3 Orbital and physical parameters of the components

3.1 Radial velocities and orbital parameters

Radial velocity determinations were made by cross-correlation of each order of the HR 7428 spectra with spectra of the bright radial velocity standard star $\alpha$ Ari, whose radial velocity is -14.3 kms^-1 (Evans 1979).

Template spectra of $\alpha$ Ari have been obtained during each observing run. From several spectra of $\alpha$ Ari and other RV standard stars we have evaluated an average accuracy for the radial velocity measurements better than $\pm$0.4 kms^-1 rms which is the result of the stability of the experimental apparatus and is consistent with the limits given by the spectral resolution.

The wavelength ranges for the cross-correlation of HR 7428 spectra were selected in order to exclude the H$\alpha$ and Na I D₂ lines, which are contaminated by chromospheric emission. The spectral regions heavily affected by telluric lines (e.g. the $\lambda~6276{-}\lambda~ 6315$ band of O₂) were also excluded.

Each radial velocity value listed in Table 1 is the weighted mean of the values obtained by the cross-correlation of each order of the target spectra with the corresponding order of the closest-in-time spectrum of $\alpha$ Ari. The errors, listed in Table 1, have been computed as the square root of the quadratic average of errors in the cross correlation of each order. The latter are computed according to the fitted peak height and the antisymmetric noise as described by Tonry & Davis (1979). This error estimate is consistent with the dispersion of data points around the mean curve.

 

 

Table 1: Radial velocities and H$\alpha$ equivalent widths of HR 7428.

HJD	Phase	RV	error	$EW^{\rm abs}_{\rm H\alpha}$	error	$EW^{\rm net}_{\rm H\alpha}$	error
- 2400000		(km$\,$s-1)		(Å)		(Å)
50631.5584	0.2724	17.9	1.0	-1.463	0.108	0.112	0.119
50632.5635	0.2817	19.3	0.4	-1.674	0.097	-0.255	0.107
50633.5440	0.2907	18.3	1.0	-1.686	0.181	-0.228	0.200
50634.5728	0.3002	15.7	0.5	-1.940	0.120	-0.558	0.132
50635.5511	0.3092	16.0	1.4	--	--	--	--
50636.5153	0.3181	15.9	2.8	-1.921	0.076	-0.432	0.084
50678.4853	0.7046	-29.8	0.7	-0.656	0.102	0.757	0.112
50679.4757	0.7137	-30.2	0.3	-0.784	0.071	0.676	0.078
50680.4779	0.7229	-30.0	1.3	-0.759	0.124	0.655	0.137
50681.5193	0.7325	-30.5	0.9	-0.905	0.097	0.481	0.107
50982.5670	0.5052	-7.2	0.7	-0.813	0.120	0.510	0.132
50992.5589	0.5972	-20.0	1.3	-0.918	0.128	0.473	0.141
51000.5715	0.6710	-27.8	0.5	-1.146	0.121	0.277	0.133
51002.5660	0.6894	-27.4	0.1	-1.113	0.113	0.284	0.124
51288.5618	0.3234	15.8	0.7	--a		--
51332.5522	0.7285	-30.1	0.4	-1.316	0.127	0.068	0.140
51381.4301	0.1787	14.9	0.5	-0.925	0.117	0.610	0.129
51382.4819	0.1884	16.1	0.6	-1.019	0.096	0.375	0.105
51383.3908	0.1968	16.2	0.6	-1.186	0.104	0.287	0.114
51385.4111	0.2154	17.6	0.5	-1.342	0.082	0.208	0.090
51386.3445	0.2240	18.0	0.5	-1.353	0.104	0.158	0.115
51387.3975	0.2337	17.7	0.5	-1.285	0.063	0.135	0.069
51387.4260	0.2339	17.7	0.5	-1.325	0.056	0.134	0.061
51388.3746	0.2427	18.4	0.5	-1.380	0.075	0.089	0.082
51407.6048	0.4198	4.0	0.6	-1.316	0.078	0.113	0.086
51417.5704	0.5115	-7.6	0.5	-0.475	0.154	0.876	0.169
51423.4962	0.5661	-16.5	0.4	-0.376	0.135	1.206	0.149
51426.4815	0.5936	-19.5	0.3	-0.750	0.087	0.607	0.096
51427.4105	0.6022	-20.9	0.4	-0.903	0.100	0.433	0.110
51465.4373	0.9524	-12.7	0.7	-1.009	0.105	0.542	0.116
51473.4580	0.0263	-3.5	0.7	-1.324	0.101	0.215	0.111
51474.2698	0.0337	-1.3	0.7	-1.387	0.076	0.153	0.083
51476.2680	0.0521	0.4	0.6	-1.342	0.069	0.160	0.076
51477.3756	0.0576	--	--	-1.259	0.067	0.207	0.074

^aCa  II H & K.

Hall et al. (1990a), from Sanford's (1925) data set, re-determined the elements for a circular orbit and derived JD $_{0}=2423455.2 \pm 0.20$ as the time of conjunction (primary in front). From their own light curve (dominated by ellipticity effect) they deduced a time of conjunction JD $_{0}=2445988.0 \pm 0.3$. Combining the photometric and spectroscopic epoch of conjunction, Hall et al. (1990a) derived a period of 108.854 days. This value is slightly different from that given by Sanford (1925) of 108.57 days but very similar to the one they deduced from Fourier analysis of V band photometry (108.83 days).

If we use the above-mentioned ephemeris of Hall et al. (1990a) to fold in phase our radial velocities, the time of conjunction (primary star in front) occurs at phase $\phi = 0.64$, indicating that the period and/or the time of conjunction are not correct.

Figure 1: The cleaned periodogram of our and literature RV data. The peak corresponds to a period of 108.578 days.

Radial velocities available from the literature (excluding the poor-quality five measures of Xue-fu & Hui-song 1986a) and our own data have been analysed applying the periodogram technique (Scargle 1982). The CLEAN iterative deconvolution algorithm (Roberts et al. 1986) has been used to eliminate the effects of the data sampling introduced by the observation spectral window in the power spectrum. The maximum of the power spectrum, plotted in Fig. 1, yields a period of 108.578 days, very close to Sanford's estimate. The new ephemeris we obtain is

$$T_{\rm conj} = 2450601.982 + 108.578\,n,$$ (1)

where the epoch refers to the conjunction with the primary star in front.

The observed RV values are displayed as a function of the orbital phase, determined from our ephemeris (Eq. (1)), in Fig. 2. Adopting our conjunction time, the epoch of Hall et al. (1990a), ${\rm JD} = 2445988.0$, appears to be shifted by half a period, i.e. this light-curve minimum epoch corresponds to the conjunction with the hot secondary star in front instead of the cool primary star, as they assumed. The epoch of the deeper minimum observed by Barksdale et al. (1985) is in good agreement with our ephemeris.

Following Lucy & Sweeney (1971), we have assumed circular orbits and deduced new orbital parameters from our own radial velocities. The best fit orbital solution for the cooler component is reported in Table 2.

Figure 2: Radial velocities of HR 7428 and circular solution. Our own data are displayed with filled circles. Other symbols represent literature RV data, not used for the orbital solution. Two radial velocity measurements for the hot star, from our own blue-violet spectrum ($\phi =0.32$) and from an IUE high resolution spectrum ($\phi =0.67$), are also shown together with a tentative orbital solution for this component.

 

 

Table 2: Orbital parameters for the cooler component of HR 7428.

Element	Present solution	Sanford solution
$K_{\rm c}$	$23.71 \pm 0.10$ kms-1	22.1 kms-1
$\gamma_{\rm c}$	$-5.85 \pm 0.09$ kms-1	-5.2 kms-1
$T_{\rm conj}$	$2450601.982 \pm 0.138$	2423455.21
P	$108.578 \pm 0.005$ days	108.5707 days
$a_{\rm c}\sin i$	$3.54 \times 10^{7} \pm 0.02 \times 10^{7}$ km	$3.304 \times 10^{7}$ km
$f(m)_{\rm c}$	$0.1500 \pm 0.0025~M_{\odot}$	0.1222 $M_{\odot}$

¹ Deduced from the time of periastron passage.

The contribution of the hotter component to the observed spectrum becomes stronger at shorter wavelengths. With the aim of detecting the spectrum of the hot component we have taken an exposure of HR 7428 in the blue-violet region with our spectrograph, equipped with the thinned back-illuminated UV-enhanced SITe CCD. We recorded 11 echelle orders with a complete wavelength coverage from 4550 Å to 3700 Å. Spectra of $\alpha$ Boo and $\alpha$ Lyr (which mimic the cool and hot component of HR 7428, respectively) were obtained on the same night and with the same instrumental setting. Our purpose was to obtain the hot star spectrum by removing, from the observed spectrum of HR 7428, the weighted contribution of the cooler component simulated by the rotationally broadened spectrum of $\alpha$ Boo.

Due to the low S/N ratio of HR 7428 spectrum at these wavelengths, the residual spectrum is rather noisy, but clearly shows the typical signatures of a hot star, like the strong and wide Balmer lines.

Figure 3: Residual spectrum of the hot component of HR 7428 in the violet region, obtained as described in the text. The spectrum of Vega shifted downward by 0.2 units is displayed for comparison. The three hydrogen Balmer lines H9, H10 and H11 are clearly visible.

The residual spectrum in the wavelength region containing the three strong Balmer lines H₉, H₁₀ and H₁₁ ( $\lambda\lambda 3835, 3798, 3771$) is shown in Fig. 3, superimposed on the spectrum of Vega obtained with the same instrument.

We have cross-correlated the orders of the residual spectrum containing the strongest lines with the corresponding ones of Vega to find the radial velocity of the hotter component at this phase ( $\phi = 0.322$). The evaluated radial velocity (- $58~\rm km\,s^{-1}$) has a rather large error ($\pm$ $14~\rm km\,s^{-1}$) because of the noise and of the large width of the hydrogen lines.

Another possibility for measuring the radial velocity of the hot component is to look for spectral signatures of the A2 star in high resolution IUE spectra. Unfortunately, no high resolution spectrum at short wavelengths (where the A2 star flux overcomes that of the cooler companion) is available. However, we could use the well exposed part of the long-wavelength region of spectrum LWR10313, acquired with IUE on 1981 April 8th ( $\phi =0.67$) and retrieved from the IUE Final Archive. We have also retrieved the spectrum LWR09830 of Vega, acquired nearly at the same time.

By cross-correlation of each order of the HR 7428 spectrum with the corresponding one of Vega, we measured the radial velocity of the A2 star, obtaining + $45 \pm 19~ \rm km\,s^{-1}$. The large error is essentially due to the low S/N ratio.

With these two RV values, we are able to give, for the first time, a tentative orbital solution for the hot component.

The orbital parameters of the hot component, including the values of $m\sin^{3}i$ we derived, are reported in Table 3.

 

 

Table 3: Orbital parameters of the hot component of HR 7428.

Element	Present solution
$K_{\rm h}$	$58 \pm 8$ kms-1
$a_{\rm h}\sin i$	$8.6 \times 10^{7} \pm 1.2 \times 10^{7}$ km
$m_{\rm h}\sin^{3}i$	$1.8 \pm 0.4$ $M_{\odot}$
$m_{\rm c}\sin^{3}i$	$4.4 \pm 1.5$ $M_{\odot}$
$m_{\rm c}/m_{\rm h}$	$2.40 \pm 0.06$

3.2 Physical parameters

Physical parameters of the components of HR 7428 are not yet well known. As a matter of fact the radius of 62 $R{_\odot }$ found by Hall (1990) for the cool primary component leads to a distance of about 600 pc (Rodonò et al. 1998) which is at odds with the value of 302 pc given by Charles (1994) and with that measured by Hipparcos spacecraft ( $d = 323 \pm 52$ pc).

Difficulties in accounting for optical and infrared magnitudes adopting stellar parameters reported in the literature were encountered by comparing black body emission with observed photometric fluxes (Rodonò et al. 1998).

As a first step, we have checked the consistency of magnitudes and colours with the radius of the cool component reported in the literature, by means of the Barnes-Evans relation (Barnes et al. 1976) between (V-R) colour index, V magnitude and angular diameter $\phi$ for late type stars:

$$\log\phi = 0.4874+0.858 (V{-}R) -0.2 V.$$ (2)

The (V-R) colour index is more appropriate than B-V because it is practically not affected by the light contribution from the hot star. Adopting (V-R) = 0.95 (Fernie 1983) and V = 6.371 (Rodonò et al. 1998) we obtain $\phi = 1.067~$milliarcsec and, from the Hipparcos distance, we find for the cool component radius, $R_{\rm cool}=37~ R{_\odot}$, which is significantly smaller than the value reported by Hall (1990), who had used a larger value of $v\sin i$.

We made a more accurate determination of physical parameters by using UBV magnitudes given in the literature and low resolution IUE spectra, whose spectral region is essentially dominated by the hot component, comparing them with synthetic theoretical spectra.

We defined an average observed UV spectrum using the available highest S/N ratio spectra, namely LWP 27954 and LWP 30925 for the long wavelength region (1850-3350 Å) and SWP 50606 and SWP 50609 for the short wavelength region (1150-2000 Å), retrieved from the IUE Final Archive (Nichols & Linsky 1996).

IUE spectra, as well as optical photometry, have been corrected for interstellar extinction. The typical value of 1 magnitude per kilo-parsec for the interstellar extinction and the Hipparcos distance, adopted for the computations, lead to an extinction magnitude A(V $) = 0\hbox{$.\!\!^{\rm m}$ }$32. Assuming the standard reddening law A(V $)=3.1\times E$(B-V), a colour excess E(B- $V)= 0\hbox{$.\!\!^{\rm m}$ }$10 has been derived. IUE spectra have been de-reddened according to the selective extinction function of Cardelli et al. (1989).

Synthetic spectra have been generated with the NextGen (Hauschildt et al. 1999a, 1999b) photosphere models for the hotter secondary and cool primary star. The bright cool star model was computed for this purpose by Dr. Peter H. Hauschildt, including a spherical symmetry solution (Hauschildt et al. 1999b).

Consistent with the previous estimates of spectral type and with our estimate of the mass, several models between $T_{\rm eff}=8400$ K and 9400 K were considered for the hot component. To take into account the uncertainties in the relation between spectral type and effective temperature of the cooler component, initially we considered models with $T_{\rm eff}= 4200$ K, 4400 K, and 4600 K. The first model turned out to be cooler and the last hotter for our observational data. Finally, in agreement with the accurate spectral type determination by Ginestet et al. (1999), two models of $T_{\rm eff}=4400$ K, with $\log g=1.5$, [M/H]=0, $M=5~M{_\odot}$ and with $\log g=2$, [M/H]=0, $M=4~M{_\odot}$ respectively, were considered.

Since synthetic spectra give fluxes at the stellar surface, the comparison of models with observations has been made by scaling the result with the angular radii of the components and the distance.

Several iterations have been performed to find the model that gives the better fit to the IUE spectrum and optical photometry, allowing the stellar parameters to vary independently. However, some constraints have been placed and tests for consistency have been made.

The distance was fixed to the Hipparcos value and the effective temperature adopted for the cool component was checked to be consistent with the spectral type classification and spectral line behaviour.

The temperature of the hotter component was mainly constrained by the shape of the UV spectral region. The temperature and the radius of the cooler component are instead mainly linked to the near-UV and optical spectral distribution.

The errors in the effective temperatures have been estimated as the minimum $\Delta T_{\rm eff}$ value that gives appreciable difference with respect to the observed spectrum. Due to the evident mismatch in the flux distribution of the two models with $T_{\rm eff}= 4200$ and 4600 K with respect to the observed one, we can estimate an uncertainty in the temperature of the cooler component, smaller than 200 K, probably of 100-150 K. For the hot component, having a lower weight in the composite spectrum, an error $\Delta T_{\rm eff} \approx 200$ K is deduced from the UV spectral region.

To evaluate the uncertainties in the radii we have considered the two main sources of errors: the error in Hipparcos distance ($\pm$52 pc) and the accuracy of the correction for interstellar extinction. The accuracy of the de-reddening procedure is affected by both the intrinsic uncertainties in the E(B-V) value and the errors in distance. We have evaluated the first one by assuming the standard deviation on colour excesses equal to the variance of the coefficients given by Johnson (1965) for different sky regions. We find that the dominant source of errors is the distance uncertainty.

The final best-fit stellar parameters are listed in Table 4 together with the estimated errors.

The errors in the parameters, listed in Table 4, do not include any intrinsic uncertainty for the synthetic spectra. However, Hauschildt et al. (1999a), from a comparison of their models with the observed UV and optical spectra of solar type stars and Vega, found discrepancies always smaller than a few percent in the continuum. This source of error should be thus negligible in comparison with the accuracy of the data themselves.

In Fig. 4 the spectrum obtained by summing the synthetic spectra for the hot and cool component, degraded to the IUE spectral resolution, is plotted superimposed on the IUE de-reddened spectrum (left) and on the UBV fluxes (right). In the latter case the synthetic spectrum was convolved with a Gaussian kernel having a semi-amplitude of 30 Å. The models displayed have $T_{\rm eff}=9000$ K, $\log g=4.0$ and $T_{\rm eff}=4400$ K, $\log g=2$ for hotter and cooler components, respectively. The radii which give the best fit to the observed spectrum are 2.25 $R{_\odot }$ and 40 $R{_\odot }$, respectively.

Figure 4: Comparison of the photospheric models of the two components of HR 7428 with the IUE low resolution spectra ( left) and UBV fluxes ( right) at Earth. Both IUE spectra and optical photometry have been de-reddened with E(B-V) = 0.10. A distance of 322.6 pc, and radii of 2.25 and 40 $R{_\odot }$ for the hotter and cooler components respectively, are considered.

 

 

Table 4: Physical parameters of HR 7428.

Element	Primary (cooler)	Secondary (hotter)
R	$40.0 \pm 6.5$ $R{_\odot }$	$2.25 \pm 0.50$ $R{_\odot }$
$T_{\rm eff}$	$4400~{\rm K} \pm 150$ K	$9000 \pm 200$ K
$\log g$	$2.0 \pm 0.5$	$4.0 \pm 0.5$

3.3 Rotational velocity and system inclination

The only measurements of the projected rotational velocity of HR 7428 that we found in the literature are $v\sin i = 21$ km$\,$s^-1 reported by Hui-song & Xue-fu (1987) and $v\sin i = 17.2$ km$\,$s^-1 measured by De Medeiros & Mayor (1995) with CORAVEL. Notwithstanding the moderate resolution, we have performed $v\sin i$ estimates on our spectra, taking advantage of the large number of observations and of the high S/N ratio. We have compared the echelle orders of HR 7428 centered at 6250 and 6400 Å with those of several slowly-rotating K-type giant stars, acquired with the same instrumentation. The standard stars that give the best reproduction of the HR 7428 spectrum are $\beta$ Oph (K2 III) and $\alpha$ Ari (K2 III), whose $v\sin i$'s are negligible in comparison to the instrumental resolution.

We have set up a procedure for the $v\sin i$ determination. First we aligned in wavelength the spectra of the active and standard star by means of cross-correlation, then the spectrum of the standard star was progressively broadened by convolution with a rotational profile of increasing $v\sin i$. At each step, the sum of residuals between the observed and template spectrum was calculated and, at the end, the minimum of the residuals was found. The $v\sin i$ value that we deduced by analysing several HR 7428 spectra is $17.4 \pm 1.6$ km$\,$s^-1, in very good agreement with the CORAVEL measurement (De Medeiros & Mayor 1995).

The projected rotational velocity is given by

$$v\sin i = 2\pi R\sin i/P,$$ (3)

where $P=108\hbox{$.\!\!^{\rm d}$ }8$ is the rotational period of the primary star from photometry (Hall et al. 1990a), and R is the radius from the model fit.

The system inclination that we derive from Eq. (3) with $v\sin i = 17.2$ km$\,$s^-1 is $i=67\hbox{$^\circ$ }$. From the errors in radius (16%) and in $v\sin i$ (6%) we estimated the error in $\sin i$, which implies a system inclination in the 50 $\hbox{$^\circ$ }$-75 $\hbox{$^\circ$ }$ range. The maximum inclination of 75$^{\circ}$ is constrained by the absence of eclipse in the HR 7428 light curve. Indeed, from radii and temperature that we determined (Table 4), a primary eclipse (hot star beyond) of 0 $\hbox{$.\!\!^{\rm m}$ }$07 in V band and 0 $\hbox{$.\!\!^{\rm m}$ }$13 in B should be observed. These inclination values lead to masses of the two components $M_{\rm H}=2.75$ $M_{\odot}$ and $M_{\rm C}=6.71$ $M_{\odot}$ for $i=67^{\circ}$, and $M_{\rm H}=2.28$ $M_{\odot}$ and $M_{\rm C}=5.57$ $M_{\odot}$ for $i=75^{\circ}$, while the lower limit of inclination (50$^{\circ}$) would lead to higher masses not consistent with the spectral types.

3.4 Position in the HR diagram

Using the effective temperatures and radii in Table 4, we have placed the two components of HR 7428 on the $\log T-\log L$ plane. Their position and the evolutionary tracks for intermediate mass stars calculated by Fagotto et al. (1994) are displayed in Fig. 5.

The cool component lies in the region of He-burning stars, and implies a mass from 4 to 5 $M_{\odot}$, more consistent with the higher inclination limit, and an age from 140 to 150 Myr. The position of the hot component is consistent with a 2.3 $M_{\odot}$ star still on the main-sequence band, but slightly evolved. Its estimated age of 100-250 Myr is in perfect agreement with the estimated age of the cooler more massive component. This suggests a normal evolution of the two stars, yet without any relevant mass exchange, as expected from the long orbital period.

Figure 5: Position of the cool (grey dot) and hot (black square) components of HR 7428 in a $\log T-\log L$ plane. The evolutionary tracks for different masses from Fagotto et al. (1994) are also shown.