  Understanding the sources of instrumental systematic noise is a must to
improve the design of future spectrographs. In this study, we alternated
observations of the well-suited pair of M-stars GJ 725A+B to delve into the
sub-night HARPS-N response. Besides the possible presence of a low-mass planet
orbiting GJ 725B, our observations reveal changes in the spectral energy
distribution (SED) correlated with measurements of the width of the
instrumental line profile and, to a lower degree, with the Doppler
measurements. To study the origin of these effects, we searched for
correlations among several quantities defined and measured on the spectra and
on the acquisition images.
  We find that the changes in apparent SED are very likely related to flux
losses at the fibre input. Further tests indicate that such flux losses do not
seriously affect the shape of the instrumental point spread function of
HARPS-N, but identify an inefficient fitting of the continuum as the most
likely source of the systematic variability observed in the FWHM. This index,
accounting for the HARPS-N cross-correlation profiles width, is often used to
decorrelate Doppler time-series. We show that the Doppler measurement obtained
by a parametric least-squares fitting of the spectrum accounting for continuum
variability is insensitive to changes in the slope of the SED, suggesting that
forward modeling techniques to measure moments of the line profile are the
optimal way to achieve higher accuracy. Remaining residual variability at ~1
m/s suggests that for M-stars Doppler surveys the current noise floor still has
an instrumental origin.
