Determining the size of the helium core of KIC 9970396 using asteroseismology: a red giant approaching the red giant bump

Mixed modes are less affected by the surface effect and carry internal information of a star. Asteroseismic modelling of mixed modes can constrain the size of the helium core of a star. Previous work has shown that the evolution stage of KIC 9970396 may be near the red giant bump phase, which is one of the most interesting regions in the red giant evolution stage. It is known that there is a clear relation between the core and mass luminosity, following homological principles on the low-luminosity giant branch. Our aim in this paper is to determine the size of the helium core and constrain the internal structure of the star. Our approach is first to identify the most p-dominated frequencies in the observational mixed modes. The calculated frequencies are then fitted to the observational frequencies, which we choose not to correct for surface effect offsets. We have computed a grid of theoretical models to fit the l = 1 modes of KIC 9970396. The size of the helium core of KIC 9970396 is determined to be M-He = 0.229 +/- 0.001 M-circle dot and R-He = 0.03055 +/- 0.00015 R-circle dot. The best-fitting model shows that KIC 9970396 is undergoing a stage that is between the first dredge-up event and the red giant bump. Compared with the results of our model, the previous core-mass luminosity relation overestimates the helium core mass by 0.009 M-circle dot when the stellar luminosity of a red giant star is known.