1. System Costs Differences for Dishes: 12 vs 15 m diameter. P
System Costs Differences for Dishes: 12 vs 15 m diameter. P. Dewdney Apr 15, 2010

2. Influence of Numbers of Dishes*
Cost per unit area of larger dishes is greater by ~17% (d2.7 rule of thumb – could be smaller for mold-based production).
Capital costs that scale with the number of antennas (n or n2) will occupy a large fraction of the dish-based system budget.
e.g. Feeds, receivers, land, power services, communications, correlation and imaging.
Operational costs that scale with n or n2 also will be large.
Depends on power efficiency of devices, but some of this was “priced in” already in the Astro2010 estimates.
These costs tend to dominate the large areal dish costs for 15-m dishes vs 12-m dishes.
Mold-based production techniques may also lower the areal cost of dishes, as well as flattening the cost curve with frequency.
This result also holds for dish costs that are higher than the Astro2010 estimates, but the contrast is not as large.
If dish costs become a very large fraction of the total system cost, then the curve flattens.
* Assuming constant total area.

3. Cost Comparison for Astro2010 Case
The areal cost difference between 12 and 15-m antennas.
One way of estimating this difference is to use the c  d2.7 rule of thumb.
yields a cost difference of 17% per unit area more for 15-m antennas.
Total difference of $117M (€90M)for 3300 dishes (10000 m2/K).

4. System Cost Function (McCool & Colegate)
T = {Cd*n}diameter + {Ca * n} + {Caop*10*n} + {Cb * n2/2} + {Cbop*10*n2/2} + F + {10*Fop}
T = Total costs for 10-yr ops;
Cd = Dish costs incl transport;
Ca = All those system costs, per dish that scale with n;
Caop = Annual operating costs, per dish, for all hardware that scales with n;
Cb = Cost per baseline of correlating and computing hardware;
Cbop = Annual operating costs, per baseline, for correlating and computing hardware;
F = Fixed costs;
Fop= Operational fixed costs.

5. Limits on System Cost Difference
T12 – T15 < €150M
€150M is “tolerance”, set at ~10% of total.
Estimates of dish costs (second term is shipping):
Cd12 = 117k + 23k = €140k & Cd15 = 214k + 42k = €256k.
Substitute Cds => Ca + 10Caop Cb Cbop < €147k
Ca << €147k per dish AND
Caop << €15k per dish per annum AND
Cb << €35 per baseline AND
Cbop << €3.5 per baseline per annum
If any of these conditions cannot be met then the scaling of cost with number of dishes implies a 15m dish choice, with €150M tolerance.
Are these strong limits?
Based on setting all but one cost coefficients to zero in turn.

6. Testing the Limits using Astro2010 EstimatesCa
€40k; €50k; signal transport & €20k.
Total €110k – not enough by itself (limit €147k).
Caop
€1.1k; €4.1k (per year).
Total €5.2k per yr – not enough by itself (limit €15k per yr). Cb
€16; €22 per baseline.
Total €48 per baseline – enough to indicate 15-m antennas (limit €35).
Cbop
Correlator €11M; Processing €44M per yr.
Total €55M per yr – enough to indicate 15-m antennas (limit €3.5 per yr).

7. Summary of Previous Page
Using the coefficients in this table the inequality,
Ca + 10Caop Cb Cbop = €877k
Should be < €147k for 12-m dish choice.

8. Constant Survey Speed with PAFs
With SPFs the SS is decreased with 15-m dishes (1.56).
With PAFs, as the dish diameter is increased, the survey speed can be made constant if the size of the PAFs (number of beams or elements) increases as d2.
The total cost of the PAFs remains constant,
cost of PAFs over the whole array is proportional to the total number of beams in the array.
Thus, with dishes equipped with PAFs the survey speed can be made invariant with increased dish diameter at approx constant antenna/feed cost.
Correlation cost, which is proportional to the product of the number of beams with the number of baselines, will be less expensive for larger PAF-equipped antennas.
In the constant survey-speed case, the number of baselines decreases as the fourth power of the diameter ratio, but the number of beams increases as only the square of the number of antennas.

9. Smaller dishes There are scenarios that favour smaller dishes:
Anything that might make the dish costs a large fraction of the total capital and operating budgets.
Semi-conductor system components that are very inexpensive and use very little power.
May happen at some point, but Astro2010 estimates already assume 20-nm technology.
Power is free to the project.

10. End