Systematic investigation of X-ray spectral variability of TeV blazars
  during flares in the $RXTE$ era
  Utilizing all the 16-year $RXTE$ observations, we analyze the X-ray spectra
of 32 TeV blazars, and perform a systematic investigation of X-ray spectral
variability for the 5 brightest sources during their major flares that lasted
several days. We obtain photon spectral index ($\alpha$), flux and synchrotron
radiation peak energy ($E_{\rm{p}}$) from empirical spectral fitting, and
electron spectral index ($p$) from theoretical synchrotron radiation modeling.
We find that both $\alpha$ and $p$ generally display a harder-when-brighter
trend, confirming the results of many previous works. Furthermore, we confirm
and strengthen the result that $p$ must vary in order to explain the observed
X-ray spectral variability during flares, which would have useful implications
for interpreting the associated higher-energy spectral variability. We see
apparent electron spectral hysteresis in many but not all $p$-flux plots that
takes a form of "loop" or oblique"8". We obtain a tight $p$-hardness ratio (HR)
relation and a tighter $p$-$\alpha$ relation using spectra of flaring periods,
both of which are also applicable to stacked data of quiescent periods. We
demonstrate that these two empirical relations can be used efficiently to
estimate $p$ from HR or $\alpha$ that is readily achieved. Finally, we find
that, when considering TeV blazars as a whole, $\alpha$ and X-ray luminosity
are positively correlated, $E_{\rm{p}}$ is negatively correlated with $p$ and
$\alpha$, and $E_{\rm{p}}$ is positively correlated with HR; all these
correlations are in line with the blazar sequence. However, after correcting
for the Doppler boosting effect, $\alpha$ and intrinsic X-ray luminosity follow
an anti-correlation.
