Evolution of accretion disc geometry of GRS 1915+105 during its χ state as revealed by TCAF solution

The evolution of the C-type low frequency quasi-periodic oscillations (LFQPOs) and associated time lag in transient black hole sources as a function of time can be explained by variation of the Compton cloud size in a Two Component Advective Flow solution (TCAF). A similar study of a persistent source, GRS~1915+105, has not been attempted. We fit the evolution of QPOs with propagatory oscillating shock (POS) solution for two sets of so-called $\chi$-state observations and find that the shock steadily recedes with almost constant velocity when QPO frequency is decreasing and the spectrum is hardening. The shock moves inward with a constant velocity $v_0=473.0$ cm s$^{-1}$ and $v_0=400.0$ cm s$^{-1}$ respectively in these two cases, when the QPO frequency is increasing and the spectrum softens. This behavior is similar to what was observed in XTE~J1550-564 during the 1998 outburst. The time lag measured at the QPO frequency varies in a similar way as the size of the Compton cloud. Most interestingly, in both the cases, the lag switches sign (hard lag to soft lag) at a QPO frequency of $\sim 2.3 - 2.5$ Hz irrespective of the energy of photons. We find, at very low frequencies $< 1$ Hz, the Comptonizing Efficiency (CE) increases with QPO frequency and at higher QPO frequencies the trend is opposite. The time lags become mostly positive at all energies when CE is larger than $\sim 0.85\%$ for both the sources.