Asteroseismology Of The ? Scuti Pulsating Variable Stars

The ? Scuti stars show rich oscillation modes in observations,both of radial oscillation modes and nonradial oscillation modes are detected.Different oscillation modes show different propagation behaviors in the star and reflect different properties of the stellar structure,thus ? Scuti stars are very promising targets for asteroseismology.In this paper,we analyze three ? Scuti pulsating stars HD50844,CoRoT 102749568,and EE Cam with asteroseismology.Some meaningful results are obtained.HD 50844 was discovered to be a ? Scuti variable star in 2005.Balona(2014)extracted a total of 59 significant oscillation modes from the CoRoT timeseries.We pay more attention to 40 of these oscillation modes.Their frequencies are near or higher than that of the fundamental radial mode.On basis of the regularities in rotational splitting,we find two sets of complete triplets,one set of incomplete triplets,seven sets of incomplete quintuplets,and three sets of incomplete septuplets.According to frequency differences in these multiplets,the rotational period of HD 50844 is estimated to be Prot= 2.44+0.13-0.08 days.When we do model fittings,we fit theoretical oscillation frequencies to three m = 0 nonradial oscillation modes(f11,f22,f29)and the fundamental radial mode f4.The physical parameters of HD 50844 are hence determined to be M = 1.81 ± 0.01M?,Z = 0.008 ± 0.001,Teff= 7508 ± 125 K,log g = 3.658 ± 0.004,R = 3.300± 0.023 R?,L = 30.98 ± 2.39 L?.Besides,we find that most of the observed frequencies belong to the mixed modes.The three nonradial oscillation modes f11,f22,andf29 propagate like g modes in the helium core while like p modes in the stellar envelope.The fundamental radial mode f4 propagates mainly in the stellar envelope,and represents the property of the stellar envelope.This fact confirms that they are mixed modes and can therefore represent the property of the helium core.In order to fit these four pulsation modes well,both of the stellar envelope and the helium core need to match the actual structure of the star.Then the size of the helium core is estimated to be MHe= 0.173 ± 0.004M?,RHe= 0.068 ± 0.001 R?.Papar (?) et al.(2013)extracted a total of 52 independent oscillation frequencies from the CoRoT timeseries.Based on the regularities in rotational splitting,we seek possible rotational splits in the observed frequencies of CoRoT102749568.There are four sets of multiplets with l = 1,nine sets of multiplets with l = 2,and eight sets of multiplets with l = 3.In particular,there are three sets of complete triplets(f10,f12,f14),(f31,f34,f35),and(f41,f43,f44).From frequency differences in these multiplets,the rotational period of HD 50844 is estimated to be Prot= 1.34+0.04-0.05 days.Our fitting results show that CoRoT10274958 is in post-main-sequence evolution stage.The physical parameters of CoRoT 102749568 are M = 1.54 ± 0.03 M?,Z = 0.006,fov= 0.004 ± 0.002,log g = 3.696 ± 0.003,Teff= 6886 ± 70 K,R = 2.916 ± 0.039 R?,and L = 17.12± 1.13 L?.Similar to the situation of HD 50844,most of pulsation frequencies of CoRoT 102749568 are mixed modes.Based on the properties of nonradial oscillation modes(f12,f34,f43)and the radial oscillation mode f13,the size of the helium core is estimated to be MHe= 0.148 ± 0.003 M?and RHe= 0.0581± 0.0007 R?.EE Cam was discovered to be a variable star in 1997.From 2006 to 2010,EE Cam was observed photometrically for 300+ nights.Breger et al.(2015)obtained 37 independent oscillation frequencies from the photometric data.We find five sets of multiplets with l = 1 and ten sets of multiplets with l = 2.The frequency spectra of EE Cam is disentangled only with oscillation modes with l= 0,1,and 2.This fact is very different from those of HD 50844 and CoRoT102749568.The rotational period is estimated to be Prot= 1.84+0.07-0.05 days.After doing model fittings,we find the best-fitting model for EE Cam.Its physical parameters are M = 2.04 M?,Z = 0.028,Teff= 6433 K,log L/L?= 1.416,R= 4.120 R?,log g = 3.518.Our results show that the two nonradial oscillation modes f11 and f32are mixed modes.They propagate like g modes in the helium core while like p modes in the stellar envelope.The fundamental radial mode f1 propagates mainly in the stellar envelope.Finally,the size of the helium core is estimated to be MHe= 0.181 M?and RHe= 0.0796 R?.