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Title: TESS Reveals a Short-period Sub-Neptune Sibling (HD 86226c) to a Known Long-period Giant Planet
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  1. Context. As primary anchors of the distance scale, Cepheid stars play a crucial role in our understanding of the distance scale of the Universe because of their period-luminosity relation. Determining precise and consistent parameters (radius, temperature, color excess, and projection factor) of Cepheid pulsating stars is therefore very important. Aims. With the high-precision parallaxes delivered by the early third Gaia data release (EDR3), we aim to derive various parameters of Cepheid stars in order to calibrate the period-luminosity and period-radius relations and to investigate the relation of period to p -factor. Methods. We applied an implementation of the parallax-of-pulsation method through the algorithm called spectro-photo-interferometry of pulsating stars (SPIPS), which combines all types of available data for a variable star (multiband and multicolor photometry, radial velocity, effective temperature, and interferometry measurements) in a global modeling of its pulsation. Results. We present the SPIPS modeling of a sample of 63 Galactic Cepheids. Adopting Gaia EDR3 parallaxes as an input associated with the best available dataset, we derive consistent values of parameters for these stars such as the radius, multiband apparent magnitudes, effective temperatures, color excesses, period changes, Fourier parameters, and the projection factor. Conclusions. Using the best set of data and the most precise distances for Milky Way Cepheids, we derive new calibrations of the period-luminosity and period-radius relations: M K S = −5.529 ±0.015   −  3.141 ±0.050 (log P   −  0.9) and log R = 1.763 ±0.003   +  0.653 ±0.012 (log P   −  0.9). After investigating the dependences of the projection factor on the parameters of the stars, we find a high dispersion of its values and no evidence of its correlation with the period or with any other parameters such as radial velocity, temperature, or metallicity. Statistically, the p -factor has an average value of p  = 1.26 ± 0.07, but with an unsatisfactory agreement ( σ  = 0.15). In absence of any clear correlation between the p -factor and other quantities, the best agreement is obtained under the assumption that the p -factor can take any value in a band with a width of 0.15. This result highlights the need for a further examination of the physics behind the p -factor. 
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  2. In this work, we aimed to derive the gri-band period-luminosity (PL) and period-luminosity-color (PLC) relations for late-type contact binaries, for the first time, located in the globular clusters, using the homogeneous light curves collected by the Zwicky Transient Factory (ZTF). We started with 79 contact binaries in 15 globular clusters, and retained 30 contact binaries in 10 globular clusters that have adequate number of data points in the ZTF light curves and unaffected by blending. Magnitudes at mean and maximum light of these contact binaries were determined using a fourth-order Fourier expansion, while extinction corrections were done using the {\tt Bayerstar2019} 3D reddening map together with adopting the homogeneous distances to their host globular clusters. After removing early-type and "anomaly" contact binaries, our derived gri-band PL and period-Wesenheit (PW) relations exhibit a much larger dispersion with large errors on the fitted coefficients. Nevertheless, the gr-band PL and PW relations based on this small sample of contact binaries in globular clusters were consistent with those based on a larger sample of nearby contact binaries. Good agreements of the PL and PW relations suggested both samples of contact binaries in the local Solar neighborhood and in the distant globular clusters can be combined and used to derive and calibrate the PL, PW and PLC relations. The final derived gr-band PL, PW and PLC relations were much improved than those based on the limited sample of contact binaries in the globular clusters. 
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  3. Abstract We present the first gri -band period–luminosity (PL) and period–Wesenheit (PW) relations for 37 Type II Cepheids (TIICs) located in 18 globular clusters based on photometric data from the Zwicky Transient Facility. We also updated BVIJHK -band absolute magnitudes for 58 TIICs in 24 globular clusters using the latest homogeneous distances to the globular clusters. The slopes of g / r / i - and B / V / I -band PL relations are found to be statistically consistent when using the same sample of distance and reddening. We employed the calibration of ri -band PL/PW relations in globular clusters to estimate a distance to M31 based on a sample of ∼270 TIICs from the PAndromeda project. The distance modulus to M31, obtained using calibrated ri -band PW relation, agrees well with the recent determination based on classical Cepheids. However, distance moduli derived using the calibrated r - and i -band PL relations are systematically smaller by ∼0.2 mag, suggesting there are possible additional systematic errors on the PL relations. Finally, we also derive the period–color (PC) relations and for the first time the period–Q-index (PQ) relations, where the Q -index is reddening free, for our sample of TIICs. The PC relations based on ( r − i ) and near-infrared colors and the PQ relations are found to be relatively independent of the pulsation periods. 
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