Multiplicity of Galactic Cepheids from long-baseline interferometry V. High-accuracy orbital parallax and mass of SU Cygni

As with many Cepheids, the hottest star in the SU Cyg system is the companion. The hottest star outshines the Cepheid in the ultraviolet. Because of this the velocity of the companion can be measured on Hubble space telescope (HST) Space Telescope Imaging Spectrograph (STIS) spectra.

We obtained ultraviolet spectra with the STIS instrument on board the HST in Cycles 22 and 23 (PI: Gallenne) and 28 and 29 (PI: Evans). The high-resolution echelle mode E140H was used in the wavelength region 1163-1357 Å. The reduction procedure is described in detail in Evans et al. (2018b). The echelle orders are combined into a 1D spectrum using an appropriate blaze function (Baer et al. 2018).

The velocity differences between the spectra were measured using an IDL cross-correlation routine, as described in Evans et al. (2018b). The first spectrum in the series was arbitrarily selected for the velocity difference anchor. Cross-correlation was done in wavelength segments typically 10 Å wide. Because SU Cyg B is a sharp-lined HgMn star with many lines, the agreement between segments is very good, as shown in Fig. 1.

On the other hand, in spectra with more typical broad lines, such as V1334 Cyg and S Mus, interstellar lines can be used to check the velocity shift zero point variation due to the positioning of the star within the aperture (Proffitt et al. 2017). In SU Cyg this cannot be done because the spectrum is full of lines with a similar width to interstellar lines. Internal errors depend largely on the number of lines available and also the rotational velocity of the star. For SU Cyg B, internal consistency between segments is excellent as shown by Fig. 1.

The observation started with a peak-up on the star. Nominally, an ACQ/Peak is accurate to 5 % of the slit width. We used the 0.2X0.09 aperture, and the plate scale of the E140H is 0.047”/pixel in the dispersion direction. A nominal centring accuracy $0.05\times 0.09$ / 0.047 is thus about 0.1 pixels. One pixel is $\lambda/228000$, so a tenth of a pixel would give about $0.1c/228000$ or about 0.13 km s^-1, which is probably not the limiting accuracy of the wavelength scale. Instead, the accuracy is probably limited by the stability of the bench and the accuracy of the wavecal measurements. We added extra deep wavecals that bracketed our exposures to help correct for any drifts in that. The CALSTIS pipeline software then interpolates the shift in the velocity alignment as measured from the two lamp exposures to the mean time of the observation. Even if the actual shift in the bench is highly non-linear over the course of an observation, any additional error should be less than half the value of the shift between the two wavecals. We find that the average shift in the position of the lamp spectrum between the leading and trailing wavecal is equivalent to a velocity change about $-0.28\pm 0.23$ km s^-1, with the most extreme shift being about $-0.7$ km s^-1. The small scatter in the STIS velocities around the best fit model in the next Section (Fig. 2) also confirms the accuracy of the velocities.

We collected long-baseline optical interferometric data at Georgia State University’s Center for High Angular Resolution Astronomy (CHARA) Array (ten Brummelaar et al. 2005) located at Mount Wilson Observatory. The CHARA Array consists of six 1m aperture telescopes in an Y-shaped configuration with two telescopes along each arm, oriented to the east (E1, E2), west (W1,W2) and south (S1, S2), offering good coverage of the $(u,v)$ plane. The baselines range from 34 m to 331 m, providing an angular resolution down to 0.5 mas at $\lambda=1.6\,\mu$m.

Data were collected with the Michigan InfraRed Combiner (MIRC, Monnier et al. 2004) before 2019 and the upgraded Michigan InfraRed Combiner-eXeter (Anugu et al. 2020) after 2019. In addition, recent improvements at CHARA include the commissioning of a new 6-telescope beam combiner MYSTIC (Michigan Young STar Imager at CHARA Monnier et al. 2018; Setterholm et al. 2023), designed alongside the MIRC-X upgrade and capable of simultaneous observations. MIRC combined the light coming from all six telescopes in the $H$ band ($\sim 1.6\,\mu$m), with three spectral resolutions ($R=42,150$ and 400). The recombination of six telescopes gives simultaneously 15 fringe visibilities and 20 closure phase measurements, that are our primary observables. MIRC-X also combines the light from six telescopes, with the spectral resolution $R=50,102$, and 190. Our MIRC and MIRC-X observations used only the lowest spectral resolution to favour the signal-to-noise ratio (S/N). MYSTIC is a $K$-band instrument working similarly to MIRC-X and offering a spectral resolution of 50. The log of our observations is available in Table 1.

We followed a standard observing procedure, i.e., we monitored the interferometric transfer function by observing a calibrator before and after our Cepheids. The calibrators were selected using the SearchCal software¹¹1http://www.jmmc.fr/searchcal (Chelli et al. 2016) provided by the Jean-Marie Mariotti Center. They are listed in Table 1.

The data were reduced with the standard MIRC and MIRC-X pipeline²²2https://gitlab.chara.gsu.edu/lebouquj/mircx_pipeline.git (Monnier et al. 2007; Anugu et al. 2020). The main procedure is to compute squared visibilities and triple products for each baseline and spectral channel, and to correct for photon and readout noises. MYSTIC data are also reduced by the MIRC-X pipeline.

RVs of the companion were determined using the cross-correlation technique. No synthetic template spectrum was used, instead we estimated a velocity difference with respect to the first observation by cross-correlating each of the spectra against it. This was typically done in 11 segments of approximately 10Å, which were inspected to optimize the features so that a feature was not divided by a boundary. Mild smoothing was included. These velocity differences are directly used in our orbital model fitting. Measurements are listed in Table 4, with internal uncertainties derived from the differences between segments.

In the ultraviolet wavelength range, the Cepheid is faint compared to its companions, therefore the STIS velocities inform us about the orbital motion of the companions around their common centre of mass (i.e. the system B = B_a+B_b) plus the motion around the centre of mass with the Cepheid. We therefore need to disentangle the centre of mass motion of the short orbital period from the long orbital period.

To disentangle the orbital motions, we used the following model, assuming a circular short-period orbit (consistent hypothesis for such short orbit) and therefore an argument of periapsis $\omega_{\mathrm{short}}=0$:

with $\gamma$ the systemic velocity, $K_{\mathrm{B}}$ the semi-amplitude of the centre of mass of B in the long orbit, $K_{\mathrm{Ba}}$ the semi-amplitude in the short orbit, $\omega$ the argument of periastron of B’s orbit (with $\pi$ subtracted to take into account the symmetry compared to the equations in Sect. 4), $\nu$ the true anomaly of the centre of mass in the long orbit, $e$ the eccentricity of the long orbit, and $\nu_{\mathrm{Ba}}$ the true anomaly in the short orbit.

The true anomaly is defined implicitly with the following Keplerian parameters $P_{\mathrm{orb}},T_{\mathrm{p}},e$ (respectively for the long- and short-period orbit), and Kepler’s equation:

where $E(t)$ is the eccentric anomaly, $T_{p}$ the time of periastron passage, $t$ the time of radial velocity observations, and $P_{\mathrm{orb}}$ the orbital period. In the following, we will use $P_{\mathrm{L}},T_{\mathrm{p,L}},P_{\mathrm{S}}$ and $T_{\mathrm{p,S}}$ to refer to the orbital period and time of periastron passage for the long and short orbit, respectively.

Due to the finite speed of light, we also adjusted at each iteration the times of observations to account for the light travel time effect in the short orbit (LTTE, Irwin 1952, 1959). The LTTE causes apparent shifts in the timing of the RVs because light from Ba takes time to travel to us as the star moves in its short orbit (LTTE for the long orbit is adjusted later in the global fit). More details can be also found in Konacki et al. (2010) or Zucker & Alexander (2007).

However, the RVs from STIS are differential velocities as they were calculated by cross-correlation with the first spectrum (see Gallenne et al. 2018), therefore our model to fit the STIS velocities is:

As first guess parameters, we used the values given in Evans & Bolton (1990). We display in Fig. 2 the disentangled long- and short-period orbital velocities and the fitted parameters in Table 3. We used a standard least-squares minimization with errors taken from the correlation matrix. As the STIS uncertainties are only internal (mean standard deviation of segments for all the spectra), we rescaled the RVs uncertainties to the mean r.m.s. of the fit, which was of $\sim 0.29$ km s^-1. Our measured parameters are in agreement with the one estimated by Evans & Bolton (1990), who performed a similar analysis using high-dispersion spectrographs from the IUE satellite. Additionally, the small residuals for the short-period orbit support our hypothesis of a circular orbit.

We then analytically removed the short-period motion from the STIS velocities which will be used in our combined fit of Sect. 4.

In the visible wavelength, where most of the ground-based spectrograph operate, the Cepheid dominates the spectra. Therefore, RVs obtain from these observations give us information about the orbital reflex motion of the Cepheid. There are several dataset available in the literature, but some suffered from large uncertainties or large scatter in the measurements so we decided to not include them in the fit, which are the case of RVs from (Hellerich 1919; Barnes et al. 1987; Wilson et al. 1989). The velocities from Borgniet et al. (2019) are of very good quality, unfortunately there are only 13 measurements with a small orbital coverage. There are a large set of observations from Evans (1989) spanning about six years, providing a good coverage of the long orbital period. However, the mean scatter of the data are on the order of 1.6 km s^-1 when fitting the orbital and pulsation motion. This is likely due to a number of changes made to the spectrograph during those six years which might result in some residual offsets between datasets, although some corrections were done by the authors. The method used to determine the RVs may also contribute to the scatter as they fitted parabola to the line cores of about 20 selected lines, which is less robust than the cross-correlation technique. Imbert (1984) obtained spectra from the CORAVEL instrument (Mayor et al. 1983) spanning about three years, i.e. about two times the long orbital period. The RVs were extracted using the cross-correlation technique, providing more accurate measurements. We found a mean scatter $\sim 0.6$ km s^-1 when fitting the orbital and pulsation motion. A comparison of Imbert and Evans dataset is displayed in Fig. 3. We preferred to use a uniform more accurate dataset, therefore we only used the measurements from Imbert (1984) for our simultaneous fit of the next section.

To detect the companion (or to be more precise the centre of light in our specific case of a binary companion), we used the interferometric tool CANDID⁵⁵5Available at https://github.com/amerand/CANDID and https://github.com/agallenne/GUIcandid for a GUI version. (Gallenne et al. 2015). The main function allows a systematic search for companions performing an $N\times N$ grid of fits, the minimum required grid resolution of which is estimated a posteriori in order to find the global minimum in $\chi^{2}$. The tool delivers the binary parameters, namely the flux ratio $f$, and the relative astrometric separation ($\Delta\alpha,\Delta\delta$), together with the uniform-disk angular diameter $\theta_{UD}$ of the primary star (the Cepheid). The angular diameter of the companion is assumed to be unresolved by the interferometer. The significance of the detection, $n\sigma$, is also given, taking into account the reduced $\chi^{2}$ and the number of degrees of freedom⁶⁶6with a maximum displayed of $8\sigma$, because $p$-values are converted into chi-squared statistics, but for values very close to 1 the computer precision is reached.. They are listed in Table 4, together with our measured astrometric positions.

For the 2016 data, we had poor seeing conditions, we therefore combined the two nights to detect the companion and used only the closure phase signal which is less sensitive to the atmospheric variations. We fixed the angular diameter to the average diameter measured from all our observations (see next paragraph). The night of 2020-06-30 also suffered from poor observing conditions, so we decided to combine with the following night to increase the S/N, again using only the closure phases. We fixed the angular diameter to the average value given by fitting the $V^{2}$. The data taken during two nights in 2022 suffered from vibrations on the delay lines systems, which affects mostly the visibilities as the order of magnitude is less than a wavelength. We therefore also only used the closure phase with a fixed flux ratio. To increase the detection level, we combine these two nights for MIRC-X and MYSTIC. Both instruments provide identical detection of the companion (see Table 4).

Uncertainties on the fitted parameters are estimated using a bootstrapping function (bootstrap on the observing time and baselines). From the distribution, we took the median value and the maximum value between the 16th and 84th percentiles as uncertainty for the flux ratio and angular diameter. For the fitted astrometric position, the error ellipse is derived from the bootstrap sample (using a principal components analysis).The angular diameters and astrometric separations were then divided by factors of $1.0014\pm 0.0006$ for MIRC, $1.0054\pm 0.0006$ for MIRC-X and $1.0067\pm 0.0007$ for MYSTIC (J.D. Monnier, private communication) to take into account the uncertainty from the wavelength calibration. This is equivalent to adjusting the respective wavelengths reported in the OIFITS files by the same factors.

Uncertainty on the angular diameter measurements was estimated using the conservative formalism of Boffin et al. (2014) as follows:

where $N_{\mathrm{sp}}$ is the number of spectral channels, $\sigma^{2}_{\mathrm{stat}}$ the uncertainty from the bootstrapping and $\delta\lambda=0.07$ %, as mentioned above. We measured a mean uniform disk diameter in the $H$ band of $\mathrm{\theta_{UD}}=0.323\pm 0.061$ mas (the standard deviation is taken as uncertainty), which is in agreement with the value of Trahin (2019) estimated from a global fit. We also estimated an average flux ratio in $H$ of $f_{\mathrm{H}}=1.68\pm 0.24$ %.

As the companion is itself a binary, the astrometry we measured is the one of the photocentre of the companion pair. For stars with equal brightness, the photocentre would be in the middle of the two stars and would follow a simple elliptical orbit around the Cepheid. However, this is not our case because $Ba$ is a B7.5V star and $Bb$ has a spectral type later than A0V, which means that the photocentre has a wobble around its elliptical orbit around the Cepheid. The amplitude of the wobble can be roughly quantified from the mass ratio and contrast between the stars. Using the spectral type calibration from Pecaut & Mamajek (2013), the companions have a mass close to $3.6\,M_{\odot}$ and $\leq 2.2\,M_{\odot}$, respectively for $Ba$ and $Bb$, and a magnitude difference in $\Delta H$ of $\sim 1.1$ mag. The semimajor axis $a_{\mathrm{ph}}$ of the photocentre orbit of the companion pair can be estimated with:

with $a_{\mathrm{short}}$ the semimajor axis of the short orbit in arcsecond, and $M_{Ba}$ and $M_{Bb}$ the mass of each component. $a_{\mathrm{short}}$ can be also estimated from the centre of mass equation:

where $a_{Ba}$ and $a_{Bb}$ are the distance between the centre of mass and the star $Ba$ and $Bb$, respectively. Using $a_{Ba}$ from Table 3, the inclination of 80^∘ (estimated a posteriori, see Sect. 4), the Gaia distance of 1000 pc and the predicted masses, we estimated $a_{\mathrm{short}}\sim 0.1$ mas, and a semimajor axis of the photocentre of $a_{\mathrm{ph}}\sim 11\,\mu$as. This is slightly below our average astrometric precision. To be conservative, we added to our astrometry an additional uncertainty of $15\,\mu$as.

This is the second Cepheid with the most precise and accurate orbital parallax measured to 1 % using a combination of interferometry and space- and ground-based spectroscopy. In Gallenne et al. (2018), we measured the orbital parallax of the Cepheid V1334 Cyg with an accuracy of 1 % using the same method and showed the disagreement with the most used P-L relations and a deviation of $3.6\sigma$ with the Gaia DR2. Here we will perform a similar analysis with the Cepheid SU Cyg. Gaia DR3 provides a parallax of $1.000\pm 0.052$ mas, including a zero point correction of $-0.028$ mas (Lindegren et al. 2021), which is in agreement at $1.5\sigma$ with our measurements, but our uncertainty is $\sim 9\times$ better. Because the DR3 parallax has large uncertainties, the zero-point correction has no effect on the agreement with our distance estimate. The Renormalized Unit Weight Error (RUWE) indicator of the Gaia astrometric fit is 3.44, meaning that the astrometric solution is problematic, likely due to the orbital motion of the Cepheid. Wahlgren & Evans (1998) identified the companion B as a HgMn type star of B8V spectral type, giving a magnitude difference with the Cepheid of $\Delta V\sim 3$ mag (Evans 1995). Using our measured orbital parameters and this contrast, we calculated the orbit of the photocentre, which we display in Fig. 6. We see that its semi-major axis is $\sim 1.4$ mas and certainly impacts the Gaia single-star model solution. Since the Cepheid is redder than its companion, the magnitude difference in the G band is slightly larger than the estimate in the V band, of the order of +0.1-0.2mag, leading us to expect a similar bias in the G-band parallax. In addition, the orbital period is close to one year. Unfortunately, there are no information in the non-single stars catalogue of Gaia for this system (Halbwachs et al. 2023). In addition, the Cepheid is in the bright-star magnitude range of Gaia, suffering from saturation effects in the detector which alter the astrometric performances of Gaia.

Our precise and independent distance measurement also provide an independent check of the P-L relations for fundamental-mode Cepheids as SU Cyg is classified as pulsating in its fundamental mode (see e.g. Clementini et al. 2019; Luck 2018). A reliable method to determine whether a star is in the fundamental or first overtone mode is to examine the period-radius relation, which is notably precise. According to the empirical relation from Gallenne et al. (2017), the prediction for a fundamental mode is $\sim 34\,R_{\odot}$, aligning well with our estimate of $31.9\pm 6.0\,R_{\odot}$. Thus, the Cepheid is consistent with being a fundamental mode pulsator. We selected a sample of published relations in the $V$- and $K$-band filters, as they are the most frequently used photometric bands (Sandage et al. 2004; Fouqué et al. 2007; Benedict et al. 2007; Storm et al. 2011; Groenewegen 2013, 2018; Breuval et al. 2020). For each tested P-L relation, we determined the predicted absolute magnitudes for $P_{0}$, that are represented in Fig. 7 as black dots. We adopted as uncertainties the scatter of each relation given by the authors, which better represents the observed intrinsic dispersion (the width of the instability strip is the dominating uncertainty in the P-L relations). We derived the absolute magnitude with the usual relation $M_{\lambda}=m_{\lambda}-A_{\lambda}-5\,\log d+5$, with $m_{\lambda}$ the apparent magnitude at wavelength $\lambda$, $A_{\lambda}$ the interstellar absorption parameter, and $d$ the distance to the star. We corrected for the interstellar extinction, using $A_{\mathrm{V}}=3.23E(B-V)$ and $A_{\mathrm{K}}=0.119A_{\mathrm{V}}$ (Fouqué et al. 2007), with $E(B-V)=0.109\pm 0.006$ (Trahin et al. 2021). For the $V$ band, we estimated the weighted mean apparent magnitude using a periodic cubic spline fit of four different published light curves (Berdnikov 2008; Kiss 1998; Moffett & Barnes 1984; Szabados 1977). The curve is shown in Fig. 8. We determined the weighted mean dereddened magnitude $m_{\mathrm{0,V}}=6.602\pm 0.024$ mag, where the total uncertainty is estimated from the standard deviation of the residual values. In the near-infrared, we used the mean apparent magnitude $m_{\mathrm{K}}=5.307\pm 0.015$ mag from Monson & Pierce (2011), determined from near-IR light curve. We estimated the dereddened magnitude $m_{\mathrm{0,K}}=5.285\pm 0.015$ mag.

We then combined with our measured distance to estimate the absolute magnitudes $M_{\lambda}$, for which the total error bar includes the uncertainties on $d,A_{\lambda}$, and $m_{\lambda}$. Fig. 7 shows the difference between $M_{\lambda}$ of SU Cyg (red area) and the predicted values from literature P-L relations (black dots). However, the contribution of the companion must be subtracted from the magnitudes. To correct from the flux contamination of the companion, we used the average magnitude difference between the components estimated in $V$ by Evans (1995), $\Delta V=3$ mag, and from our measured $H$-band average flux ratio from interferometry for the $K$ band, i.e. we assumed $\Delta K=\Delta H=4.44\pm 0.16$ mag. With our distance we can estimate the mean corrected absolute magnitudes of the Cepheid to be $M\mathrm{{}_{V}(cep)}=-3.23\pm 0.03$ mag, $M\mathrm{{}_{H}(cep)}=-4.50\pm 0.02$ mag and $M\mathrm{{}_{K}(cep)}=-4.55\pm 0.02$ mag. Note that we assumed a similar flux ratio in $H$ and $K$, but this a a good approximation as the companion is rather faint in the near-IR and our MIRCX and MYSTIC observations provide similar flux ratios in $H$ and $K$ band, respectively (see Table 4). From the magnitude differences, absolute magnitudes for the companion pair can also be derived, we found $M\mathrm{{}_{V}(B)}=-0.22\pm 0.11$ mag, $M\mathrm{{}_{H}(B)}=-0.07\pm 0.16$ mag and $M\mathrm{{}_{K}(B)}=-0.11\pm 0.16$ mag. The spectral type of B8V identified by Evans (1995) for the companion Ba corresponds to an absolute magnitude $M\mathrm{{}_{V}(Ba)}\sim 0.0$ mag (Pecaut & Mamajek 2013)¹⁰¹⁰10see also http://www.pas.rochester.edu/~emamajek/EEM_dwarf_UBVIJHK_colors_Teff.txt, which when combined with $M\mathrm{{}_{V}(B)}$ gives a flux ratio of $f_{\mathrm{Bb}}/f_{\mathrm{Ba}}\sim 22$ % in $V$, and so an absolute magnitude for the component Bb of $M\mathrm{{}_{V}(Bb)}\sim 1.6$ mag. This would correspond to a A3V-A4V spectral type (Pecaut & Mamajek 2013), in agreement with the suggestion of Evans (1995). The expected mass from such spectral types is about $3.4\,M_{\odot}$ and $1.9\,M_{\odot}$, respectively for Ba and Bb.

The blue area in Fig. 7 shows the absolute magnitudes of the Cepheid corrected from the flux contamination. We see that only the relations based on direct distance estimates are in acceptable agreement ($<1.4\sigma$) with our measured corrected absolute magnitude of the Cepheid. This small discrepancy for Gaia can be explained by the fact that the measurements are still not accurate and suffer from various effect for pulsating stars (chromaticity correction, saturation, etc.), in addition to the binarity effect which is not taken into account yet. The relation from Benedict et al. (2007) based on HST Fine Guidance Sensor parallax is at $1.8\sigma$ and $2.4\sigma$, respectively in $V$ and $K$. Breuval et al. (2020) shows a very good agreement, where we used the Gaia DR2 parallaxes of wide companions of Cepheids or their host open cluster as proxy of the Cepheid distances, which are thought to be less prone of systematics. The other relations based on photometric measurements, calibrated using the Baade-Wesselink method (Baade 1926; Wesselink 1946) to determine the distance of Cepheids, are in agreement at $1.5-3.3\sigma$ in $V$ and $1.2-2.5\sigma$ in $K$.

SU Cyg seems brighter than expected, by $\sim 0.15$ mag in $V$ and $\sim 0.08$ mag in $K$. One explanation would be the presence of a circumstellar envelope (CSE), which we know exist around other Cepheids (Kervella et al. 2006; Mérand et al. 2006; Gallenne et al. 2012, 2013; Hocdé et al. 2020a, 2021; Gallenne et al. 2021). The infrared excess is similar to Hocdé et al. (2020b) who estimated for this Cepheid an excess of $\sim 0.04$ mag using a multi-wavelength global analysis with the SPIPS code (SpectroPhoto-Interferometry of Pulsating Stars Mérand et al. 2015) and a simple power-law parametrization. However, this model assumes dust emission and no excess for $\lambda<1.2\,\mu$m. In Hocdé et al. (2020b), we also modelled the CSE emission with a shell of ionized gas to allow a deficit or excess in the visible. However, this model showed a deficit in the visible, i.e. an absorption instead of an emission, no excess in $K$. Therefore, no CSE models could explain the possible offset to reconcile the photometrically-based P-L relations.

Another explanation would be a wrong estimate of the pulsation period, in the sense that the Cepheid does not pulsate in its fundamental mode but instead in its first overtone mode. To test this hypothesis, we can assume that the pulsation period we measure is the first overtone and we can convert it to a fundamental mode. Several works exit on this topic for Milky-Way and Magellanic Clouds Cepheids but there is still no consensus about a good empirical or theoretical relation to ”fundamentalize” the first overtone period (Alcock et al. 1995; Feast & Catchpole 1997; Kovtyukh et al. 2016; Szilàdi et al. 2018; Pilecki et al. 2021, 2024). The most used is from (Alcock et al. 1995) using LMC Cepheids, who estimated the ratio of the first overtone to the fundamental period as $P_{1}/P_{0}=0.720-0.027\log P_{0}$, while the latest relation is $P_{0}=P_{1}(1.367+0.079\log P_{1})$ from Pilecki (2024) for MW Cepheids. All published relations provides a ”fundamentalized” period for SU Cyg between 5.434 and 5.757 days, but they do not provide either a consistent agreement with our measured value. We can see in Fig. 7 the recalculated values of the P-L relations using 5.434 days and 5.757 days, represented by down and up triangles, respectively. In $V$, the photometric-based relation would be in agreement, however, the relations based on the Gaia parallaxes are now not consistent. In $K$, all relations would be at several $\sigma$ from the measurements. Based on this, we can confirm that SU Cyg must pulsate in its fundamental mode, the pulsation mode cannot be the reason of the discrepancy. We do not have any other explanation about why the P-L relations not calibrated from the Gaia measurements are systematically offset.

We measured a very precise dynamical mass for the Cepheid and the companion pair, respectively at 1.2 % and 0.8 % level. To compare the dynamical mass of the Cepheid with the evolutionary models, we used the average effective temperature $T_{\mathrm{eff}}=6165\pm 50$ K and radii $R=40.6\pm 0.8\,R_{\odot}$ determined by Trahin (2019), who used the SPIPS algorithm to combine all types of available data for a variable star (multi-band and multicolour photometry, radial velocity, effective temperature, and interferometric measurements) in a global modelling of the stellar pulsation. We determined a luminosity $L=2138\pm 109\,L_{\odot}$, which we compared to PARSEC (Bressan et al. 2012; Nguyen et al. 2022; Costa et al. 2019) and Geneva stellar evolutionary tracks. Both models follow standard mixing-length theory (MLT) for convective mixing with mixing-length parameter $\alpha_{\mathrm{MLT}}=1.6$ and 1.74, respectively for Geneva and Parsec, and both using the Schwarzschild criterion to determine convective boundaries. For the convective core overshoot, Geneva uses step overshooting with overshoot parameter $\alpha_{\mathrm{ov}}=0.1$, while PARSEC uses the overshoot formalism of Bressan et al. (1981) which would correspond to a step overshooting parameter of 0.25.

The metallicity of SU Cyg is $\sim 0$ dex (Andrievsky et al. 2013), which converts to $Z\sim 0.014$ for both models. We also compared models with metallicities encompassing this value, i.e. 0.01 and 0.017. In Fig. 9 we display tracks for a stellar mass of $4.8\,M_{\odot}$, without rotation and with a moderate rotation ($\Omega/\Omega_{\mathrm{crit}}=0.3$), to see any impact from additional mixing due to rotation. We see that at the expected metallicity of 0.014, both models are inconsistent, including or not rotation effects. Models predict a higher mass than expected, as already reported in our previous works (see e.g. Evans et al. 2018a). PARSEC models predict a mass between $5.1\,M_{\odot}$ and $5.6\,M_{\odot}$, while Geneva tracks predict $5.4-6.0\,M_{\odot}$, $>0.3\,M_{\odot}$ with our measurement. The only agreement would be with the PARSEC models, with and without rotation, with a metallicity of $Z=0.01$ (see upper right panel of Fig. 9). However, this would mean $Fe/H\sim-0.18\,$dex which is in a large disagreement with the measured value.

To check if the discrepancy could come from our measured mass, we added 5 % error to our astrometric measurements and normalized all RVs to 1 km s^-1. The MCMC analysis gave a similar Cepheid mass with $4.83\pm 0.17\,M_{\odot}$, although a larger uncertainty, making us confident about the robustness of our measurement.

The mass of SU Cyg can be compared with other well-determined masses for the Milky Way (MW) Cepheids, V1334 Cyg (Gallenne et al. 2018) and Polaris (Evans et al. 2024b), and also Cepheids in the Large Magellanic Cloud (LMC) (Pilecki et al. 2018). Fig. 10 shows them in comparison with predictions from evolutionary tracks from Bono et al. (2016) and Anderson et al. (2014). The Bono tracks cover MW and LMC metallicities (for the MW we used the relations with no main sequence convective core overshoot and with moderate overshoot). The main sequence rotation and moderate convective overshoot are shown for the Anderson relation (Equ. 5 with solar metallicity). The MW Cepheids are consistently brighter than the predictions, as are the LMC Cepheids except for one on the first instability crossing (LMC 1812), and a system which has possibly had binary interaction (LMC 1718 A and B).

The mass for each companion can also be determined if we do the reasonable hypothesis that their orbital inclination is identical to their orbit around the Cepheid. We previously determined the mass function $f_{\mathrm{B}}$ of the companion pair (see Table 3), which combined with the inclination and the total mass $M_{\mathrm{B}}$ gives:

We determined a mass for the secondary component of $M_{\mathrm{Ba}}=3.595\pm 0.033\,M_{\odot}$ and for the tertiary star $M_{\mathrm{Bb}}=1.546\pm 0.009\,M_{\odot}$. This would correspond to stars with spectral types of a B8V and F1V star, respectively. This is in good agreement with Evans & Bolton (1990) who suggested a B7.5V star from ultraviolet spectra for the secondary companion, and set an upper limit for the spectral type of the tertiary companion to be later than A0V.

The temperature of the hottest star in the SU Cyg system can be determined by comparing the IUE spectrum SWP 14773 with the BOSZ Synthetic Stellar Spectral Library (Bohlin et al. 2017). This spectrum has been discussed before (Evans 1995), however the model comparisons provide a direct link between temperatures and the masses of detached eclipsing binaries (Evans et al. 2023). Details of the comparisons are provided in Evans et al. (2024a).

The first step in the temperature determination is obtaining the reddening $E(B-V)$. Since this was discussed by (Evans 1995) using colours corrected for the companion, we start with the $E(B-V)=0.109$ from Trahin (2019). Using this, we dereddened the IUE spectrum and ran it thought the program which compares it with models of a series of temperatures. Details of the fitting are provided by the figures in Appendix A.

The best fit temperature is $13500\pm 920$ K. from the parabola fit to the standard deviations of the spectrum-model differences (Fig. 13). As in the discussion of the DEBs, the uncertainty was also estimated visually from the spectral differences (Fig. 12) to be 500 K. This corresponds approximately to a B7.3V in the calibration of Pecaut & Mamajek (2013), which is in good agreement with our estimate from the previous Section. To estimate its mass, we can use the formula relating temperature and mass derived from detached eclipsing binaries (DEBs) by Evans et al. (2023). As for the FN~Vel system (Evans et al. 2024a), because SU Cyg Ba is the companion of a massive young Cepheid, it is at the younger (hotter) half of the relation. Thus the mass is decreased by 0.02 in $\log{M}$. SU Cyg Ba is a HgMn star and in fact the DEBs study provides some information about those. In the sample of DEBs there are 5 Am (metallic lined) stars. They sit above (at larger masses) than normal stars. Although they are all cooler than SU Cyg Ba, this is an indication that chemically peculiar stars have cooler temperatures than normal stars. We can make an approximate correction of 0.02 in $\log{T}$ for this, the mass of Ba becomes $3.75\,M_{\odot}$, and the mass of Bb is $1.37\,M_{\odot}$. They are similar to our previous determination of Sect. 5.2.