M.A. Holdaway
National Radio Astronomy Observatory
Socorro, NM 87801
February 20, 1996
% of
the time for simultaneous 86 GHz and 690 GHz observations, or 
% of the time for time shared multiband observations. Masers
fainter than 4 Jy will seldom be very useful for either technique
because the time required to detect them with sufficient SNR is
greater than the atmospheric stability time scale. We estimate there
are about 100 SiO masers with peak fluxes greater than 4 Jy. With an
86 GHz beam width of 
, simultaneous dual band calibration
will be relevant to about 0.04 square degrees. And finally, since SiO
masers are usually associated with evolved stars, most types of MMA
observations would not be aided by calibrating on an SiO maser in the
beam.
Because of the very limited usefulness of simultaneous dual band calibration, and because the Chilean atmosphere plus fast switching or radiometric phase correction will permit observing at 690 GHz for a substantial fraction of the time, we argue against incorporating simultaneous dual band observations with the MMA.
Several working groups at the October 1995 MMA Science Workshop expressed a desire to be able to observe with two or more bands simultaneously. The technical working group countered that dual band capability on the MMA would complicate the optics, and that dual band capability should be avoided unless absolutely required by the science. The meeting came to an understanding that dual band observing was scientifically justified when the source structure changes on time scales of seconds, as in the case of solar observing, and perhaps when the atmosphere changes on time scales of seconds, as in the case of submillimeter observing where simultaneous low frequency observations may be used for phase calibration.
The solar group was the only group who could justify the need for simultaneous dual band observing on the basis of the astronomical source structure changing over seconds, but they have since stated that subarrays chosen to provide comparable resolution is a more appropriate solution to their problem.
For the submillimeter calibration issue, we compare here the effectiveness of true simultaneous dual band observation on an 86 GHz SiO maser for calibration and a target source at 690 GHz, and time sharing between 86 GHz and 690 GHz.
If an SiO maser could be observed at 86 GHz simultaneously with a target source at 690 GHz, we could transfer the atmospheric phase from the lower frequency to the higher. This is attractive because the SiO maser is much brighter than the target source at 690 GHz, and the system sensitivities are much better at the lower frequency.
Lets say our maser has an average flux of 10 Jy over a bandwidth of
3 MHz (10 km/s), or a peak flux of about 20 Jy. With 
, the noise per visibility for a single visibility (both
polarizations included) will be 1.14 Jy in one second. (A 
of 40 K at 86 GHz is much more optimistic than we assumed in the recent
exploration of fast switching in MMA Memo 139.) The error in the gain
solution will be about
or about 0.02 for our 10 Jy maser, leading to 
rms phase
errors. When scaled to 690 GHz, these will become 
rms phase
errors per antenna, or 
per baseline. However, this is only
part of the contribution of the phase error. We must also determine
how much the atmospheric phase changes over our 1 s of integration.
The residual phase error after simultaneous dual frequency calibration will be approximately
where 
is the phase structure function, v is the
atmospheric velocity, and 
is the time it takes to detect the
antenna based phase on the SiO maser calibrator with the required SNR.
In terms of the temporal structure function 
,
which we measure with the site test interferometer,
Use of the temporal structure function for this application is desirable as it bypasses the need to know the atmospheric velocity.
The alternative to simultaneous dual frequency calibration is
to time share between the 86 GHz and 690 GHz bands. For this
analysis, we have two extra parameters, the time required
to switch between frequencies 
, and the fraction
of time spent on the target source f. The residual phase
errors are now
or for 
and 
s,
With the target source fraction of 
, the noise in our
time sharing observations will increase by 
.
For our 20 Jy SiO maser,
we plot the distributions of residual phase errors for dual band
calibration and for time sharing calibration for the May-August Chile
atmospheric phase stability conditions in Figure 1.
Table 1 indicates the fraction of time the atmosphere will be good
enough to use 20, 10, and 4 Jy SiO maser calibrators.
Table: What fraction of the time will calibration on an SiO maser
of some flux result in residual phase errors of 
or less at 690 GHz?
These numbers are calculated using the May-August 1995 Chile phase stability
database. The cumulative source count estimates are derived from the
arguments in the next section.
We can draw a few important conclusions on phase calibration of a 690 GHz target source with an SiO maser at 86 GHz:
How many SiO masers are there? Considering the data from Jewell et al. (1991), which includes information on all previously known masers as well as masers they identified, in the v=1, J=1-0 line at 43 GHz, The survey of Izumiura et al. (1993) of SiO masers in the Galactic bulge does not find any masers as bright at 4 Jy peak flux. We also assume that the 43 GHz peak fluxes and velocity widths are comparable to those of the 86 GHz line which would be observed by the MMA. These data, if complete down to 25 Jy, imply a cumulative distribution function like
or about 100 SiO masers with peak flux brighter than 4 Jy. (Jewell et al. only cover about 75% of the sky, so there will be somewhat more SiO masers which can be viewed from the MMA, which can see about 90% of the sky if located in Chile or Mauna Kea.) Jewell et al. will be incomplete below some flux level, but there is no strong indication of incompleteness at the 25 Jy level in the log(N), log(S) plot shown in Figure 2.
The 
power law is a bit odd for the source counts. If known
SiO masers were dominated by local sources of similar intrinsic
luminosities, we would expect an isotropic distribution and a 3-D
integrated source count law like 
. If known SiO masers were
dominated by sources along our spiral arm alone with similar
luminosities, we expect a 1-D integrated source count law like
. If known SiO masers were scattered uniformly through the
Galactic plane, we would expect a 2-D integrated source count law like
. The distribution of known SiO masers with brighter than 
greater than 20 Jy km/s (our 4 Jy, 10 km/s minimum
useful calibrator) is shown in galactic coordinates in
Figure 3, and in celestial coordinates in
Figure 4. The distribution of masers does not cry
out for any of these possibilities. The 1-D source count law, which
comes closest to the power law which we derive from the data, can be
ruled as SiO masers come at all galactic longitudes. Accounting for
the part of galactic coordinates which was not seen by the SiO surveys
(ie, south of declination -30), about 75% of the masers brighter than
4 Jy are distributed quasi-uniformly, while the band within 
of
the plane contains an additional 25% of the masers. Masers brighter
than 50 Jy are not clumped towards the plane, indicating that at this
level the source counts really are dominated by local objects. That
the masers are more clumped towards the plane at lower levels
indicates that we have run out of local sources, or that we have
integrated beyond the scale height of SiO masers, and the more distant
plane sources have kicked in. The odd power law exponent of 
is probably due to a combination of effects such as the local sources
showing up only as very bright masers, the adding together of local
and plane sources, and possibly different populations of SiO masers
with different intrinsic luminosities. This all casts a bit of doubt
on the true number of masers there may be which are brighter than
4 Jy, but we expect that our estimated number is in error by well
under an order of magnitude.
The 
40 SiO masers brighter than 20 Jy will permit phase stable
observations at 690 GHz of several interesting galactic regions in and
out of the plane, even under marginal phase stability during the
Chilean winter. On the other hand with a 
primary beam at
86 GHz, all of the usable SiO masers will make less than one tenth of
a square degree of of sky accessible to 690 GHz observations, and
mostly in the regions about evolved stars where SiO masers tend to
live. Clearly, a more general technique, such as radiometric
correction or fast switching, will be required to phase calibrate the
typical source at 690 GHz. Preliminary investigations of fast
switching phase calibration at 690 GHz indicates that it should work
about 25% of the time during the Chilean winter at 690 GHz. Since
methods which do not require dual band capability will work a
reasonable fraction of the time, it seems that simultaneous dual band
observations are not justified.
Acknowledgements
Many thanks to Phil Jewell for useful discussions on the topic of SiO maser source counts.
References
Izumiura, H., et al., 1994, ApJ, 437, 419.
Jewell, P.R., et al., 1991, A&A, 242, 211.
Figure: 1 Residual phase errors per baseline on a target
source at 690 GHz for simultaneous dual band calibration
on a 20 Jy peak flux SiO maser (solid line) and for time sharing
observations spending half the time on the target source
(dashed line). The sharp cutoff at 
is due to the
effect of system noise on tha gain solution in 1 s of
integration on the SiO maser. Our rule of thumb is that
residual phase errors will result in very good
images.
Figure 2: 
plot for known SiO masers brighter than 2 Jy peak
flux. The straight line is a fit to the data for masers with peak flux
brighter than 25 Jy. There does not appear to be any significant problem
with completeness in Jewell et al.s's sample at the 25 Jy level.
Figure 3: Distribution of known SiO masers brighter than 4 Jy peak flux
in Galactic coordinates, sin of latitude preserves sky area. The solid
line indicates the region in galactic coordinates which is below declination
-30, and was not accessible to the Effelsberg telescope.
Figure 4: Distribution of known SiO masers brighter than 4 Jy peak flux
in celestial coordinates.