A Study of the Magnetic Activity and Variability of GJ 436

We present a magnetic activity study of GJ 436 using spectroscopic data from HARPS, spanning over 14 years, and additional data from NARVAL, falling within the HARPS observations. We study the CaII H&K, HeID3, NaI doublet, H$\alpha$ and CaII IRT triplets lines and explore linear correlations between them. Using the full HARPS dataset, we found indices H$\alpha$ vs CaII H&K & H$\alpha$ vs HeI to correlate positively. From the NARVAL dataset, covering one observing epoch, we found CaII IRT$_{3}$ vs CaII IRT$_{2}$ & CaII IRT$_{3}$ vs H$\alpha$ index to correlate negatively. We investigate long and short-term periodicity in these index variations using the Generalised Lomb-Scargle periodogram. For CaII H&K, NaI and H$\alpha$ indices, we detect long-term periods of 2470.7d (~ 6.8 years), 1861.6d (~ 5.1 years) and 2160.9d (~ 5.9 years) respectively, consistent with GJ 436's photometric cycle of ~ 7.4 years. Applying the "Pooled Variance" technique to H$\alpha$ & NaI indices, we found ~ 2500d to be the period of an activity cycle mechanism, in good agreement with the detected 2470.7d period. For CaII H&K and H$\alpha$ indices, we detect short-term periods of $39.47^{+0.11}_{-0.15}$d and $40.46^{+0.44}_{-0.52}$d respectively, identified as the stellar rotation period. The stellar rotation is detected after prewhitening the long-term periodicity. It is detected as well in the analysis of individual observing epochs.