The purpose of an absolute calibration is the determination of the values G (hereafter we refer to a given channel and drop the subscript n)
according to Eq. (1) through the measurement of
(at least) two signals corresponding to well known intrinsic antenna
temperatures 
and 
. If their difference 
is a priori known with an uncertainty
, from Eq. (1) we find the uncertainty on G
to be
The measured voltage difference has a spread
Here 
is the rms noise associated with
the measurement of 
or 
in an integration
time 
:
where K is a constant of order unity depending on the type of receiver (for the differential baseline LFI receivers 
),
is the bandwidth,
and 
is the system noise temperature.
Therefore, to first
order, the uncertainty in the absolute calibration is given by:
or, according to Eq. (6),
From Eq. (7) it appears that a large calibrating signal can produce higher accuracies for a given integration time and sensitivity. However, the useful amplitude of the signal is limited by saturation effects occurring in the amplification chains of the radiometers.
Because of our peculiar motion relative to the comoving frame, the CMB Planckian radiation
field of temperature is observed to have a temperature
Here
K is the CMB temperature (Mather et al.
1994) while 
and 
are the observing line of
sight and the direction of our motion relative to the comoving frame,
respectively. The amplitude (
mK)
and direction 
of the CMB dipole have been accurately measured by the
COBE-DMR instruments (Kogut et al. 1993; Lineweaver et al. 1996). The
dipole emission will show up in the LFI data as a continuous
modulation of the signals V(t) with accurately known amplitude. The
CMB dipole is thus a particularly attractive source as it can be used for calibration without reducing the efficiency of the observations.
For a first-order evaluation of the achievable accuracy, we have calculated the antenna
temperature difference, 
, between the maximum
and the minimum of the dipole modulation for each spin rotation during
the first year of the mission (2003), as seen by each channel at the
four LFI frequencies. We then estimated 
from Eq. (7)
assuming 
in a portion of 
around 
and 
,
respectively. This rough calculation suggests that the CMB dipole
allows for absolute calibration accuracies better than 3% for all LFI
channels, at a rate of about one day. This estimate, however, besides
the simplifying assumption in the fit, neglects the presence of the
Galactic emission which is mixed to the dipole signal. The Galactic
contribution is known with relatively poor accuracy in the frequency
range of interest (30-140 GHz), making the regions with strong
Galactic signal useless for accurate absolute calibration.
We performed a more refined study by simulating the
microwave sky (CMB dipole and Galaxy) as observed
at each LFI frequency, and by
reproducing the actual 
offset, 1 rpm scan pattern of the LFI array. We model the Galactic contribution including the
synchrotron, free-free and dust components.
The synchrotron emission at the LFI frequencies is obtained by
properly rescaling the 408 MHz full-sky survey of Haslam et al. (1982):
where 
, and the spectral index 
is assumed to have a constant value over the sky. Although
pixel-to-pixel variations of the spectral index are
expected, as suggested by a comparison of the Haslam 408 MHz
and Reich and Reich 1.4 GHz maps (Lawson et al. 1987), our
results are not affected significantly by this effect. We
consider the two values 
, the upper
limit recently derived by Kogut et al. (1996) from the COBE
data, and 
, a more ``conservative"
value commonly used in the literature. the results The angular
resolution of the synchrotron emission has been artificially increased by interpolating the 408
MHz map on pixels of sizes equal to the FWHM of the antenna beams.
To account for free-free emission we considered both a diffuse component
[with a cosec(
) dependence]
and a compilation of 
HII regions
near the Galactic plane (Witebsky 1988).
The HII regions have been
first convoluted with the angular response of the antenna beam, approximated with
a Gaussian
of dispersion 
. We extrapolated free-free emission
to the observing frequencies by assuming a spectral index
. The dust emission has been modeled by
extrapolating the IRAS 
map with a spectral index
.
Accurate calibration needs to take into account
the presence of the CMB fluctuations themselves.
High order multipoles should not affect significantly
the dipole calibration, since their global contribution
to the dipole reconstruction will tend to average out.
On the other hand, ignoring the quadrupole and other
low-
CMB components
would result in a significant error in the calibration
(of order of 1%), and
the a-priori accuracy in the value of these terms as
determined by COBE is relatively poor.
However, it will be possible to construct
an iterative algorithm yielding the CMB sky fluctuations and the
calibration by successive approximations.
In these simulations the presence of the CMB cosmological
quadrupole and higher order multipoles
was not included.
The following algorithm was constructed to simulate the observations.
We sample the circle scanned by the beam with N independent
elements (or ``pixels"), where N=720 at
31.5 GHz, N=1200 at 53 GHz and N=1800 at 90 and 125 GHz.
Thus, the simulated output of a given scan is a set
of N voltages 
corresponding to the expected sky temperature (dipole plus Galaxy)
in each direction, to which we add a Gaussian noise
characteristic of each radiometer at the
proper integration time:
For each pixel
in a given circle, we then calculate the simulated signal difference
between opposite pixels on the circle. To minimize
Galactic contamination, the procedure eliminates all
the pixels for which the expected Galactic temperature difference,
, is a
significant fraction, 
, of the CMB dipole temperature
difference, 
. The value of this
threshold 
was left as an adjustable parameter. We
assume here that G is stable over the time scale of the absolute calibration
(in the next sections
we will discuss correction of long-term gain drifts).
Thus every 120 minutes we have a set of N/2
differences, 
, with a dispersion
,
where 
is the instrumental noise per pixel
(Eq. 6). We perform the absolute calibration by
minimizing
where 
is the a-priori-known value of the dipole temperature
difference corresponding to the observed directions.
It is clear that we can neither increase nor decrease too much the threshold 
. In fact,
in the first case the number of surviving pixel pairs tends
to zero; in the second case the number of pixel pairs
contaminated by the Galactic emission will be large, and the
minimization of the 
will yield answers affected by
systematic errors. It turns out that in order to achieve
accurate calibration one needs 
,
depending on the frequency. Our results are shown in Fig. 1 (click here).
Figure 1: Absolute calibration against the CMB dipole
at the four LFI frequencies. Each point represents the ratio between the reconstructed
value, G, and the assumed input value, 
, of the calibration constant.
The results are shown for
two assumptions of the synchrotron spectral index:
(left hand panels) and
(right hand panels).
The horizontal limits show the a-priori uncertainty of
the CMB dipole amplitude from the COBE-DMR data
The frequency of the calibration and the choice of
the threshold 
have been set at each frequency aiming at
an upper limit to the uncertainty 
, which
corresponds to the intrinsic limit of
the accuracy of 
as measured by COBE-DMR.
At the highest frequencies (90 and 125 GHz) the Galaxy is
relatively unimportant. The calibration constant is recovered within
every 10 days for most of the mission
lifetime, except in two periods (approximately around days 70 and 250), which correspond
to those circles where the differential dipole signal,
as observed by the LFI
detectors, has a minimum. This confirms the comment to Eq. (7) above.
At 31.5 GHz, in order to avoid contamination from the
Galaxy a severe threshold 
is needed. For this reason one
can derive G at the same accuracy level only once every 
days.
The fit is achieved again by minimizing Eq. (13), where the sum is
now extended over the ``clean" pixels of 264 adjacent circles.
A comparison between the results for 
and 
indicates that the dependence
on the assumed Galactic model is weak. Note that we have
not attempted any subtraction of the Galactic components.
New accurate observations of the Galactic foregrounds,
expected as part
of a mission like COBRAS/SAMBA, will allow to construct an
iterative subtraction procedure which will improve the calibration
accuracy by further reducing systematic errors.
Performing the fits over more extended
time periods can largely reduce the statistical error.
For example,
at 31.5 GHz, with a threshold 
the statistical
error is below 0.1%
when considering 1 year of observations (i.e. just one calibration per year).
However,
the ultimate limit to the overall accuracy in the absolute calibration
is set by the uncertainty of a-priori knowledge of the CMB dipole
amplitude and direction. The
four-year COBE-DMR data yield a best fit dipole amplitude
with a 
accuracy (Lineweaver et al. 1996).
Position uncertainties
will be dominated by the COBE-DMR 
error in the
absolute (
) dipole direction, which propagates to 
error. Due to the slow angular gradient of the dipole,
the COBRAS/SAMBA pointing uncertainties (2.5' in 30 minutes)
will result in a negligible contribution (less than 0.03%) to the total
error.
We can conclude that the CMB dipole will allow frequent (
days)
absolute calibration reaching the limit
set by the intrinsic uncertainty on the calibrating signal.
This uncertainty induces a 
K
uncertainty in the amplitude of the small scale CMB anisotropy, which is
adequate to the mission goal.
To push the long-term (
1 year) calibration accuracy beyond the
COBE-DMR uncertainty in the
CMB dipole amplitude, we can use the Doppler effect due to the
satellite orbital velocity.
This effect was also used by COBE-DMR for absolute calibration
(Kogut et al. 1996). In the case of COBRAS/SAMBA,
after correction for the small (
)
component of the satellite motion in the
L2 rest frame, the seasonal velocity component
around the Sun (
. on average,
or 
) produces a modulation in the
observed CMB dipole with an expected amplitude of
.
We have simulated, accordingly, the reconstruction of the dipole
amplitude during one year mission. In Fig. 2 (click here) we show the simulated
Figure 2: Simulation of the observed modulation in the value of the
CMB dipole due to the orbital velocity of the satellite.
This modulation can be used for overall calibration of the maps with accuracy
modulation at 90 GHz, after subtraction of the Galactic
emission assuming 
.
The ``observed" signal modulation is then fitted to
the expected periodic curve which, to very good approximation, is a cosine.
The recovered amplitude is 
.
This result indicates that
an overall calibration accuracy of 
at 90 GHz can be
achieved with this method
(somewhat degraded accuracies are expected
at lower frequencies, due to second order effects from
the stronger Galactic emission). This will represent the most
accurate overall calibration of the maps on very long time
scales (
year), and it will be complementary to the
shorter term
CMB dipole calibration discussed in the previous section.
External planets are good candidates as calibration sources since they provide a signal with suitable intensity and high stability (after correction for slowly varying seasonal effects). Each planet will fall in the COBRAS/SAMBA field of view only about twice per year. Dedicated re-pointing of the satellite is not currently included in the mission plan since it would significantly complicate the observing strategy and increase the required consumables. However, even if occurring occasionally, observations of the external planets will yield a valuable cross check of the CMB Dipole calibration.
The measured antenna temperature of a planet, 
can
be written as:
where 
is the frequency-dependent brightness
temperature of the planet, 
is the antenna beam pattern, and 
and 
are the line of sight and planet directions,
respectively. Since the planet's solid angle is small compared to the
beam size, 
, we can
approximate Eq. (14) as follows:
where J is now assumed to be symmetric.
Assuming the spacecraft at the Lagrangian point L2 in anti-sun configuration, the distance
at which the planets will be when in the field of view of the LFI is given approximately by:
where 
, 
is the
average planet-Sun distance, and 
AU is the spacecraft-Sun distance
(see Fig. 3 (click here)). The accurate spacecraft-planet
distances and relative angular sizes, calculated from the
Lissajous orbit around L2 including orbit eccentricity and
inclination on the ecliptic, are given in Table 1 (click here) for all the
pointing events in the first two-years mission. All the listed
calibrations will have the planets within 2' of the beam center.
In Table 1 (click here) the reference beam is assumed at the center
of the array. The entire array will be calibrated within a few days
from the indicated time, as the spacecraft spin axis is moved.
Table 1: Calibration signals from external planets
Figure 3: Spacecraft-Planet geometry when an external
planet, P, is in the field of view of the COBRAS/SAMBA
detectors
Several authors
reported measurements of the planets brightness temperature at millimeter wavelengths
(e.g. Ulich et al. 1980; Sholherb et al. 1980; Epstein et al. 1980;
Ulich 1981;
De Pater & Matisse 1985;
Dowling et al. 1987; Rudy et al. 1987;
De Pater 1989, 1990;
Muhleman et al. 1991).
The accuracy of planetary absolute calibration is limited by the uncertainty on
the intrinsic radiometric temperatures 
.
We estimate accuracies of 
on 
,
on 
and 
, and 
on 
and 
.
Based on published data and on the above determination
of the planets solid angles we derived estimates of the antenna temperatures
during calibration as detected by the LFI channels (last four columns
of Table 1 (click here)).
The results show that Mars, Jupiter and Saturn have useful signal for calibration.
The higher angular resolution at the higher frequencies
makes the planets signal at detection much stronger at 90 and 125 GHz.
However, even at the level of the largest signals (
K in the
higher resolution channels) no significant saturation effects are expected in the LFI receivers.
Uranus and Neptune will also be detectable
sources with low (few mK) signal level, while Pluto will
give a 
signal.
In Fig. 4 (click here) we report the accuracy achieved for absolute
calibration using Mars, Jupiter and Saturn for the 31.5, 53 and 90 GHz receivers, as a 2-hours set of observations (120 spins) takes place.
Accuracies of 3% to 5% (i.e. the limit imposed
by the assumed intrinsic uncertainty in
) are generally obtained
in the 120 minutes time-scale. In the case of
Jupiter the large signal allows to reach such limit
in less than 5 minutes.
Figure 4: Absolute calibration accuracies using external
planets. The plot shows the convergence to the assumed
intrinsic uncertainties
(3%, 5%, and 5% for Mars Jupiter and Saturn respectively)
during a 2-hours set of observations (120 spins).
For each planet, the upper, middle and lower curves refer to the
31.5 GHz, 53 GHz, and 90 GHz channels, respectively. At 125 GHz
the results are very close to those at 90 GHz
The CMB dipole is an extended source filling completely the antenna beam, so that the calibration is not sensitive to the details of the beam pattern. On the other hand, when calibrating with sources with small angular size compared to the instrument resolution, such as planets, it is of primary importance to know the beam shape and to take into account pointing errors. The main lobes of the LFI detectors will be measured on ground as part of the testing procedures. In addition, in-flight measurements of the main lobes will be done exploiting the plantes emission itself, as the field of view crosses their positions in the sky. We have simulated the main lobe reconstruction process assuming a Gaussian beam for the 90 GHz channel, and using Jupiter's signal as reference. The results, shown in Fig. 5 (click here), demonstrate that the main lobe can be accurately recovered down to about -20 dB (see also Delabrouille et al. 1996).
Figure 5: Reconstruction of the 12' main lobe of the
90 GHz system using Jupiter as a source.
The values of Table 1 (click here) are calculated assuming
that the planets will fall exactly
in the beam center for each scan in the 120 minutes set of observations.
Pointing errors can be estimated from the
a-posteriori spacecraft
pointing stability, which is specified to be 2.5'.
Assuming Gaussian beams,
this will introduce additional errors of 
and 
for beam size of 12', 18' and 30', respectively.
The spacecraft will be re-pointed every 120 minutes by 5'.
If the planet is in
the beam center at a given time, after re-pointing the planet's signal will be attenuated by up to
(for the 12' beams).
In principle, one can integrate the signal from the planets
(or from other point sources) for a time interval longer than
120 minutes by taking into account
the change in the beam response due to the re-pointing of the system
by using their measured beam shapes.
