model for Cataclysmic Variable stars. The model includes a white
dwarf primary surrounded by a disc accreted from a Roche-Lobe filling late-type secondary
star. The stream of material, coming through the L1 point intersects the edge of the disc
producing a bright spot, which can contribute a large fraction of the visual flux. The bright
spot is observed as a strong hump in the light curves of U Gem and precedes a partial eclipse
of the accretion disk and bright spot themselves (the white dwarf is not eclipsed in this
object).
A mean recurrence time for U Gem outbursts of ≈ 118 days, with ∆mV =5 and out-
burst width of 12 d, was first found by Szkody & Mattei (1984). However, recent analysis
shows that the object has a complex outburst behavior (Cook 1987; Mattei et al. 1987;
Cannizo, Gehrels & Mattei 2002). Smak (2004), using the AAVSO data on the 1985 out-
burst, has discovered the presence of super-humps, a fact that challenges the current theories
of super-outbursts and super-humps for long period system with mass ratios above 1/3. The
latter author also points out the fact that calculations of the radius of the disc – obtained
from the separation of the emission peaks (Kraft 1975) in quiescence – are in disagreement
with the calculations of the disc radii obtained from the photometric eclipse data (Smak
2001).
Several radial velocity studies have been conducted since the first results published by
Kraft (1962). In the visible spectral range, where the secondary star has not been detected,
their results are mainly based on spectroscopic radial velocity analysis of the emission lines
arising from the accretion disc (Kraft 1962; Smak 1976; Stover 1981; Unda-Sanzana et al.
2006). In other wavelengths, works are based on absorption lines: in the near-infrared,
on the Na I doublet from the secondary star (Wade 1981; Friend et al. 1990; Naylor et al.
2005) and in the ultraviolet, on lines coming from the white dwarf itself (Sion et al. 1998;
Long & Gilliland 1999).
Although the research work on U Gem has been of paramount importance in our under-
standing of cataclysmic variables, the fact that it is a partially-eclipsed and – in the visual
range – a single-lined spectroscopic binary, make the determination of its physical param-
– 3 –
eters difficult to achieve through precise measurements of the semi-amplitudes K1,2 and of
the inclination angle i of the orbit. Spectroscopic results of K1,2 differ in the ultraviolet,
visual and infrared ranges. Therefore, auxiliary assumptions have been used to derive its
more fundamental parameters (Smak 2001). In this paper we present a value of K1, obtained
from our high-dispersion Echelle spectra, which is in agreement with the ultraviolet results,
and of K2 from a new method applicable to optical spectroscopy. By chance, the system was
observed at a peculiar low state, when the classical hot spot was absent.
2. Observations
U Geminorum was observed in 1999, January 15 with the Echelle spectrograph at the
f/7.5 Cassegrain focus of the 2.1 m telescope of the Observatorio Astrónomico Nacional at San
Pedro Mártir, B.C., México. A Thomson 2048×2048 CCD was used to cover the spectral
range between λ5200 and λ9100 Å, with spectral resolution of R=18,000. An echellette
grating of 150 l/mm, with Blaze around 7000 Å , was used. The observations were obtained
at quiescence (V ≈ 14), about 20 d after a broad outburst (data provided by the AAVSO:
www.aavso.org). The spectra show a strong Hα emission line. No absorption features were
detected from the secondary star. A first complete orbital cycle was covered through twenty-
one spectra, each with 10 min exposure time. Thirteen further spectra were subsequently
acquired with an exposure of 5 min each. The latter cover an additional half orbital period.
The heliocentric mid-time of each observation is shown in column one in Table 1. The
flux standard HR17520 and the late spectral M star HR3950 were also observed on the same
night. Data reduction was carried out with the IRAF package1. The spectra were wavelength
calibrated using a Th-Ar lamp and the standard star was also used to properly subtract the
telluric absorption lines using the IRAF routine telluric.
3. Radial Velocities
In this section we derive radial velocities from the prominent Hα emission line observed
in U Gem, first by measuring the peaks, secondly by using a method based on a cross-
correlating technique, and thirdly by using the standard double-Gaussian technique designed
to measure only the wings of the line. In the case of the secondary star, we were unable to
detect any single absorption line in the individual spectra; therefore it was not possible to
1IRAF is distributed by the National Optical Observatories, operated by the Association of Universities
for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
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use any standard method. However, here we propose and use a new method, based on a co-
adding technique, to derive the semi-amplitude of the orbital radial velocity of the companion
star. In this section, we compare our results with published values for both components in
the binary. We first discuss the basic mathematical method used here to derive the orbital
parameters and its limitation in the context of Cataclysmic Variables; then we present our
results for the orbital parameters – calculated from the different methods – and finally discuss
an improved ephemeris for U Gem.
3.1. Orbital Parameters Calculations
To find the orbital parameters of the components in a cataclysmic variable – in which
no eccentricity is expected (Zahn 1966; Warner 1995) – we use an equation of the form
V(t)(em,abs) = γ +K(em,abs)sin[(2π(t− HJD⊙)/Porb)],
where V(t)(em,abs) are the observed radial velocities as measured from the emission lines
in the accretion disc or from the absorption lines of the red star; γ is the systemic velocity;
K(em,abs) are the corresponding semi-amplitudes derived from the radial velocity curve; HJD⊙
is the heliocentric Julian time of the inferior conjunction of the companion; and Porb is the
orbital period of the binary.
A minimum least-squares sinusoidal fit is run, which uses initial values for the four
(Porb, γ, Kem,abs, and HJD⊙) orbital parameters. The program allows for one or more of
these variables to be fixed, i.e. they can be set to constant values in the initial parameters
file.
If the orbital period is not previously known, a frequency search – using a variety of
methods for evenly- or unevenly-sampled time series data (Schwarzenberg-Czerny 1999) –
may be applied to the measured radial velocities in order to obtain an initial value for Porb,
which is then used in the minimum least-squares sinusoidal fit. If the time coverage of the
observations is not sufficient or is uneven, period aliases may appear and their values have to
be considered in the least-squares fits. A tentative orbital period is selected by comparing the
quality of each result. In these cases, additional radial velocity observations should be sought,
until the true orbital period is found unequivocally. Time series photometric observations
are usually helpful to find orbital modulations and are definitely important in establishing
the orbital period of eclipsing binaries. In the case of U Gem, the presence of eclipses and
the ample photometric coverage since the early work of Kreminski (1965), has permitted to
establish its orbital period with a high degree of accuracy (Marsh et al. 1990). Although in
eclipsing binaries a zero phase is also usually determined, in the case of U Gem the variable
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positions of the hot spot and stream, causes the zero point to oscillate, as mentioned by
the latter authors. Accurate spectroscopic observations are necessary to correctly establish
the time when the secondary star is closest to Earth, i.e in inferior conjunction. Further
discussion on this subject is given in section 4.
To obtain the real semi-amplitudes of the binary, i.e K(em,abs)=K(1,2), some reasonable
auxiliary assumptions are made. First, that the measurements of the emission lines, produced
in the accretion disc, are free from distortions and accurately follow the orbital motion of the
unseen white dwarf. Second, that the profiles of the measured absorption lines are symmetric,
which implies that the brightness at the surface of the secondary star is the same for all its
longitudes and latitudes. Certainly, a hot spot in the disc or irradiation in the secondary
from the energy sources related to the primary will invalidate either the first, the second, or
both assumptions. Corrections may be introduced if these effects are present. In the case of
U Gem, a three-body correction was introduced by Smak (1976) in order to account for the
radial velocity distortion produced by the hot spot, and a correction to heating effects on
the face of the secondary star facing the primary was applied by Friend et al. (1990) before
equating Kabs = K2.
As initial values in our least-squared sinusoidal fits, we use Porb = 0.1769061911 d and
HJD⊙ = 2, 437, 638.82325 d from Marsh et al. (1990), a systemic velocity of 42 km s
−1 from
Smak (2001), and K1 = 107 km s
−1 and K2 = 295 km s
−1 from Long & Gilliland (1999) and
Friend et al. (1990), respectively. In our calculations, the orbital period was set fixed at the
above mentioned value, since our observations have a very limited time coverage. This allow
us to increase the precision for the other three parameters.
3.2. The Primary Star
In this section we compare three methods for determining the radial velocity of the
primary star, based on measurements of the Hα emission line. Although, as we will see
in the next subsections, the last method results in far better accuracy and agrees with the
ultraviolet results, we have included all of them here because the first method essentially
provides an accurate way to determine the separations of the blue and red peaks, which is
an indicator of the outer radius of the disc (Smak 2001), and the second yields a Kem value
much closer to that obtained from UV results than any other published method. This cross-
correlation method might be worthwhile to consider for its use in other objects. Furthermore,
as we will see in the discussion, all three methods yield a consistent value of the systemic
velocity, which is essential to the understanding of other parameters in the binary system.
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Table 1: Measured Hα Radial Velocities.
HJD φ∗ Peaksa Fxcb Wingsc
(240000+) (km s−1)
51193.67651 0.68 166.1 139.1 121.1
51193.68697 0.75 183.4 130.0 133.8
51193.69679 0.80 181.9 125.0 126.9
51193.70723 0.86 167.9 102.0 101.1
51193.71744 0.92 137.1 81.7 90.9
51193.72726 0.97 90.0 46.8 41.7
51193.73581 0.02 14.0 -17.9 6.9
51193.74700 0.09 -47.9 -48.1 -27.1
51193.75691 0.14 -67.1 -66.7 -48.2
51193.76743 0.20 -99.6 -84.6 -79.3
51193.77738 0.26 -132.3 -86.1 -75.7
51193.78900 0.32 -152.6 -60.2 -48.8
51193.80174 0.39 -77.9 -32.9 -33.6
51193.81211 0.45 9.0 10.9 14.5
51193.82196 0.51 104.3 79.2 65.1
51193.83176 0.56 134.6 113.7 107.0
51193.84175 0.62 141.0 142.8 124.9
51193.85156 0.67 159.3 158.6 147.6
51193.86133 0.73 165.6 148.0 131.7
51193.87101 0.79 192.9 142.8 130.3
51193.88116 0.84 175.0 120.7 110.6
51193.88306 0.91 154.6 106.5 91.1
51193.90530 0.98 90.6 32.3 31.9
51193.91751 0.05 -70.5 8.0 -23.1
51193.93029 0.12 -88.5 -71.8 -51.6
51193.94259 0.19 -97.1 -79.0 -66.7
51193.95483 0.26 -114.4 -88.8 -75.6
51193.95955 0.29 -142.2 -70.9 -67.9
∗Orbital phases derived from the ephemeris given in section 4
aVelocities derived as described in section 3.2.1
bVelocities derived as described in section 3.2.2
cVelocities derived as described in section 3.2.3
– 7 –
To match the signal to noise ratio of the first twenty-one spectra, we have co-added,
in pairs, the thirteen 5-minute exposures. The last three spectra were added to form two
different spectra, in order to avoid losing the last single spectrum. A handicap to this
approach is that, due to the large read-out time of the Thomson CCD, we are effectively
smearing the phase coverage of the co-added spectra to nearly 900 s. However, the mean
heliocentric time was accordingly corrected for each sum. This adds to a total sample of
twenty-eight 600 s spectra.
3.2.1. Measurements from the double-peaks
We have measured the position of the peaks using a double-gaussian fit, with their sepa-
ration, width and position as free parameters. The results yield a mean half-peak separation
Vout of about 460 km s
−1. The average value of the velocities of the red and blue peaks, for
each spectrum, is shown in column 3 of Table 1. We then applied our nonlinear least-squares
fit to these radial velocities. The obtained orbital parameters are shown in column 2 of
Table 2. The numbers in parentheses after the zero point results are the evaluated errors of
the last digit. We will use this notation for large numbers throughout the paper. The radial
velocities are also shown in Figure 1, folded with the orbital period and the time of inferior
conjunction adopted in the section 4. The solid lines in this figure correspond to sinusoidal
fits using the derived parameters in our program. Although we have not independently tab-
ulated the measured velocities of the blue and red peaks, they are shown in Figure 1 together
with their average. The semi-amplitudes of the plotted curves are 154 km s−1 and 167 km
s−1 for the blue and red peaks, respectively.
3.2.2. Cross Correlation using a Template
We have also cross-correlated the Hα line in our spectra with a template constructed
as follows: First, we selected a spectrum from the first observed orbital cycle close to phase
0.02 when, in the case of our observations, we should expect a minimum distortion in the
double-peaked line due to asymmetric components (see section 5). The blue peak in this
spectrum is slightly stronger than the red one. This is probably caused by the hot spot
near the L1 point (see section 3.3.2), which might be visible at this phase due to the fact
that the binary has an inclination angle smaller than 70 degrees. The half-separation of the
peaks is 470 km s−1, a value similar to that measured in a spectrum taken during the same
orbital phase in next cycle. The chosen spectrum was then highly smoothed to minimize
high-frequency correlations. The resulting template is shown in Figure 2. A radial velocity
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Fig. 1.— Radial velocity curve of the double peaks. The half-separation of the peaks, shown
at the top of the diagram has a mean value of about 460 km s−1. The curve at the middle
is the mean from blue (bottom curve) and red (top curve).
– 9 –
Fig. 2.— Hα template near phase 0.02. The half-separation of the peaks has a value of 470
km s−1.
– 10 –
for the template was derived from the wavelength measured at the dip between the two peaks
and corrected to give an heliocentric velocity. The IRAF fxc task was then used to derive
the radial velocities, which are shown in column 4 of Table 1. As in the previous section, we
have fitted the radial velocities with our nonlinear least-squares fit algorithm. The resulting
orbital parameters are given in column 3 of Table 2. In Figure 3 the obtained velocities and
the corresponding sinusoidal fit (solid line) are plotted.
3.2.3. Measurements from the wings and Diagnostic Diagrams
The Hα emission line was additionally measured using the standard double Gaussian
technique and its diagnostic diagrams, as described in Shafter, Szkody and Thorstensen
(1986). We refer to this paper for the details on the interpretation of our results. We
have used the convolve routine from the IRAF rvsao package, kindly made available to us by
Thorstensen (private communication). The double peaked Hα emission line – with a sepa-
ration of about 20 Å – shows broad wings reaching up to 40 Å from the line center. Unlike
the case of low resolution spectra – where for over-sampled data the fitting is made with
individual Gaussians having a FWHM of about one resolution element – in our spectra, with
resolution ≈ 0.34 Å, such Gaussians would be inadequately narrow, as they will cover only a
very small region in the wings. To measure the wings appropriately and, at the same time,
avoid possible low velocity asymmetric features, we must select a σ value which fits the line
regions corresponding to disc velocities from about 700 to 1000 kms−1.
As a first step, we evaluated the width of the Gaussians by setting this as a free pa-
rameter from 10 to 40 pixels and for a wide range of Gaussian separations (between 180 and
280 pixels). For each run, we applied a nonlinear least-squares fit of the computed radial
velocities to sinusoids of the form described in section 3.1. The results are shown in Figure 4,
in particular for three different Gaussian separations: a = 180, 230 and 280 pixels. These
correspond to the low and upper limits as well as to the value for a preferred solution, all of
which are self-consistent with the second step (see below). In the bottom panel of the figure
we have plotted the overall rms value for each least-squares fit, as this parameter is very sen-
sitive to the selected Gaussian separations. As expected at this high spectral resolution, the
parameters in the diagram change rapidly for low values of σ, and there are even cases when
no solution was found. At low values of a (e.g. crosses) there are no solutions for widths
narrower than 20 pixels. The rms values increase rapidly with width, while the σ(K)/K,
γ and phase shift values differ strongly from the other cases. For higher values of a (open
circles) we obtain lower values for σ(K)/K, but the rms results are still large, in particular
for intermediate values of the width of the Gaussians. For the middle solution (dots) the
– 11 –
Fig. 3.— Radial velocities obtained from cross correlation using the template. The solid line
correspond to the solution from column 3 in Table 2.
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results are comparable with those for large a values, but the rms is much lower. Similar
results were found for other intermediate values of a, and they all converge to a minimum
rms for a width of 26 pixels at a = 230 pixels.
For the second step we have fixed the width to a value of 26 Å and ran the double-
Gaussian program for a range of a separations, from about 60 to 120 Å. The results obtained
are shown in Figure 5.
If only an asymmetric low velocity component is present, the semi-amplitude should
decrease asymptotically as a increases, until K1 reaches the correct value. Here we observe
such behavior, although for larger values of a, there is a K1 increase for values of a up to
40 Å, before it decreases strongly with high values of a. This behavior might be due to
the fact that we are observing a narrow hot-spot near the L1 point (see section 5). On
the other hand, as expected, the σ(K)/K vs a curve has a change in slope, at a value
of a for which the individual Gaussians have reached the velocity width of the line at the
continuum. For larger values of a the velocity measurements become dominated by noise.
For low values of a, the phase shift usually gives spurious results, although in our case it
approaches a stable value around 0.015. We believe this value reflects the difference between
the eclipse ephemeris, which is based mainly on the eclipse of the hot spot, and the true
inferior conjunction of the secondary star. This problem is further discussed in section 5.
Finally, we must point out that the systemic velocity smoothly increases up to a maximum
of about 40 km s−1 at Gaussian separation of nearly 42 Å, while the best results, as seen
from the Figure, are obtained for a = 31Å. This discrepancy may be also be related to the
narrow hot-spot near the L1 point and might be due to the phase-shift between the hot-spot
eclipse and the true inferior conjunction. This problem will also be address in section 4. The
radial velocities, corresponding to the adopted solution, are shown in column 5 of Table 1
and plotted in Figure 6, while the corresponding orbital parameters – obtained from the
nonlinear least-squares fit – are given in column 4 of Table 2.
3.3. The Secondary Star
We were unable to detect single features from the secondary star in any individual
spectra, after careful correction for telluric lines. In particular we found no radial velocity
results using a standard cross-correlation technique near the NaI λλ8183.3, 8194.8 Å doublet.
As we will see below, this doublet was very weak compared with previous observations (Wade
1981; Friend et al. 1990; Naylor et al. 2005). We have been able, however, to detect the NaI
doublet and the TiO Head band around λ7050 Å with a new technique, which enables us to
derive the semi-amplitude Kabs of the secondary star velocity curve. We first present here
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the general method for deriving the semi-amplitude and then apply it to U Gem, using not
only the absorption features but the Hα emission as well.
3.3.1. A new method to determine K2
In many cataclysmic variables the secondary star is poorly visible, or even absent, in the
optical spectral range. Consequently, no V (t) measurements are feasible for this component.
Among these systems are dwarf novae with orbital periods under 0.25 days, for which it
is thought that the disc luminosity dominates over the luminosity of the Roche-Lobe filling
secondary, whose brightness depends on the orbital period of the binary (Echevarŕıa & Jones
1984). For such binaries, the orbital parameters have been derived only for the white-dwarf-
accretion disc system, in a way similar to that described in section 3.1.
In order to determine a value of Kabs from a set of spectra of a cataclysmic variable, for
which the orbital period and time of inferior conjunction have been already determined from
the emission lines, we propose to reverse the process: derive V (t)abs using Kpr as the initial
value for the semi-amplitude, and set the values of Porb and HJD⊙, derived from the emission
lines, as constants. The initial value for the systemic velocity is set to zero, and its final value
may be calculated later (see below). The individual spectra are then co-added in the frame
of reference of the secondary star, i.e. by Doppler-shifting the spectra using the calculated
V (t)calc from the equation given in section 3.1, and then add them together. Hereinafter we
will refer to this procedure as the co-phasing process. Ideally, as the proposed Kpr is changed
through a range of possible values, there will be a one for which the co-phased spectral
features associated with the absorption spectrum will have an optimal signal-to-noise ratio.
In fact, this will also be the case for any emission line features associated with the red
star, if present. In a way, this process works in a similar fashion as the double Gaussian
fitting used in the previous section, provided that adequate criteria are set in order to select
the best value for Kabs. We propose three criteria or tests that, for late type stars, may
be used with this method: The first one consists in analyzing the behavior of the measured
depths or widths of a well identified absorption line in the co-phased spectra, as a function of
the proposed Kpr; one would expect that the width of the line will show a minimum and its
depth a maximum value at the optimal solution. This method could be particularly useful for
K-type stars which have strong single metallic lines like Ca I and Fe I. The second criterion is
based upon measurements of the slope of head-bands, like that of TiO at λ7050 Å. It should
be relevant to short period systems, with low mass M-type secondaries with spectra featuring
strong molecular bands. In this case one could expect that the slope of the head-band will be
a function of Kpr, and will have a maximum negative value at the best solution. A third test
– 14 –
is to measure the strength of a narrow emission arising from the secondary. This emission,
if present, would be particularly visible in the co-phased spectrum and will have minimum
width and maximum height at the best selected semi-amplitude Kpr.
We have tested these three methods by means of an artificial spectrum with simulated
narrow absorption lines, a TiO-like head band and a narrow emission line. The spectrum with
these artificial features was then Doppler shifted using pre-established inferior conjunction
phase and orbital period, to produce a series of test spectra. An amount of random Gaussian
noise was added to each Doppler shifted spectrum, sufficient to mask the artificial features.
We then proceeded to apply the co-phasing process to recover our pre-determined orbital
values. All three criteria reproduced back the original set of values, as long as the random
noise amplitude was of the same order of magnitude as the strength of the clean artificial
features.
3.3.2. Determination of K2 for U Gem
We have applied the above-mentioned criteria to U Gem. The time of the inferior
conjunction of the secondary and the orbital period were taken from section 4. To attain the
best signal to noise ratio we have used all the 28 observed spectra. Although they span over
slightly more than 1.5 orbital periods, any departure from a real K2 value will not depend
on selecting data in exact multiples of the orbital period, as any possible deviation from the
real semi-amplitude will already be present in one complete orbital period and will depend
mainly on the intrinsic intensity distribution of the selected feature around the secondary
itself (also see below the results for γ).
Figure 7 shows the application of the first test to the NaI doublet λλ 8183,8195 Å. The
spectra were co-phased varying Kpr between 250 to 450 km s
−1. The line depth of the blue
and red components of the doublet (stars and open circles, respectively), as well as their mean
value (dots) are shown in the diagram. We find a best solution for K2 = 310 ± 5 km s
The error has been estimated from the intrinsic modulation of the solution curve. As it
approaches its maximum value, the line depth value oscillates slightly, but in the same way
for both lines. A similar behavior was present when low signal to noise features were used
on the artificial spectra process described above. Figure 8 shows the co-phased spectrum
of the NaI doublet of our best solution for K2. These lines appear very weak as compared
with those reported by Friend et al. (1990) and Naylor et al. (2005). We have also measured
the gamma velocity from the co-phased spectrum by fitting a double-gaussian to the Na I
doublet (dotted line in Figure 7) and find a mean value γ = 69± 10 km s−1 (corrected to
the heliocentric standard of motion). We did a similar calculation for γ by co-phasing the
– 15 –
selected spectra used in section 5, covering a full cycle only. The results were very similar
to those obtained by using all spectra.
The second test, to measure the slope of the TiO band has at λ7050 Å was not suc-
cessful. The solution curve oscillates strongly near values between 250 and 350 km s−1. We
believe that the signal to noise ratio in our spectra is too poor for this test and that more ob-
servations, accumulated during several orbital cycles, have to be obtained in order to attain
a reliable result using this method.
However, we have co-phased our spectra for K2 = 310 km s
−1, with the results shown in
Figure 9. The TiO band is clearly seen while the noise is prominent, particulary along the
slope of the head-band. We have used this co-added spectrum to compare it with several
late-type M stars extracted from the published data by Montes et al. (1997) fitted to our
co-phased spectrum. A gray continuum has been added to the comparison spectra in order
to compensate for the fill-in effect arising from the other light sources in the system, so as
to obtain the best fit. In particular, we show in the same figure the fits when two close
candidates – GJ406 (M6 V, upper panel) and GJ402 (M4-5 V, lower panel) – are used. The
best fit is obtained for the M6 V star, to which we have added a 95 percent continuum. For
the M4-5 V star the fit is poor, as we observe a flux excess around 7000 Å and a stronger
TiO head-band. Increasing the grey flux contribution will fit the TiO head band, but will
result in a larger excess at the 7000 Å region. On the other hand, the fit with the M6 V
star is much better all along the spectral interval. There are a number of publications which
assign to U Gem spectral types M4 (Harrison et al. 2000), M5 (Wade 1981) and possibly as
far as M5.5 (Berriman et al. 1983). Even in the case that the spectral type of the secondary
star were variable, its spectral classification is still incompatible with its mass determination
(Echevarŕıa 1983).
For the third test, we have selected the region around Hα, as in the individual spectra
we see evidence of a narrow spot, which is very well defined in our spectrum near orbital
phase 0.5. In this test we have co-phased the spectra as before, and have adopted as the test
parameter the peak intensity around the emission line. The results are shown in Figure 10. A
clear and smooth maximum is obtained for Kpr = 310 ± 3 km s
−1. The co-phased spectrum
obtained from this solution is shown in Figure 11. The double-peak structure has been
completely smeared – as expected when co-adding in the reference frame of the secondary
star, as opposed to that of the primary star- and instead we observe a narrow and strong
peak at the center of the line. We have also fitted the peak to find the radial velocity of the
spot. We find γ = 33± 10 km s−1, compatible with the gamma velocity derived from the
radial velocity analysis of the emission line, γ = 34± 2 km s−1 (see section 3.2.3). This is a
key result for the determination of the true systemic velocity and can be compared with the
– 16 –
values derived from the secondary star (see section 7).
4. Improved Ephemeris of U Gem
As mentioned in section 3.1, the presence of eclipses in U Gem and an ample photo-
metric coverage during 30 years has permitted to establish, with a high degree of accuracy,
the value of orbital period. This has been discussed in detail by Marsh et al. (1990). How-
ever, as pointed by these authors, this object shows erratic variations in the timing of the
photometric mid-eclipse that may be caused either by orbital period changes, variations in
the position of the hot spot, or they may even be the consequence of the different methods
of measuring of the eclipse phases. A variation in position and intensity of the gas stream
will also contribute to such changes. A date for the zero phase determined independently
from spectroscopic measurements would evidently be desirable. Marsh et al. (1990) discuss
two spectroscopic measurements by Marsh & Horne (1988) and Wade (1981), and conclude
that the spectroscopic inferior conjunction of the secondary star occurs about 0.016 in phase
prior to the mean photometric zero phase. There are two published spectroscopic studies
(Honeycutt et al et al. 1987; Stover 1981), as well as one in this paper, that could be used
to confirm this result. Unfortunately there is no radial velocity analysis in the former paper,
nor in the excellent Doppler Imaging paper by Marsh et al. (1990) based on their original
observations. However, the results by Stover (1981) are of particular interest since he finds
the spectroscopic conjunction in agreement with the time of the eclipse when using the pho-
tometric ephemerides by Wade (1981), taken from Arnold et al. (1976). The latter authors
introduce a small quadratic term which is consistent with the O-C oscillations shown in
Marsh et al. (1990).
It is difficult to compare results derived from emission lines to those obtained from
absorption lines, especially if they are based on different ephemerides. Furthermore, the
contamination on the timing of the spectroscopic conjunction – either caused by a hot spot,
by gas stream or by irradiation on the secondary – has not been properly evaluated. However,
since our observations were made at a time when the hot spot in absent (or, at least, is along
the line between the two components in the binary) and the disc was very symmetric (see
section 5), we can safely assume that in our case, the photometric and spectroscopic phases
must coincide. If we then take the orbital period derived by Marsh et al. (1990) and use the
zero point value derived from our measurements of the Hα wings, (section 3.2.3), we can
improve the ephemeris:
HJD = 2, 437, 638.82566(4) + 0.1769061911(28) E ,
– 17 –
for the inferior conjunction of the secondary star. These ephemeris are used throughout
this paper for all our phase folded diagrams and Doppler Tomography.
5. Doppler Tomography
Doppler Tomography is a useful and powerful tool to study the material orbiting the
white dwarf, including the gas stream coming from the secondary star as well as emission
regions arising from the companion itself. It uses the emission line profiles observed as a
function of the orbital phase to reconstruct a two-dimensional velocity map of the emitting
material. A detailed formulation of this technique can be found in Marsh & Horne (1988).
A careful interpretation of these velocity maps has to be made, as the main assumption
invoked by tomography is that all the observed material is in the orbital plane and is visible
at all times.
The Doppler Tomography, derived here from the Hα emission line in U Gem, was
constructed using the code developed by Spruit (1998). Our observations of the object
cover 1.5 orbital cycles. Consequently – to avoid disparities on the intensity of the trailed
and reconstructed spectra, as well as on the tomographic map – we have carefully selected
spectra covering a full cycle only. For this purpose we discarded the first 3 spectra (which
have the largest airmass) and used only 18 spectra out of the first 21, 600 s exposures, starting
with the spectrum at orbital phase 0.88 and ending with the one at phase 0.86 (see Table 1).
In addition, in generating the Tomography map we have excluded the spectra taken during
the partial eclipse of the accretion disc (phases between 0.95 and 0.05). The original and
reconstructed trailed spectra are shown in Figure 12. They show the sinusoidal variation
of the blue and read peaks, which are strong at all phases. The typical S-wave is also seen
showing the same simple sinusoidal variation, but shifted by 0.5 in orbital phase with respect
to the double-peaks. The Doppler tomogram is shown in Figure 13; as customary, the oval
represents the Roche-Lobe of the secondary and the solid lines the Keplerian (upper) and
ballistic (lower) trajectories. The Tomogram reveals a disc reaching to the distance of to the
inner Lagrangian point in most phases. A compact and strong emission is seen close to the
center of velocities of the secondary star. A blow-up of this region is shown in Figure 14.
Both maps have been constructed using the parameters shown at the top of the diagrams
and a γ velocity of 34 km s−1. The velocity resolution of the map near the secondary star is
about 10 km s−1. The V (x, y) position of the hot-spot (in km s−1) is (-50,305), within the
uncertainties.
The tomography shown in Figure 13 is very different from what we expected to find
and from what has been observed by other authors. We find a very symmetric full disc,
– 18 –
reaching close to the inner Lagrangian point and a compact bright spot also close to the L1
point, instead of a complex system like that observed by Unda-Sanzana et al. (2006), who
find U Gem at a stage when the Doppler Tomographs show: emission at low velocity close
to the center of mass; a transient narrow absorption in the Balmer lines; as well as two
distinct spots, one very narrow and close in velocity to the accretion disc near the impact
region and another much broader, located between the ballistic and Keplerian trajectories.
They present also tentative evidence of a weak spiral structure, which have been seen as
strong spiral shocks during an outburst observed by Groot (1991). Our results also differ
from those of Marsh et al. (1990), who also find that the bulk of the bright spot arising from
the Balmer, He I and He II emission come from a region between the ballistic and Keplerian
trajectories. We interpret the difference between our results and previous studies simply by
the fact that we have observed the system at a peculiar low state not detected before (see
sections 1 and 7) . This should not be at all surprising because, although U Gem is a well
observed object, it is also a very unusual and variable system.
Figure 14 shows a blow-up of the region around the secondary star. The bright spot is
shown close to the center of mass of the late-type star, slightly located towards the leading
hemisphere. Since this is a velocity map and not a geometrical one, there are at two possible
interpretations of the position in space of the bright spot (assuming the observed material is
in the orbital plane). The first one is that the emission is been produced at the surface of the
secondary, i.e. still attached to its gravitational field. The second is that the emission is the
result of a direct shock front with the accretion disc and that the compact spot is starting
to gain velocity towards the Keplerian trajectory. We believe that the second explanation
is more plausible, as it is consistent with the well accepted mechanism to produce a bright
spot. On the other hand, at this peculiar low state it is difficult to invoke an external source
strong enough to produce a back-illuminated secondary and especially a bright and compact
spot on its leading hemisphere.
6. Basic system parameters
Assuming that the radial velocity semi-amplitudes reflect accurately the motion of the
binary components, then from our results –Kem = K1 = 107±2 km s
−1; Kabs = K2 = 310±5
km s−1 – and adopting P = 0.1769061911 we obtain:
= 0.35± 0.05,
– 19 –
M1 sin
3 i =
PK2(K1 +K2)
= 0.99± 0.03M⊙,
M2 sin
3 i =
PK1(K1 +K2)
= 0.35± 0.02M⊙,
a sin i =
P (K1 +K2)
= 1.46± 0.02R⊙.
Using the inclination angle derived by Zhang & Robinson (1987), i = 69.7◦ ± 0.7, the
system parameters become: MWD = 1.20 ± 0.05M⊙; MRD = 0.42 ± 0.04M⊙; and a =
1.55± 0.02R⊙.
6.1. The inner and outer size of the disc
A first order estimate of the dimensions of the disc – the inner and outer radius – can
be made from the observed Balmer emission line. Its peak-to peak velocity separation is
related to the outer radius of the accreted material, while the wings of the line, coming
from the high velocity regions of the disc, can give an estimate of the inner radius (Smak
2001). The peak-to-peak velocity separation of the 31 individual spectra were measured (see
section 3.2.1), as well as the velocity of the blue and red wings of Hα at ten percent level of
the continuum level. ¿From these measurements we derive mean values of Vout = 460 km s
and Vin = 1200 km s
These velocities can be related to the disc radii from numerical disc simulations, tidal
limitations and analytical approximations (see Warner (1995) and references therein). If we
assume the material in the disc at radius r is moving with Keplerian rotational velocity V (r),
then the radius in units of the binary separation is given by (Horne, Wade & Szkody 1986):
r/a = (Kem +Kabs)Kabs/V (r)
The observed maximum intensity of the double-peak emission in Keplerian discs occurs
close to the velocity of its outer radius (Smak 1981). From the observed Vout and Vin values
we obtain an outer radius of Rout/a = 0.61 and an inner radius of Rin/a = 0.09. If we take
a = 1.55±0.02R⊙ from the last section we obtain an inner radius of the disc Rin = 0.1395R⊙
– 20 –
equivalent to about 97 000 km. This is about 25 times larger than the expected radius of the
white dwarf (see section 7). On the other hand, the distance from the center of the primary
to the inner Lagrangian point, RL1/a, is
RL1/a = 1− w + 1/3w
2 + 1/9W 3,
where w3 = q/(3(1 + q) ((Kopal 1959)). Using q = 0.35 we obtain RL1/a = 0.63. The
disc, therefore, appears to be large, almost filling the Roche-Lobe of the primary, with the
matter leaving the secondary component through the L1 point colliding with the disc directly
and producing the hot spot near this location.
7. Discussion
For the first time, a radial velocity semi-amplitude of the primary component of U Gem
has been obtained in the visual spectral region, which agrees with the value obtained from
ultraviolet observations by Sion et al. (1998) and Long & Gilliland (1999). In a recent paper,
Unda-Sanzana et al. (2006) present high-resolution spectroscopy around Hα and Hβ and
conclude that they cannot recover the ultraviolet value for K1 to better than about 20
percent by any method. Although the spectral resolution at Hα of the instrument they used
is only a factor of two smaller than that of the one we used, the diagnostic diagrams they
obtain show a completely different behavior as compared to those we present here, with
best values for K1 of about 95 km s
−1 from Hα and 150 km s−1 from Hβ (see their Figures
13 and 14, respectively). We believe that the disagreement with our result lies not in the
quality of the data or the measuring method, but in the distortion of the emission lines due
to the presence of a complex accretion disc at the time of their observations, as the authors
themselves suggest. Their Doppler tomograms show emission at low velocity, close to the
center of mass, two distinct spots, a narrow component close to the L1 point, and a broader
and larger one between the Keplerian and the ballistic trajectories. There is even evidence
of a weak spiral structure. In contrast, we have observed U Gem during a favorable stage,
one in which the disc was fully symmetric, and the hot-spot was narrow and near the inner
Lagrangian point. This allowed us to measure the real motion of the white dwarf by means
of the time-resolved behavior of the Hα emission line.
Our highly consistent results for the systemic velocity derived from the Hα spot (γ =
33± 10 km s−1 and those found from the different methods used for the radial velocity
analysis of the emission arising from the accretion disk (see section 3.2 and Table 2), give
strong support to our adopting a true systemic velocity value of γ = 34± 2 km s−1. If we are
– 21 –
indeed detecting the true motion of the white dwarf, we can use this adopted value, to make
an independent check on the mass of the primary: The observed total redshift of the white
dwarf (gravitational plus systemic)– found by Long & Gilliland (1999) – is 172 km s−1, from
which, after subtraction of the adopted systemic velocity, we derive a gravitational shift of
the white dwarf of 138 km s−1. From the mass-radius relationship for white dwarfs (Anderson
1988), we obtain consistent results for Mwd = 1.23M⊙ and Rwd = 3900 km (see Figure 7
in (Long & Gilliland 1999)). This mass is in excellent agreement with that obtained in this
paper from the radial velocity analysis.
¿From our new method to determine the radial velocity curve of the secondary (sec-
tion 3.3.2), we obtain a value for the semi-amplitude close to 310 km s−1. Three previous
papers have determinations of the radial velocity curves from the observed Na I doublet
in the near-infrared. In order to evaluate if our method is valid, we here compare our re-
sult with these direct determinations. The published values are: Krd = 283 km s
((Wade 1981)); Krd = 309 km s
±3, (before correction for irradiation effects, (Friend et al.
1990)); and Krd = 300 km s
−1 (Naylor et al. 2005). Wade (1981) notes that an elliptical
orbital (e = 0.086) may better fit his data, as the velocity extremum near phase 0.25 ap-
pears somewhat sharper than that near phase 0.75 (see his Figure 3). However, he also
finds a very large systemic velocity, γ = 85 km s−1, much larger than the values found by
Kraft (1962) (γ = 42 km s−1) and Smak (1976) (γ = 40± 6 km s−1), both obtained from
the emission lines. Since the discrepancy with the results of these two authors was large,
Wade (1981) defers this discussion to further confirmation of his results. Instead, and more
important, this author discusses two scenarios that may significantly alter the real value of
K2: the non-sphericity and the back-illumination of the secondary. In the latter effect, each
particular absorption line may move further away from, or closer to the center of mass of
the binary. He estimates the magnitude of this effect and concludes that the deviation of
the photocenter would probably be much less than 0.1 radii. Friend et al. (1990) further
discusses the circumstances that might cause the photocenter to deviate, and concludes that
their observed value for the semi-amplitude should be corrected down by 3.5 percent, to yield
K2 = 298 km s
± 9. Although they discuss the results by Martin (1988) – which indicate
that the relatively small heating effects in quiescent dwarf novae always lead to a decrease in
the measured Krd for the Na I lines – they argue that line quenching, produced by ionization
of the same lines, may also be important, and result in an increased Krd. Another disturbing
effect, considered by the same authors, is line contamination by the presence of weak disc
features, like the Paschen lines. In this respect we point out here that a poor correction for
telluric lines will function as an anchor, reducing also the amplitude of the radial velocity
measurements. Friend et al. (1990) also find an observed systemic velocity of γ = 43± 6
km s−1 and a small eccentricity of e = 0.027. Naylor et al. (2005) also discuss the distortion
– 22 –
effects on the Na I lines and, based on their fit residuals, argue in favor of a depletion of the
doublet in the leading hemisphere of the secondary, around phases 0.4 and 0.6, as removing
flux from the blueward wing of the lines results in an apparent redshift, which would explain
the observed residuals. However, they additionally find that fitting the data to an eccentric
orbit, with e = 0.024, results in a significant decrease in the residuals caused by this deple-
tion, and conclude that it may be unnecessary to further correct the radial velocity curve.
We must point out that a depletion of the blueward wing of the Na I lines will results in
a contraction of the observed radial velocity curves, as the measured velocities – especially
around phases 0.25 and 0.75 – will be pulled towards the systemic velocity. Naylor et al.
(2005) present their results derived from the Na I doublet and the K I/TiO region (around
7550-7750 Å), compared with several spectral standards, all giving values between 289 and
305 km s−1 (no errors are quoted). Based on the radial velocity measurements for Na I,
obtained by these authors in 2001 January (115 spectra), and using GJ213 as template (see
their Table 1), we have recalculated the circular orbital parameters through our nonlinear
least-squares fit. We find K2 = 300 km s
± 1, in close agreement with their published
value.
It would be advisable to establish a link between the observed gamma velocity of the
secondary and the semi-amplitude K2, under the assumption that its value may be distorted
by heating effects. We take as a reference our results from the radial velocity analysis of the
broad Hα line and the hot-spot from the secondary, which support a true systemic velocity
of 34 km s−1. However, we find no positive correlation in the available results derived from
the Na I lines, either between different authors or even among one data set. In the case of
Naylor et al. (2005), the gamma values show a range between 11 and 43 km s−1, depending on
the standard star used as a template, for K2 velocities in the range 289 to 305 km s
−1. Wade
(1981) finds γ = 85± 10 km s−1 for a low K2) value of 283 km s
−1, while Friend et al. (1990)
finds γ = 43± 6 km s−1 for K2 about 309 km s
−1, and we obtain a large gamma velocity of
about 69 km s−1 for a K2 value of 310 km s
−1. We believe that further and more specific
spectroscopic observations of the secondary star should be conducted in order to understand
the possible distortion effects on lines like the Na I doublet, and their implications on the
derived semi-amplitude and systemic velocity values.
Acknowledgments
E. de la F wishes to thank Andrés Rodriguez J. for his useful computer help. The
Thomson detector, used in our observations, was obtained through PACIME-CONACYT
project F325-E9211.
– 23 –
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– 25 –
Table 2: Orbital parameters derived from several radial velocities calculations of the Hα
emission line.
Orbital Peaks (a) Fxc (b) Wings (c)
Parameters
γ (km s−1) 38 ± 5 35 ± 3 34 ± 2
K (km s−1) 162 ± 7 119 ± 3 107 ± 2
HJD⊙ 0.8259(2) 0.82462(6) 0.82152(9)
(+2437638 days)
Porb (days) (d) (d) (d)
σ 25.2 12.2 9.1
aDerived from measurements of the double-peaks
bDerived from cross correlation methods
cResults from the fitting of fixed double gaussians to the wings
dPeriod fixed, P=0.1769061911 d
– 26 –
Fig. 4.— Diagnostic Diagram One. Orbital Parameters as a function of width of individual Gaussians for
several separations. Crosses correspond to a = 180 pixels; Dots to a = 230 pixels (≈ 34 Å) and Open circles
to a = 280 pixels;
– 27 –
Fig. 5.— Diagnostic Diagram Two. The best estimate of the semi-amplitude of the white
dwarf is 107 km s−1, corresponding to a ≈ 34 Å.
– 28 –
0 0.5 1 1.5 2
Fig. 6.— Radial velocities for U Gem. The open circles correspond to the measurements
of the first 21 spectra single spectra, while the dots correspond to those of the co-added
spectra (see section 3.2). The solid line, close to the points, correspond to the solution with
Kem = 107 km s
−1 (see text), while the large amplitude line correspond to the solution found
for K2 (see section 3.3.1).
– 29 –
Fig. 7.— Maximum flux depth of the individual NaI lines λ8183.3 Å (top), λ8194.8 Å (bot-
tom) and mean (middle) as a function of Kpr.
– 30 –
Fig. 8.— Co-phased spectrum around the NaI doublet.
– 31 –
Fig. 9.— U Gem TiO Head Band near 7050 Å compared with GJ406, an M6V star (upper
diagram), and GJ402, an M4 V star (lower diagram) (see text).
– 32 –
Fig. 10.— Maximum peak flux of the co-added Hα spectra as a function of Kpr
– 33 –
Fig. 11.— Shape of the co-added Hα spectrum for K2 = 310 km s
– 34 –
Fig. 12.— Trailed spectra of the Hα emission line. Original (left) and reconstructed data
(right).
– 35 –
Fig. 13.— Doppler Tomography of U Gem. The various features are discussed in the text.
The vx and vy axes are in km s
−1. A compact hot spot, close to the inner Lagrangian point is
detected instead of the usual bright spot and/or broad stream, where the material, following
a Keplerian or ballistic trajectory strikes the disc. The Tomogram reveals a full disc whose
outer edge is very close to the L1 point (see text).
– 36 –
Fig. 14.— Blow-up of the region around the hot spot. Note that this feature is slightly
ahead of the center of mass of the secondary star. Since this is a velocity map and not a
geometrical one, its physical position in the binary is carefully discussed in the text.
	Introduction
	Observations
	Radial Velocities
	Orbital Parameters Calculations
	The Primary Star
	Measurements from the double-peaks
	Cross Correlation using a Template
	Measurements from the wings and Diagnostic Diagrams
	A new method to determine K2
	Determination of K2 for U Gem
	Improved Ephemeris of U Gem
	Doppler Tomography
	Discussion
