In Sect. 2 we have shown that, in the absence of an accurate a-priori knowledge of the Galactic emission, absolute calibration can be done on time scales of several days. It is important then to address the question of the relative calibration, i.e., to estimate at what level one can control the radiometer gain stability over a time scale of hours or days.
For relative calibration, the accurate knowledge of the observed signals is not critical, as
long as it is highly stable within the time scale of interest. By repeatedly observing a stable
signal difference 
one can monitor long-term gain changes. If a gain drift
in a time scale 
is occurring, the instrument will measure at
time 
a signal difference 
, and at time 
a signal
difference 
. Assuming a well balanced
differential system, the measured change in the signal difference is related to the gain
change by
From Eq. (5), the
accuracy with which one can account for variations in the gain is:
where 
is the rms noise associated with the
measurement of 
, according to Eq. (6). So we obtain:
Equation (8), as expected, reduces to
Eq. (20) when 
, since in this case the accuracy of
the calibration load temperatures is not relevant.
The time scale of the expected gain drifts depends on the source of instability. Gain
variations can be due to thermal changes or can be intrinsic of the detectors.
The differential receiver minimizes the effect of intrinsic 1/f noise. In
addition, residual effects on time scales < 2 hours are averaged out by the large number
(typically 
) of successive measurements of the same sky pixel (observed by
different radiometers at a given frequency), with the net effect of a small
increase of the nominal white noise level of the receivers (e.g. Janssen et al. 1996). Our main
concern here is then our ability of tracing slow drifts due to residual 1/f noise on
time-scales >2 hours. Laboratory measurements suggest that one may expect gain
variations of order 
in few hours, or 
in 
hours (Weinreb 1996).
Possible thermal effects on the gain are quite small, thanks to the choice of the L2 orbit and
scanning technique. Thermal variations can only arise due to re-pointing maneuvers away
from the anti-sun direction. A thermal study
(Bersanelli et al. 1996) shows that any temperature drifts at the focal
plane will be dominated by a slowly varying component
with a time-scale of 
hours, so that for time scales 
hours thermal effects can be considered linear. Thermal
effects on the gain are expected only at a level 
in the 100 hours period.
We have simulated the presence of a linear drift in the noise-added data stream with a rate
of 
per hour. As shown in Sect. 1, in order
to disentangle a gain drift from a baseline drift in the internal load
(which will be discussed in the next section), we need to monitor
signal differences rather than the signal intensity in a fixed pixel.
We follow the circle described by the beam in two hours. The voltage associated to each
pixel is
where G is the gain at an arbitrary time t=0,
min, k=1,2,...120, and 
is the temperature relative to
the internal load of the i-th pixel at time 
due to the Galaxy
and the CMB dipole. We then evaluate 
, the difference
between 
and the signal of that pixel in the circle where
the dipole is expected to provide the minimum contribution. We assume
that the instrument response is stable during each spin period (1
minute). Since 
,
we can write:
Note that the
load temperature (and its possible drift occurring on time scales
min) has been
eliminated. Now we can average the previous expression over a circle obtaining
where the notation 
, N being the number of elements in a given circle. Note that in this case
the Galactic emission provides additional
modulation to the signal observed in each circle, thus contributing to the detection of gain drifts. We
have then a set 
(and associated noise) with k=1,120, that we
can fit to a linear law 
. The ratio between the angular coefficient b and the zero-value
a provides the estimate of the gain drift. Due to the slow gradient of the dipole, the time
scale can be expanded up to 
successive circles with basically no additional
uncertainty on the result. This allows to further averaging the values of 
already evaluated for each circle.
Figure 6: Reconstruction of long-term gain drifts (relative
calibration) from the closure pattern of the scanning technique.
A drift 
hour was added to the data. Here the results
are presented for the drift as reconstructed in each 2-hour circle,
corresponding to 
(dashed horizontal lines).
Each point represents the simulation results based on 6 hours of observation.
The regions of increased spread correspond to the periods
when the scan geometry is such to have minimum
sky signal modulation within the observed circle
The results of our calculations are presented in Fig. 6 (click here)
for the four LFI frequencies. Each
point in the plot represents the algorithm reconstruction of a drift of 0.25%
per hour (or 0.5% per circle) based on 6
consecutive circles (corresponding to 12 hours of data). Again, the accuracy of the results is
degraded significantly in the two periods of the year when 
is smaller due to the geometry of the CMB dipole, which dominates in general
over the Galaxy. On average, however, it is apparent that the relative calibration allows to
recover the gain drift with high accuracy at all frequencies, with a typical uncertainty of less
than 
on G for the assumed 12 hours time scale.
Our simulations show that
this result is nearly independent of the amplitude of
the assumed drift.
This demonstrates that even in the absence of frequent absolute calibration, relative calibration can provide an accurate monitor of the long-term gain stability.