Thoughts on the LSA/MMA telescope diameter

David Woody, Oct 21, 1998

The selection of the telescope diameter for the LSA/MMA is a complex issue
with many different parameters.

Some of the science parameters are
I. sensitivity
	A. point source sensitivity
	B. wide field sensitivity
II. imaging
	A. dynamic range
	B. fidelity
	C. etc.
We want these parameters to be as large as possible.

Some of the system parameters are
I. cost
	A. antennas
	B. antenna instrumentation
	C. correlator
	D. maintenance
II. risk or complexity
	A. new technology
	B. sheer quantity
We want these parameters to be minimized

The problem is manageable if you can come up with a metric or set of metrics
for measuring the science parameters and a set of scaling laws for the
system parameters.  The problem can then be reduced to a simple mathematical
optimization.  I used to believe this was the way to proceed and have
developed spread sheets and MATHCAD programs to perform the optimization.
The results produced broad optima in the range from 6 to 13 or even 15
meters for a wide range of cost functions.  This procedure shows that we are
probably in the right ballpark.

A simple calculation shows the type of results you get from a parametric
optimization procedure.  The simplest form of the cost equation is
C=N(a+bD^p), where C is the total cost, N the number of antennas, a the
fixed receiver and electronics cost per antenna, D the antenna diameter, and
b and p are the coefficient and exponent in the antenna cost function.  This
analysis ignores the correlator cost which is usually a small part of the
total budget and C is only the cost of the antennas and electronics, i.e.
the available money is the amount left after subtracting the fixed
infrastructure costs.  The number of antennas is then simply N=C/(a+bD^p).
Maximizing ND for wide field imaging sensitivity gives
Dwide=[a/((p-1)b)]^(1/p) while maximizing ND^2 for point source sensitivity
gives Dpoint=[2a/((p-2)b)]^(1/p).  Using rough numbers of a=1M$, b=2M$ for
D[10m], i.e. a 10m dish costs 2M$, and p=2.5 you get Dwide=6.4m and
Dpoint=13.2m.  A 100M$ antenna and electronics budget would then give you 20
13.2 dishes with a total collecting area of 2,700m^2 or 60 6.4maller dishes
with a total collecting area of 1,900m^2.  Including the
correlator costs will push D to slightly larger values.  There are other
more sophisticated analysis that have been done but the essence of the
parametric optimization is contained in the three simple parameters a, b,
and p and whether you optimize ND or ND^2.

Interestingly, the optimum diameter in this simple budget is independent of
the total available budget.  Thus we should in theory be able to come up
with a near optimal diameter independent of whether or not the Europeans
join the project.  Of course image quality considerations may require a
certain minimum number of antennas which won't be reached if we pick a large
diameter and the European don't join us.

After having participated in the antenna design working group, worked on the
imaging problem from a new perspective, and listened to the correlator
engineers I conclude that the cost scaling law and parametric optimization
approach is of limited value in resolving the antenna diameter question.
Scaling laws apply to the fundamental physics of a problem but do not
necessarily apply to problems which are limited by our current technology or
engineering expertise.  Von Hoerner derived the well known scaling laws for
the thermal, gravitational and wind "limits" of antenna performance as a
function of diameter.  These are not really limits but indicate how well you
could do if you simply do things correctly, but you could do much worse with
a bad design and much better using clever designs or new technology.  Even
the cost functions are not very reliable.  They apply to a particular design
approach but do not deal very well with designs using different approaches.
This was well illustrated by Dietmar's 15m antenna design.  This design
deviated significantly from the expected cost curve because it took an
entirely new approach to several of the design problems and made some
intelligent tradeoffs in the specifications.  Adhering to rigorous antenna
specifications helps somewhat in finding an approximate cost function but
does not necessarily maximize the science return.

I now view the problem of optimizing the design of the LSA/MMA as one of
deciding how far you can push the technology without excessively increasing
the project risk.  I think the original 40 8m MMA proposal is a baseline
system which has little risk.  But the science yield of this instrument can
probably be significantly improved by pushing the technology with only a
small increase in the risk.  Following is a list the areas were we might
want to
push the technology together with a few comments on their impact on the
system.

1.  Imaging techniques as they relate to the antenna specs.
	The current specs are very conservative in that they are based on producing
excellent images on very bright sources using existing algorithms.  Better
algorithms may do equally well with antennas with slightly poorer pointing
and phase specs.  Mark Holdaway has already proposed new schemes to handle
degraded pointing.  (There is also the whole issue of the whether the
science projects demanding the best imaging justifies compromising the
science that could be obtained with an instrument optimized for different
science goals.)  Relaxing the antenna specs lowers the antenna cost curve
which will increase the optimum size.

2.  Antenna surface error.
	The wavefront phase error is one of the more fundamental specifications and
because of the exponential dependence of the high frequency aperture
efficiency on the surface error this spec should be kept at 25microns
equivalent surface error.  You could allow active surface correction, but I
believe this would be too complicated and risky.  A large part of the
wavefront error can be corrected by simple optimization of the secondary
position.  James Lamb has already shown that by using several dozen
temperature sensors you could correct for most of the temperature induced
pointing and surface errors even in a steel structure.  A similar use of
strain gauges might allow you to correct for wind errors.  These ideas are
pretty new and should probably not be included in the baseline LSA/MMA
design until they are proven to work elsewhere.

3.  Antenna pointing error.
	There are a whole range of techniques which could be applied to improve the
pointing of the antennas beyond that achieved on existing telescopes.  The
addition of tiltmeters and some kind of CFRP pointing reference structure
seems to be an easy low risk step.  These devices allow you to meet the
current pointing requirements on a classical 10m antenna.  Using the
slightly more elaborate reference structure proposed in the OVRO MMA design
should allow the pointing spec to be met at 12m and possible even at 15m.
This system fails gracefully in that if the new features don't work you
still have an excellent classical pointing system, but it does have a few
more parts and thus will require a little more maintenance.  There are also
much more advanced technologies which could be applied to pointing problem,
but for the most part they are much higher cost and risk.

4.  Antenna fast switching.
	In general you would expect the position switching time to scale inversely
with the lowest resonant frequency and that this frequency would scale as
something like the diameter to the first power.  But different design
approaches produce quite different resonant frequencies.  We only have a
rough estimate of what the resonant frequency requirements are for a given
switching time.  A complete servo system analysis including the detailed
antenna design is required to come up with a realistic answer to this
question. My personal feeling is that the switching requirement will not be
a cost driver for D < 10m but you may end up spending extra money to meet
the spec for D > 12m.  Note that a high resonant frequency indicates a very
stiff structure which will probably also have better wind distortion and
pointing performance.

5.  Antenna fabrication and transportation.
	The larger antennas will clearly be somewhat more difficult to transport.
This will limit the fabrication options in San Pedro but will not seriously
effect reconfiguration on site where the roads can be very wide if
necessary, at added cost of course.  The base of the OVRO design is
significantly wider than the Tucson design and this will have an impact on
the transporter and road cost.

6.  Antenna receiver and instrumentation.
	The cost of the receivers and antenna based instrumentation is on the order
of 1M$.  This cost is independent of the antenna size and is the cost driver
which pushes you towards fewer and larger antennas.  A major technological
effort to significantly decrease this cost would push you back down towards
smaller antennas.  Unfortunately we will not be using a large enough
quantity of any item to justify real commercial scale production and cost
savings (companies want to produce thousands of items to justify a real
"production line".)  But we could consider trading off receivers with the
ultimate in illumination efficiency and Trec against cost and ease of
fabrication.  There are all planar receiver systems (Jonas Zmuidzinas's twin
slot mixers for example) which are have Trec and aperture efficiencies close
to and even better than the best feedhorn systems.  Also limiting the number
of bands or decreasing the frequency coverage directly reduces the fixed
cost per antenna.  Note that the antenna costs will be accurately determined
once we have completed the D&D prototyping because we will go out for a
fixed cost commercial contract.  The receivers and instrumentation will
probably be done mostly in house and hence will have a much larger cost
uncertainty.  If we end up with a system in which the receivers and antenna
electronics are a large fraction of the total project costs, as will be the
case if we have a lot of small antennas, we will have to carry a larger
contingency into the construction phase.

7.  Pad costs.
	The antenna pad cost will increase for the larger antennas but fewer will
be needed.  The configuration studies seem to indicate that you will have
something like five configurations independent of the number of antennas in
the array.  So each antenna will require ~5 pads and the costs could be
significant if an expensive construction technique is required.

8.  Correlator.
	The correlator is not a large fraction of the budget but it does seem to
present a hidden limit on the number of antennas.  The engineers seem to be
unwilling to consider a correlator much larger than one for the number of
elements we are already talking about, i.e. 40-80 antennas.  Ray Escoffier's
"deep memory" design does an excellent job of getting around the wire
interconnect problem which normally scales as N^2 by doing all of the
interconnects on a single board.  He has a board design which will handle
~40 antennas on one card.  Going beyond ~40 antennas puts you back at N^2
scaling of the interconnects and boards.  The cost may be manageable for a
much larger number of antennas but the volume and power consumption of the
correlator goes beyond what we are "willing to consider" but not beyond what
is routinely done in industries such as telephone companies.  The engineers
have come up with major design improvements to get from the current
correlators to the new generation required by the LSA/MMA but have not
worked on another generation beyond that.  This will change in the future
but counting on it during the three year D&D phase is risky.  The argument
about risk and contingency in item 6 above also applies to the correlator
because we will not have a final design ready for commercial bidding until
after the three year D&D phase (it would be a mistake to freeze the
correlator design five or ten years before it is used).

9.  Maintenance.
	Generally the maintenance costs is on the order of 10% of the capital
costs.  My experience indicates that this percentage should be increased for
receivers and electronics and decreased for mechanical components.
Decreasing the diameter will increase the fraction of money spent on
receivers and electronics and thus will increase the maintenance costs.

10. Imaging wide fields.
	One of the strongest arguments for smaller antennas is the number of
pointings required to cover a given wide field.  There will be a limit to
the number of pointings which can be obtained before earth rotation moves
you out of the minimum UV cell required for imaging.  Presumably the image
quality degrades if you have to wait a day or two to get the rest of the
pointings.  How serious of a limitation is this and are there algorithms
which get around this problem?

As I see it we have two choices:

1.  Push towards more small antennas by working on the receiver technology
to lower the cost (or decrease the frequency coverage) and putting more
effort into the correlator development to make the much larger correlator
manageable.

2.  Push toward fewer large antennas by working on the antenna design and
working on the wide field imaging problem by understanding the limitations
for probable science projects and possibly developing new algorithms.

Option two will carry less risk and hence less contingency into the
construction phase of the project.


David Woody
Owens Valley Radio Observatory
Caltech
P.O. Box 968, Big Pine, CA 93513
Phone: (760)-938-2075 ext 111, FAX (760)-938-2075
E-mail: dwoody@caltech.edu