or
better requires the use of two black-body loads.
Accurate radiometer calibration for millimeter wave telescopes is an important goal. Well calibrated observations better permit the inference of physical properties of the observed sources. In particular, meaningful comparisons of observations taken at different wavelengths or with different instruments depend on accurate calibrations of the various instruments.
The technique presently used at most millimeter telescopes is
the "Chopper Wheel Method" introduced by Penzias and Burrus (ARAA, 1973, 11,51),
in which the output deflection of an observed source is compared with the
difference between the sky brightness and that of an ambient temperature load.
The physical temperature of the absorbing atmosphere is then taken to be the
same as the ambient load or a mean model of the atmosphere is used. The
method is convenient to use, and absolute accuracies of a few percent are
typically obtained. Obtaining accuracies at the level of or
better requires the use of two black-body loads.
We describe here a scheme in which the calibration loads are introduced
in sequence at the center of the secondary mirror of a symmetric
Cassegrain telescope. The loads are coupled through a small hole in the
center of the secondary which has a diameter equivalent to the aperture
blockage of the secondary with respect to the primary. The two loads have
temperatures of 300K and 400K, respectively, and are alternately presented
to the Cassegrain focus feed by a 
flat mirror behind the secondary hole.
After calibration has been accomplished, the mirror is rotated to a third
position which presents a concave conical mirror to scatter ambient radiation
from the vertex window away from the feed. The coupling to the feed horn will
be 
, providing a difference signal of 1-2K to the radiometer on top of
the antenna temperature. Because the fluctuating brightness of the atmosphere
will be present during the calibration, the mirror may be spun rapidly
for a rapid comparison between the loads.
In the following sections we describe the various components of the calibration system and conclude with a discussion of expected errors. We also describe a procedure for measuring the atmospheric extinction.
Figure 1 shows the Cassegrain optics with a ray leading from the edge of
the vertex window to the feed after reflection from the secondary. The locus
of other rays that couple the vertex window to the feed reflect from the
secondary closer to the optical axis and are usually scattered to the
outer parts of the primary by the cone showed as a dotted line. We use the
part of the secondary that is normally covered by the cone as the coupling
hole for
the calibration loads. We assume that when the mirror images one of the
loads in the hole that the hole has a uniform brightness at that temperature.
The radius of the hole is b and that of the secondary is 
. If g(r)
is the axially symmetric feed horn gain projected onto the secondary, then
the coupling of the small aperture to the feed, k, is given by:
This expression ignores the spillover past the secondary which is typically
. A simple approximation to the feed pattern is given by
which gives the following for the coupling.
For the BIMA 6.1m antennas, the secondary mirror diameter is 60.96 cm
and the coupling hole diameter is 5.84cm. The illumination at the edge of
the secondary is about -10 db (i.e., a = 0.68). Thus the we find k=.0195, or
about 
.
Figure 3 shows one of the loads, a simple metal cone with absorbing
material glued to its inner surface. The conical shape insures high
emissivity. Relative to the normal to the cone base, the slope of the inner
surface is at an angle close to the Brewster angle, so that an incident ray
with its polarization
normal to the surface would be absorbed upon reflection. The other polarization
would be absorbed after many scatterings before reaching the vertex. Castable
Eccosorb from Emerson 
Cumming would be a good candidate for the absorbing
material. Note that because the conical load is such a good match,
its brightness is not critically dependent on its orientation.
The metal cone will be embedded in insulating material which is transparent
to the millimeter/submillimeter waves emitted by the load. The metal part of
the conical load will have
its temperature regulated to an accuracy and stability of .01K (RMS). This
expected accuracy is based on our experience with regulating the temperature
of the electronics on our present front ends. The
temperature will be sensed by an array of thermistors, and the metal will be
heated by a wraparound heater. The servo control will be analog, using a
circuit already developed for control of the receiver electronic components.
Emerson & Cumming have an insulator, Eccosorb Q, made of fused quartz fibers
which has two important features for this application. First, the fused
quartz is transparent to millimeter/submillimeter wavelengths. Second,
it can withstand very high temperatures, easily higher than the 400K planned
for one of the two loads, which many insulators cannot. For the
coupling for the BIMA antennas discussed above, the difference between the two
loads at temperatures of 300K and 400K will provide a calibrate signal of
=2K. Note that we expect the temperature regulation of the metal
parts of the loads to be stable to .01K. The ambient temperature of the space
where the loads will be located will vary over a large range, on the order of
50K. This variation will produce a small variation in the radiating
surface temperature
of the load, and, as a result, the effective black body temperature will be
stable only to 0.1K. Nevertheless, the 
should be stable to
.002K, that is, to a fractional accuracy of 
.
Figure 4 shows two views of the neighborhood of the secondary mirror. One
is a side view, and the other is an axial view. The mirror can rotate
about the axis of the telescope. Depending on its angle of rotation, it
brings an image of either of the two loads or a scattering mirror (to be
discussed below) to the hole in the secondary. The mirror will be
rotated by a stepping motor and can be stopped at each load or the scattering
mirror. It can also rotate continuously for a rapid comparison between the
loads. For example, 600 RPM corresponds to a 10 Hz chop. Because the
fluctuating brightness of the atmosphere is present during this calibration,
it will be usually necessary to make a rapid comparison between the loads.
The scattering mirror, which can be imaged in the secondary mirror hole,
takes the place of the scattering cone discussed above. This mirror is
conical and concave with a cone angle of 
for the BIMA optics. Figure 2
shows how rays leaving the Cassegrain feed fall on the the primary mirror
outside the vertex window after being reflected from the conical mirror. As
rays below the axial line are reflected above the line toward the primary,
there is no vignetting by the hole in the secondary. The scheme works with
sufficient margin that it is also effective for the off axis feeds.
The calibration signal will be approximately 2K at all wavelengths. It's precise value will be determined by occasional comparison with large sized loads which can be placed over the vertex window. One could be at ambient temperature and the other at, say, liquid nitrogen temperature. With care, this primary calibration can be made to an accuracy of a few tenths of a percent. As a result, the secondary calibration loads should provide that level of precision.
Consider the following expected system numbers for the MMA: BW = 8 GHz, and
Tsys(300GHz) = 50K. In a one second measurement, 
=.0006K. In that
time the calibration signal is measured with S/N =4000, more than adequate
for continuum measurements. In a ten second measurement, a one MHz spectral
channel can be calibrated to an accuracy of . Thus this calibration
scheme should be useful for spectral bandpass calibration, as well as wide
band calibration.
Unlike the Chopper Wheel method which provides a calibration with
atmospheric extinction automatically taken into account, the present plan
requires a separate observation of the extinction by means of tipping
curves. The secondary mirror calibration fixes the radiometer scale. The
output due to an input power 
is
with
and 
, the zero point, is determined by simply turning off the input
amplifiers.
The input to the receiver has several components.
is the bare antenna temperature with no atmosphere or
background
radio sources. It will have a small zenith angle dependence due to variable
ground pickup. It must be separately calibrated by means of numerous tipping
curves which allow the other components, including the time variable
atmospheric emission, to be removed. It should be constant, except for the
scaling to the current ambient temperature, for zenith angles less than about
. At longer zenith angles, the antennas will begin to see each other in
the spillover, especially in the more compact arrays. 
is the
receiver noise temperature. 
is the effective temperature of the
2.74K background. The last term is the contribution of the atmosphere with
the assumption that the air temperature is constant with altitude. A more
comprehensive model could be used but probably is unnecessary. With small
optical depth, the last term is approximately 
.
Tipping curves, measuring 
, should be made with a large number
of antennas in the array to produce an accurate average curve, one in which
the atmospheric brightness fluctuations are averaged out. The range of z should
be to z= from the zenith. With known, can be
found
from the intercept on the secz=0 axis along with the factor 
by the least squares fitting
of the equation. The goal is to find 
so that the measured visibilities
and antenna temperatures can be corrected by the factor 
.
The sensitivity discussion above suggests that the basic receiver
calibration should be accurate to about . We now consider the probable
errors resulting from the extinction correction. The sensitivity in the
atmospheric emission term is more evident if we use the approximate form:
The fit of the tipping curve is with respect to the variable secz. The term
is small, and is known. Either an average value of
may be used for this term, or it may be found by iteration in the fit. Suppose
the fitting uncertainty in the factor is 
. Then
Since the errors are uncorrelated,
Representative values for the uncertainties are: =0.5K for the
fitting, 
10K, 
280K and
0.2, which lead
to 
= .009. For z=, the correction error is
This is an error of about .
It is instructive to work out the corresponding errors in the use of the
"Chopper Wheel" calibration. In the observation of a radio source which
yields an antenna temperature 
, we make three measurements. 
is the
output from an observation of the sky near the source.
is the output with an ambient load in front of the vertex window.
is the output with the antenna pointed at the source.
Then
We can find k from 
.
Finally, the source antenna temperature, , is given by
The important point is that the main term in the right hand braces is
, which does not depend on the extinction. In addition, it is usually assumed that 
and are approximately the same and
equal to about 280K, a mean air temperature. The other terms in the right
hand braces are indeed small, but they are also not well known, at least in
part because the
extinction is not known. Altogether, the unknown quantities are , ,
, and 
. An estimate of the fractional error in using equation (17) 
, is:
where 
. For the various terms we estimate the following
values which are consistent with the errors used in the calculation above for
the calibration scheme proposed in this memo. =0.4,
=5K, 
=10K, 
=10K,
280K, 
=0.4, and
15K. The final error is 
. This
is about 5 times larger than the other estimate above. Some of these
parameters could be measured, which would improve the accuracy. At the same
time, a measurement of the air temperature for the scheme proposed in this
memo could reduce its uncertainty significantly. It seems that the greater
error for the "chopper wheel" method is inherent in the lack of having two
loads to establish the temperature scale independent of using a model for
the atmosphere. The penalty is the need to make the tipping
curves. However, that measurement should only take a few minutes, and the
improved accuracy is probably worth the effort.
The proposed plan, in which we have two black body loads weakly coupled at the secondary mirror has a number of advantages.
level or better.
