M. S. Yun, J. Mangum, T. Bastian, M. Holdaway
National Radio Astronomy Observatory
J. Welch
University of California, Berkeley
May 15, 1998
The 10% amplitude calibration accuracy achievable
with the current standard calibration techniques may be sufficient
to produce images with dynamic range of 
, but achieving
a dynamic range of 
or higher with the MMA requires better
than 1 percent accuracy in amplitude calibration. A self-calibration
technique may be applied to improve images, but it may not be possible
in all cases. Therefore, an emphasis is given to achieving
accurate initial calibration. Multi-transition spectroscopic studies
and multi-array synthesis also require high accuracy in absolute
sense as well.
The conventional ``chopper wheel" and a two temperature load method for amplitude and flux calibrations are examined. The two temperature load calibration offers a potential to achieve the 1% accuracy in amplitude and flux calibration, but it is technically challenging. In comparison, the chopper wheel gain calibration and astronomical flux calibration cannot provide better than 5% accuracy. Whether the more complex two temperature load system is justifiable for the MMA may ultimately dependent on how well the radiometric phase correction will work. Several other relevant issues including establishment of astronomical flux standards are discussed, and engineering goals are identified.
The aim of amplitude and flux calibration is to convert the output voltage or counts from the correlator into brightness temperature or flux density by carefully tracking the instrumental and atmospheric variations and determining accurate conversion factors. Because the adverse effects of instrumental and atmospheric variations grow rapidly with frequency, standard calibration procedures will not work well at submillimeter wavelengths. The high design goals of the MMA (e.g. high sensitivity and imaging with large dynamic range) also demand a much higher calibration accuracy than achieved by the conventional technique used at the existing millimeter arrays (about 10%).
In this memo, we first evaluate the required amplitude calibration accuracy in terms of the effects of amplitude error on the dynamic range of the images produced. In the following sections, the conventional ``chopper wheel" calibration method and a two temperature load calibration method are described. In §4, several potential approaches for flux calibration using astronomical sources and a direct calibration using the two temperature load system are considered. Flux calibration is discussed here in conjunction with amplitude calibration to bring attention to the fact that large amplitude gain variations expected at mm and submm wavelengths make the flux calibration inseparable from gain calibration. Additionally, the two temperature load calibration scheme can potentially offer the 1% accuracy needed in both amplitude and flux calibration simultaneously. Some special concerns for the MMA calibration such as polarization and solar observations are addressed and several engineering goals are identified in §5 and §6. Pointing error is another contributor to the gain drift, but it is already addressed extensively in the context of a more stringent requirement for mosaic imaging by Holdaway (1997b).
The dynamic range (DR) achievable from an observation of
length 
, calibrated at interval t using an interferometer
array with N antennas is
where 
is the number of samplings of the
atmosphere, and 
and 
are random
Gaussian errors in phase and amplitude (Perley 1989). As pointed out
by Perley, an important consequence of Eq. 1 is that ``a 10
phase error is as bad as 20% amplitude error". Therefore,
the dynamic ranges achievable with the MMA in most cases
- particularly at high frequencies - will critically depend on
accurate radiometric and astronomical phase calibration.
Figure 1: Plot of dynamic range achievable as a function of amplitude
error. The dynamic ranges are determined from the simulated
observations of a single point source at the phase center using
a 40 element array with a 20 minute duration. For the upper pair of
the lines, antenna based Gaussian amplitude gain error are assumed
with a calibration intervals of 40 seconds (solid line) and 200
seconds (broken line). The lower pair corresponds to the compact
array case where the amplitude fluctuations are 100% correlated
among the antennas.
Self-calibration (Cornwell & Fomalont 1989) can be used to
improve dynamic range if the source visibility is well determined
with high S/N so that both M and 
can
be improved using the on-source data.
While the phase self-calibration is relatively easy to
achieve, amplitude self-calibration generally demands a higher S/N
so that achieving high accuracy in the initial amplitude calibration
is highly desirable. Using an idealistic assumption that
no phase error exists, the effect of amplitude error on the
achieved dynamic range can be estimated from a set of simulated
observations (see Figure 1). The results agree well with the
estimates using Eq. 1, and we find that a modest dynamic range of
can be achieved even with 10% amplitude error. On the
other hand, achieving a dynamic range better than may
require an amplitude calibration accuracy of 1% or better.
For a compact configuration where the amplitude fluctuation
due to atmosphere is correlated among the antennas, Eq. 1
is no longer valid. In the extreme case of 100%
correlated fluctuations, the benefit of having N independent
samples goes away, and the dynamic range is given by
Therefore, achieving high dynamic range requires more frequent
amplitude calibration than in larger configurations.
A more realistic scenario must lie between the two extreme cases
considered (two sets of lines in Figure 1), and better than 5%
calibration accuracy is needed to achieve
a dynamic range of 
and better than 1% accuracy for 
.
In all cases, minimizing error in each calibration measurements is
essential in achieving high dynamic range.
The conventional amplitude calibration methods used
by existing millimeter arrays are similar to the methods used
at lower
frequencies (e.g. the VLA) with a few important exceptions.
The overall gain variation is measured and corrected
by frequent observations of a nearby calibrator assuming
the system and the atmosphere are stable over
several minutes and over 
10 degree separation
in the sky.
A major additional concern at mm and submm wavelengths is the
contribution to the noise (or gain) by the atmosphere.
The amplitude gain calibration is in essence the tracking of
system (noise) temperature
where
is antenna efficiency including
ohmic losses and rearward spillover and scattering,
is forward spillover and scatter efficiency,
is receiver noise temperature,
is noise due to
the atmosphere,
is noise from the antenna,
is noise from
the cosmic microwave background.
The atmospheric opacity is used to compute the
effective gain above the atmosphere so that as 
,
(no sensitivity).
In practice, 
is determined by
comparing the receiver output power from an ambient temperature
load with that of the sky (see Kutner & Ulich 1981)
where 
and 
are measured power on the
sky and on the load, and 
is estimated from
antenna efficiency and an atmospheric model.
is measured frequently (e.g. every 5 minutes) to calibrate
the atmosphere, and the effects of varying opacity, both in time and
in elevation, are then removed from the measured fringe amplitude by
The main advantage of the chopper wheel method is its
simplicity (see Eq. 4). Even in the worst case scenario,
one can achieve a dynamic range of 100 or better (see Fig. 1),
and this is the reason why it is successfully used with all
existing millimeter arrays. The main disadvantage is that
it has an internal consistency of only about 10% because
and 
are not well determined and varying
in time. At millimeter and submillimeter wavelengths, the
atmosphere contributes significantly to the ,
but (in Kelvin) and atmospheric opacity estimated from weather
data and an atmospheric model are only accurate to 5-10%.
The median opacity at 450, 650, and 850 GHz are expected to be
about unity at the Chajnantor site, and tracking amplitude gain using
an ambient load (Eq. 4) may pose a sensitivity problem.
Achieving a more accurate amplitude calibration may
require computing directly from Eq. 3
by determining each of the individual noise terms
explicitly. First, all antenna terms
(
) should be measured and tracked.
Even for the chopper wheel calibration, the antenna efficiencies,
which are usually assumed to be constant but may vary significantly
with temperature and elevation, need to be tracked for an
improved calibration. Frequent monitoring of antenna efficiencies
using a nearby transmitter and holography as well as
structural analysis may be highly desirable for optimum operations
at high frequencies anyhow, and blockage, scattering, and
sidelobe responses of the antennas should be well understood
in the first place as part of the antenna design study.
The efficiencies of the BIMA antennas were measured in the
early 80's at 1 cm wavelengths with about 4% accuracy.
The new MMA antennas with strong emphasis on the design
should be better understood and measured more accurately.
[A case for a smaller diameter antenna design may be made here
if understanding the antenna gain responses
at high frequencies become the ultimate limit in achieving high
calibration accuracy.]
Determining requires measuring both and 
.
The atmospheric opacity at the
observing frequency can be obtained either directly
by antenna tipping or indirectly from radiometry
(see Carilli et al. 1998). In principle the radiometric
measurements can be used to infer the opacity at the observed
frequency, but the current atmospheric model is
inadequate to transfer a radiometric measurement from one
frequency to another with 1% accuracy.
Direct measurement by tipping is more accurate as
long as the receiver system is stable, and it is favored at
the moment. One or more antennas
may be dedicated to continuous monitoring of
opacity at the observed frequency (see calibration
subarray discussion in §5.2).
Determining still requires an atmospheric
model, but the error in (typically 
K)
is not as critical as in the ``chopper wheel" case because
its contribution to the overall error is reduced by a factor
[
for 
].
Accurate measurement of can be achieved if well
calibrated loads at two different temperature are used and the
receiver response is linear. The output power from the receiver is given by:
where 
. The coefficient K
can be determined directly from having two different temperature loads,
where T and P are the effective load temperature
and measured output power. The constant offset 
can be
measured by turning off the amplifier on the detector, and now
(and thus ) can be computed.
Bock et al. (1998) have considered a two temperature load system
for the BIMA array that uses a rotating
mirror assembly located behind the subreflector
and tabs the two temperature controlled loads
with about 2% coupling. The small central area in
the subreflector normally reflects the ambient
radiation from the room behind the vertex window back
into the Cassagrain feed, and if not scattered away,
this can introduce significant additional noise. In place of putting a
scatter cone, a small hole just the image of the vertex
window is made in the center of the secondary and
a rotating mirror system is placed so that the hole is
effectively filled with
(a) a 300 K load, (b) a 400 K load, or (c) a scattering
mirror which functions the same way as the scattering cone.
For BW = 8 GHz and 
K,
the continuum sensitivity of the MMA, 
, is about
0.0006 K with 1 second integration, and the 
of 6-8 K can be measured with S/N 
. Unlike the chopper
wheel case where the ambient load can saturate the detector,
the resulting load is much closer to the signal from astronomical
sources and operates within the same linear regime of the detectors.
Among the terms contributing to in Eq. 3, the
antenna and the CMB terms are expected to be varying slowly
over time, and only and terms may vary on
short time scales. The 225 GHz opacity
and 11.2 GHz phase stability data at Chajnantor and atmospheric
transmission models suggest that the amplitude fluctuations over
short time scales (10-30 seconds) will typically be
well below 1% level at 345 GHz, rising to a few percent at
650 GHz (Holdaway 1998a). Therefore, the term also has
a minor contribution to amplitude gain fluctuation except
at the highest frequencies. The quantitative understanding of
the stability of the receiver system () has not been demonstrated
yet, but the two temperature load system with a rotating mirror
provides both the means to measure the receiver temperature and
its stability and also possibly a way to track and remove
its effects (by observing with the spinning mirror system continuously on)
so that the 1% relative calibration accuracy can
be achieved. If all antenna terms are accurately known, achieving
an absolute calibration at 1% accuracy level in total system gain
may be possible. An important additional benefit
is that it automatically provides
absolute flux calibration of the system without resorting to any
astronomical source (see below). It can also be used to inject
a well calibrated signal to test and calibrate the entire system.
For example, a one MHz spectral channel can be calibrated to
1% accuracy with a 10 second integration.
For a well calibrated system where all the gain terms are measured
and tracked, a direct conversion from measured counts in total
power and interferometric modes can be translated directly into flux
units so that many difficulties associated with the
astronomical calibration (see below) will disappear.
The effective radiation temperature 
of a source with excitation
temperature 
and optical depth 
is given by
where 
, h and k are
Planck's constant and Boltzmann's constant, and 
is
the microwave background temperature. The observed source
antenna temperature 
for a normalized source brightness
distribution 
in the direction of 
on the sky is then
where 
is the normalized antenna power pattern (
),
and 
is the solid angle subtended by the source
(see Eq. 3 of Kutner & Ulich 1981). In general, the source
distribution is not known a priori, and the corrected radiation
temperature 
is commonly used because
it is a source and telescope independent quantity and a good estimate
of as the source coupling factor 
in most cases. One can further show
that the source antenna temperature corrected for atmospheric
attenuation 
is related to 
as
If the antenna temperature is measured as a function of airmass
with antenna tipping, 
can be determined along with .
(The corrected antenna temperature 
is related to
by the relation 
.)
One of the important advantages of the two temperature
load calibration scheme (§3.2)
is that it requires knowing and tracking all antenna
efficiency terms needed for a direct flux conversion.
Both the VLA and VLBA uses an internal calibration signal,
``
", and estimated antenna gains to achieve good
relative calibration, and an astronomical source is used to
set the flux scaling. The two load calibration system
provides an extremely accurate signal, which can be
used directly to convert the measured power directly to flux
density (Jy). Since is independent of observed telescope,
the accuracy of the method can be checked by comparing
with measurements made at other telescopes or by examining with
a detailed model of Mars (see below).
The conventional flux calibration scheme with existing telescopes is to track the relative instrumental gains and determine the flux scaling using ``known" astronomical standards, whose fluxes are tied to a small number of careful measurements using a well calibrated horn or a small antenna. Even if the MMA adopts the two temperature load system for flux and gain calibration, establishing a set of astronomical flux standards will be necessary for calibration verification. In the event that the antenna terms are not easily measured or tracked to the needed accuracy, observations of astronomical calibrators may be used to determine the antenna gain terms.
A good astronomical flux calibrator has the following properties: (1) unresolved size; (2) constant or theoretically predictable flux; and (3) bright. At millimeter and submillimeter wavelengths, few if any sources meet all of these criteria. The current generation millimeter interferometers calibrate flux using variants of the following procedure (also see MMA Memo 149 by Holdaway 1996):
This calibration system is an extension of the flux calibration system used with millimeter and submillimeter single dishes. The key step in this calibration scheme is the determination of the flux of the primary calibrator (the planets). We discuss below potential uses of various astronomical objects for flux calibration of the MMA.
Moon.
Because of its large size compared with the primary
beam of the MMA, the limb of the moon offers essentially a one-dimensional
knife edge, the Fourier transform of which is
where 
is the temperature of the moon. The real part of
the Fourier transform is a delta function, but there is also an
imaginary component that decreases as 
. While this
offers an interesting application in the interferometric mode,
is a poorly determined quantity and has a well known
dependence on the details of the surface features. Given these
uncertainties, flux calibration using the limb of the
moon is expected to be less reliable than using the planets.
Planets. Nearly all flux scaling for commonly used flux standards at millimeter and submillimeter wavelengths are based on measurements of planets (see Ulich 1974, Ulich et al. 1980, also Muhleman & Berge 1991). The planets for which the brightness temperatures are best known in the millimeter and submillimeter range are Mars and Jupiter, with Mars probably best understood. Unfortunately, these two planets are heavily resolved by the MMA, particularly at the higher frequencies. This may not be as much of a problem if accurate single dish total power measurements are available for the MMA antennas, however. In any case, the brightness temperatures (and their distribution across the visible disk) for these bodies are not as precisely known as desired.
For Mars, the whole-disk brightness temperatures predicted at millimeter
and submillimeter wavelengths by the best current models are probably only good
to 10-20%. This is due to uncertainties in regolith dielectric and roughness
properties, and uncertainties in ice cap thermal properties (both
the residual H
O and the seasonal CO caps).
Interferometric observations of Mars by the fully functional MMA will help
constraining some of these uncertainties (e.g., the polarized flux density
can be used to help constrain the dielectric and roughness properties),
so when the MMA has matured to some degree, this situation should improve
considerably. In addition to these uncertainties, there may be unmodeled
temporal variations due to local and global dust storms, which may affect
the flux density at the shortest wavelengths. These storms
can be tracked through monitoring of the CO line,
and some care should be taken. Modeling gaseous giant planets is much
harder as their continuum spectra are not well understood.
Astreroids.
Asteroids are compact and bright blackbody emitters and thus
potentially promising primary flux calibrators. The bolometer observations
at 250 GHz of 15 nearby asteroids (heliocentric distance r
= 2.0-3.5 au, geocentric distances 
= 1-5 au) by Altenhoff
et al. (1994) found strong continuum emission (50-1200 mJy;
= 150-200 K), which agrees with the blackbody model within
the uncertainty of calibration on Mars. They are compact,
- an order of
magnitude smaller than Uranus or Neptune. Their flux density
changes significantly due to their and Earth's orbital motion
around the Sun, but the changes are highly predictable. Because
they are not perfectly round, small oscillation in observed flux is
also expected from their rotation - about 4% peak to peak over
9 hour period for Ceres (Altenhoff et al. 1996).
As is the case for the VLBA, the high spatial resolution achievable with
the MMA presents a fundamental problem in that most of these
``primary" flux calibrators are highly resolved at the
maximum resolution of the array - for example, the 3 km baseline
corresponds to 
at 850 GHz or an angular
resolution of 24 mas.
The thermal emission from the photosphere in the nearby stars
offer an interesting possibility for MMA flux calibration.
Main Sequence Stars. The Sun at a distance of 10 pc is about 1 mas in diameter and will have about 1.3 mJy of thermal continuum flux at 650 GHz. Active regions on the Sun will cause some flux variations, perhaps at the few percent level or less. The zodiacal dust in the solar system may be at the level of 1 percent or more, depending on how much cool dust resides in the outer parts of the solar system. Predicting the precise flux (likely to be somewhat higher because of the higher effective temperature at mm wavelengths) will require fairly detailed models of stellar atmospheres.
By searching the HIPPARCOS data set, Richard Simon has found that
there are 250 stars which will be brighter than 2 mJy at 650 GHz.
The number of non-variable, non-binary,
main-sequence stars visible from Chajnantor is much smaller -
22 stars, listed in Table 1. There are probably other
suitable stars which are not listed as main sequence. The
integration times needed to achieve SNR=20 are about 10 minutes
in most cases, computed assuming an rms noise of
0.50
mJy (sensitivity for a 
-m
array, corrected for the collecting area from the sensitivity
calculation by Holdaway 1997a).
Giant and Supergiant Stars.
While G, K, and M giant and supergiant stars are cooler on
average compared to the main sequence stars, they are generally
larger and thus brighter at longer wavelengths.
We have searched the Bright Star Catalog (Hoffleit 1982) for all giant
and supergiant stars with V band magnitude smaller than 5. After excluding all
binaries and variables, stars with estimated and measured diameters
published by Ochsenbein & Halbwachs (1982) are tabulated in Table 2.
Using the effective temperature taken from Allen (1973), the 650 GHz
fluxes are estimated assuming thermal emission from a face-on stellar
disk (
). On average, these stars are nearly 10
times brighter at submillimeter wavelengths
than the main sequence stars in Table 1 so that
SNR=20 detection can be achieved in less than a minute in most cases.
Some of these may still turn out to be variable, but the MMA can establish
this quickly and accurately with a modest monitoring program.
This list is only partially complete as stellar diameters
are available only for a subset of our original list.
Compact Galactic thermal sources such as ultra-compact HII regions such as W3(OH) or stellar sources such as CRL618 and IRC+10216 are commonly used secondary flux standards used by existing submillimeter telescopes such as CSO and JCMT. These objects are typically 5-10'' in size and have observed fluxes of 1-10 Jy at 230 GHz and 20-200 Jy at 650 GHz (see Sandell 1994). These objects may be used for flux calibration by the MMA in the total power mode (in place of the planets when they are not available), but they are not likely to be useful in the interferometric mode because of their large sizes and complex structures.
Lastly, one serious concern for astronomical flux calibration of the MMA is accurate bootstrapping of the flux scaling from the primary calibrator to the secondary or gain calibrators if they are observed hours apart in time. An accurate accounting of the temporal gain variation should be applied before any flux scale factors are applied. For tracks covering only a small range of hour angle (e.g. shadowing, transit at low elevations, snapshot imaging - see Holdaway 1998b), observing a primary flux calibrator at the same elevation range as the gain calibrator and the program sources may not always be possible.
Atmospheric phase fluctuations and resulting amplitude decorrelation
are serious concerns for the MMA operating at higher frequencies
as the amplitude loss scales as 
and
the rms phase error increases linearly
with frequency (see MMA Memo 136, Holdaway & Owen 1995 and references
therein). The amplitude loss due to rms phase
fluctuations of 
and 
are 13% and 50%,
respectively, with corresponding effective losses in sensitivity.
The affected fraction of time due to atmospheric phase fluctuations
and expected loss of sensitivity as function of frequency are
shown in Table 3.
Fortunately the short integration time required for the phase
calibration (see Holdaway & Owen 1995, Rupen 1997) will simultaneously
address the amplitude decorrelation problem
as the phase coherent time scale for the array at the highest
frequency is still several seconds long. The radiometric phase correction
may further increase the coherence time by a factor of 3 to 4.
Can the radiometric opacity measurements be successfully transferred to the observed frequency with better than a few percent accuracy needed? The analysis of the radiometric measurements for the MMA by Carilli, Lay, & Sutton (1998) suggests that absolute calibration of the radiometer may not be adequate for a reliable estimation of opacity at other frequencies, primarily because of the shortcomings in the atmospheric model.
Alternatively one or more antennas may be dedicated to monitoring opacity of the atmosphere at the exact observing frequency. The double load calibration and direct flux calibration relies on accurate knowledge of opacity at the observed frequency. There may be other beneficial uses of a dedicated calibration subarray such as the calibration of the radiometric system as discussed by Carilli et al. (1998).
A large fraction of MMA tracks may be too short in duration to derive the elevation-dependent gain or to obtain its own flux and opacity data (see Holdaway 1998b). Since the primary observing mode of the MMA is service observing so that only the final calibrated data are delivered to the proposers, some of the calibration may be done in a less traditional way. For example, instrumental gain terms such as antenna deformation, spillover and scatter, and ground pick-up may be corrected using an analytic model or a look-up table, disjointed from the temporal changes due to the receiver gain and atmospheric variations. Also a good database of secondary flux calibrators (quasars, stars - see §4.2) may be maintained and utilized rather than requiring an observation of a primary calibrator in each track. A direct calibration approach using model antenna gains and temperature calibrated loads (§4.1) offers many attractive aspects.
Measuring polarized continuum and spectral line emission is one of the important scientific goals for the MMA. Linear polarizer is commonly used by existing millimeter arrays for its simplicity, but linear polarizers can complicate the flux and amplitude calibration as many quasars (gain and secondary flux calibrators) are intrinsically polarized (a few to 10%). The amplitude variation as a function of parallactic angle due to polarized emission is confused with instrumental gain variation unless the intrinsic polarization of the calibrator is known and accounted for. Polarized flux may also be time variable. Therefore, the degree of polarized emission should be included as an additional consideration for MMA calibrators. Alternatively, circular polarization scheme may be considered instead.
Modern radio telescopes are designed to observe sidereal sources with
the best sensitivity that present-day technology allows. This means
minimizing by designing low-noise receivers, minimizing
antenna spillover, blockage, etc. These efforts are for naught when
observing the Sun. The Sun is much bigger and brighter than any other
source in the sky at all frequencies above 100 MHz. Consequently, the
contribution to by the Sun dominates all others, usually by
a wide margin. For example, at a wavelength of 20 cm, 
is
roughly 50,000 K, far greater than the 35 K one normally
encounters. At the VLA, the gain is reduced by using 20 dB,
phase-constant, switched attenuators. is measured with the
attenuators in place by means of high amplitude 
. These two
modifications enable one to phase-calibrate the array in the normal
way (by referencing the observations to a phase calibrator source),
and to flux calibrate by referencing the signal to the high
.
The MMA will face similar problems. will be about 4800 K for
quiet Sun conditions at a wavelength of 1 mm. Again, this is much
larger than the system temperatures anticipated. The solar signal
must therefore be attenuated by 15-20 dB. With the attenuation in
place, one cannot observe a calibrator. Hence, the attenuation must be
switched out of the signal path when observing a calibrator source.
How will gain reduction be achieved at the MMA? Through one or more
fixed attenuators? Or the use of automatic gain control with a large
dynamic range? In either case, we must be able to accurately measure
and correct for the gain change and possible phase shifts introduced
by fixed attenuators or AGC. In AGC is employed, it must operate on a
time constant less than that of possible transient activity (
second). And, like the VLA, it is likely that a known signal
will need to be injected. A possible option for monitoring gain and
phase variations is the use of a pulse cal, as used to some extent in
the VLBA (at least at low frequencies).
If the solution lies in AGC, we need to know how much dynamic range is needed. For quiet Sun observations, the AGC would need to insert 15-20 dB of additional attenuation. And for solar bursts, an additional 20 dB of attenuation may be needed, for a total of 35-40 dB over and above the normal operating point of the ALC.
We have examined the flux and amplitude gain calibration requirements for the MMA. The standard calibration techniques are compared with the two temperature load calibration system. Special topics relevant for the MMA are also considered. The conclusions are:
1. The standard calibration techniques routinely used by the
current generation of millimeter interferometers are good enough
to provide dynamic ranges of 
to , but a more accurate
technique is needed to achieve a dynamic range better than ,
especially for the compact configurations where the amplitude fluctuation
is correlated for the entire array. It should also be noted that
the dynamic range is limited more severely by the phase error
(Eq. 1) - ``a 10 phase error is as bad as 20% amplitude
error" (Perley 1898).
2. The conventional ``chopper wheel" method of gain calibration
is simple and sufficiently accurate for the current millimeter
arrays. However, it is reliable to only about 10% accuracy because of
large errors in and . High opacity in the
high frequency windows of the MMA may also pose a problem.
3. A strong case may be made for a two temperature load
calibration scheme such as proposed by Bock et al. (1998).
By tracking the individual components of (Eq. 3)
explicitly to 1% accuracy level, both absolute and relative
gain calibration of 1% accuracy may be achievable. An
extensive bookkeeping of parameters such as antenna gains
and atmospheric opacity is needed to a high accuracy,
but such an effort is necessary to achieve high dynamic
range (
). In addition, such a system also offers
a capabilty for direct flux calibration and internal calibrations
(e.g. bandpass measurements).
4. A need for establishing astronomical flux calibrators
exists in any event, and several potential calibrators are
examined. The thermal emission from giant and supergiant
stars offers many attractive features such as high brightness
and compactness (
mas). Planets such as Mars and
asteroids are also highly promising. In all cases, some additional
work such as flux monitoring or modeling is needed to
improve calibration reliability.
5. Some special considerations such as amplitude decorrelation, use of calibration subarray and dedicated calibration runs, polarization observations, and solar observations are also briefly discussed.
Lastly, the engineering issues identified for achieving highly accurate amplitude and flux calibration and speical observations include:
signal may also be needed.
Catalog Name RA Dec Parallax V Spec 
Diam. 
No. (B1950) (B1950) ('') (mag) Type (K) (mas) (mJy) (min) 113368 24Alp PsA 343.73 -29.89 0.130 1.17 A3 8720 2.2 11.8 0.7
7588 Alp Eri 23.97 -57.49 0.023 0.45 B3 18700 1.5 7.9 1.6
8102 52Tau Cet 25.42 -16.19 0.274 3.49 G8 5570 2.1 5.1 3.8
49669 32Alp Leo 151.43 12.21 0.042 1.36 B7 13000 1.4 5.0 4.0
108870 Eps Ind 329.97 -57.02 0.276 4.69 K5 4350 2.3 4.9 4.2
66459 203.81 35.97 0.092 9.06 M9 2500 5.3 4.3 5.4
22449 1Pi 3Ori 71.79 6.88 0.125 3.19 F6 6360 1.6 4.2 5.7
9236 Alp Hyi 29.31 -61.81 0.046 2.86 F0 7200 1.5 4.0 6.3
54872 68Del Leo 167.87 20.80 0.057 2.56 A4 8460 1.3 3.5 8.2
8903 6Bet Ari 27.97 20.56 0.055 2.64 A5 8200 1.3 3.5 8.2
57757 5Bet Vir 177.03 2.05 0.092 3.59 F8 6200 1.5 3.2 9.8
71908 Alp Cir 219.60 -64.76 0.061 3.18 F1 7045 1.3 3.1 10.4
84143 Eta Sco 257.14 -43.18 0.046 3.32 F3 6740 1.4 3.1 10.4
19849 40Omi2Eri 63.22 -7.77 0.198 4.43 K1 5080 1.6 3.1 10.4
27072 13Gam Lep 85.59 -22.47 0.111 3.59 F7 6280 1.4 3.0 11.1
65109 Iot Cen 199.44 -36.45 0.056 2.75 A2 8970 1.0 2.6 14.8
15510 49.53 -43.25 0.165 4.26 G8 5570 1.5 2.5 16.0
109176 24Iot Peg 331.18 25.10 0.085 3.77 F5 6440 1.2 2.4 17.4
78072 41Gam Ser 238.54 15.81 0.090 3.85 F6 6360 1.2 2.3 18.9
69701 99Iot Vir 213.35 -5.77 0.047 4.07 F7 6280 1.1 2.0 25.0
64394 43Bet Com 197.38 28.14 0.109 4.23 G0 6030 1.2 1.9 27.7
28103 16Eta Lep 88.53 -14.17 0.066 3.71 F1 7045 1.0 1.9 27.7
Required integration time to achieve SNR=20 assuming
rms sensitivity of 0.50 mJy.
0.1cm
Table 1. A list of 22 bright main sequence stars visible from
Chajnantor that are non-variable and non-binary with expected 650 GHz
flux 
mJy, compiled by Richard Simon.
They are unresolved by the 3 km baseline of the MMA, and
the thermal blackbody emission from the 5 brightest stars can be detectable with
SNR=20 in 5 minutes of integration.
HD No. Name RA Dec Parallax V Spec
Diam.
No. (J2000) (J2000) ('') (mag) Type (K) (mas) (mJy) (sec) 3627 31Del And 01:39 +30:51 0.028 3.27 K3 III 4000 4.6 19. 17
12274 59Ups Cet 02:00 -21:04 0.007 4.00 M0 III 3200 6.4 31. 6.2
24512 Gam Hyi 03:47 -74:14 0.005 3.24 M2 III 3000 9.8 67. 1.3
25422 Del Ret 03:58 -61:24 -0.001 4.56 M2 III 3000 4.6 15. 27
28028 43 Eri 04:24 -34:01 -0.008 3.96 K4 III 3900 4.8 21. 14
39425 Bet Col 05:50 -35:46 0.023 3.12 K2 III 4300 4.1 17. 21
45348 Alp Car 06:23 -52:41 0.028 -0.7 F0 II 7100 6.5 70. 1.2
50310 Tau Pup 06:49 -50:36 0.007 2.93 K1 III 4400 4.4 20. 15
50778 14The CMa 06:54 -12:02 0.022 4.07 K4 III 3900 4.1 15. 27
63700 7Xi Pup 07:49 -24:51 0.003 3.34 G6 I 4800 3.9 17. 21
76294 16Zet Hya 08:55 +05:56 0.035 3.11 G9 II 4400 3.5 13. 36
82150 Eps Ant 09:29 -35:57 0.005 4.51 K3 III 4000 4.3 17. 21
87835 31 Leo 10:07 +09:59 0.000 4.37 K3.5 III 3600 3.3 9.1 72
90432 42Mu Hya 10:26 -16:50 0.018 3.81 K4.5 III 3750 5.1 23. 11
93813 Nu Hya 10:49 -16:11 0.028 3.11 K2 III 4300 4.7 22. 12
98262 54Nu Uma 11:18 +33:05 0.020 3.48 K3 III 4000 4.9 22. 12
98430 12Del Crt 11:19 -14:46 0.024 3.56 G8 III 4700 3.4 13. 36
99998 87 Leo 11:30 -03:00 0.015 4.77 K3.5 III 3600 4.0 13. 36
129078 Alp Aps 14:47 -79:02 0.029 3.83 K3 III 4000 4.3 18. 19
129456 14:43 -35:10 0.014 4.05 K3 III 4000 4.57 19. 17
129989 36Eps Boo 14:44 +27:04 0.016 2.70 K0 III 4500 4.4 21. 14
139063 39Ups Lib 15:37 -28:08 0.049 3.58 K3 III 4000 4.5 19. 17
139663 42 Lib 15:40 -23:49 0.049 4.96 K3 III 4000 2.4 5.4 206
140573 24Alp Ser 15:44 +06:25 0.053 2.65 K2 III 4300 5.2 27. 8.2
150798 Alp TrA 16:48 -69:01 0.031 1.92 K2 III 4300 11.6 134. 0.3
152786 Zet Ara 16:58 -55:59 0.044 3.13 K3 III 4000 7.6 53. 2.1
152980 Eps1 Ara 16:59 -53:09 0.005 4.06 K4 III 3900 4.0 15. 27
157244 Bet Ara 17:25 -55:31 0.034 2.85 K3 II 3700 6.2 33. 5.5
161096 60Bet Oph 17:43 +04:34 0.033 2.77 K2 III 4300 4.9 24. 10.4
163376 17:57 -41:42 -0.005 4.88 M0 III 3200 4.2 13. 36
167818 18:18 -27:02 0.033 4.65 K3 II 3800 4.1 15. 27
168454 19Del Sgr 18:20 -29:49 0.047 2.70 K3 III 4000 6.8 42. 3.4
169916 22Lam Sgr 18:27 -25:25 0.053 2.81 K1 III 4400 4.2 18. 19
173764 Bet Sct 18:47 -04:44 0.019 4.22 G4 II 5000 2.4 6.7 134
175575 37Xi 2Sgr 18:57 -21:06 0.011 3.51 K1 III 4400 3.7 14. 31
192876 5Alp1Cap 20:17 -12:30 0.007 4.24 G3 I 5300 2.0 6.0 167
197989 53Eps Cyg 20:46 +33:58 0.057 2.46 K0 III 4500 4.5 21. 14
204867 22Bet Aqr 21:31 -05:34 0.006 2.91 G0 I 5700 2.8 10. 60
219215 90Phi Aqr 23:14 -06:02 0.010 4.22 M1.5 III 3050 5.5 11. 50
Required integration time to achieve SNR=20 assuming
rms sensitivity of 0.50 mJy.
0.1cm Table 2. A list of 39 bright giant and supergiant stars visible from Chajnantor that are non-variable and non-binary. These are selected from all giant and supergiant stars in the Bright Star Catalog with visible magnitude less than 5. The 650 GHz fluxes are computed assuming thermal emission from a star of given effective temperature (Allen 1973) with given estimated or measured diameter (Ochsenbein & Halbwachs 1982). These stars are brighter than the main sequence stars listed in Table 1 primarily because of their larger sizes.
Fraction of 
at which at which
time at 11.2 GHz 
0.75 
113 GHz 266 GHz
0.50 
209 GHz 487 GHz
0.25 
361 GHz 843 GHz
0.10 
509 GHz 1190 GHz
Table 3. Estimate of the fraction of time when the
atmospheric phase rms error are less than 30 and 70
(13% and 50% loss due to de-correlation) at
the Chajnantor site (from Holdaway & Owen 1995). Active phase
correction may increase the maximum frequency quoted in this
table by a factor of 3 to 4.