Picking a Mark IV Lens

[Tom Droege <droege@FNAL.GOV>]

This note will be concerned with the process for making a lens choice
for the Mark IV.  Paczynski [1] plots the expected occurrence of 
various types of variable stars.  For reference he uses data from 
Allen [2] for the expected number of stars that can be observed above 
a given magnitude limit.  A very rough fit to this data is:

1) N = 10^(0.44*M+0.69)

Where N is the number of stars in the sky brighter than some magnitude 
M.  This is pretty close to the numbers given by Paczynski in our area 
of interest. 

I am asking this question:  How much does it pay to spend for a better 
lens?

Let us compare the proposed 400mm f/4 with the 350mm f/4 Mak that I 
have just purchased.  The pieces needed to make use of this lens come 
to about $300.  This includes the focal reducer that changes it from a 
500mm f/5.6 to a 350mm f/4.

Looking at the Mak, I see that it has a clear aperture of 90 mm.  The 
central obstruction is 38 mm in diameter.  So the area of the aperture 
is Pi*(45)^2 - Pi*(19)^2 = 5228 mm sq.  =  81.5 mm effective aperture.

Hmmm!  I think they cheat a little when they call this an f/4.

Let us assume that the proposed 400 mm f/4 actually has a 96 mm clear 
aperture after the lenses are mounted.  This then produces an area of
7238 mm sq.  

The proposed lens is good over the whole chip.  At least that is the 
design.  We suspect that the Mak will not be so good in the corners.  
I arbitrarily propose that it will have a 70% good area.  This is not 
so far from my (limited) experience with the 135mm f/2 I have used for 
tests of the Mark IV.  

Comparing the two areas, we find that the 400mm f/4 gathers 7238/5228 
= 1.38 more light.  Since our application is sky brightness limited,
sensitivity is improved by sqrt(1.38) or 1.18.  This is then a 
magnitude improvement of 0.18.

Let us assume that the 400mm f/4 will measure to magnitude 16.  We
find from  1) that the Mark IV will be able to measure 5.3E7 stars.
Using the Mak, we will measure to 15.82 or 4.4E7 stars.  However, with
the Mak, we lose 30% of these, so we get only 3.1E7 stars.  

If we consider a Mark IV with 3 telescopes, I estimate that it will 
cost about $10,000 to get it out the door, less optics.  With 3 Mak's, 
it will cost $10,900.  This set up will measure 3.1E7 stars (but not 
from one location).  This is 2584 stars per dollar.  

We ask ourselves, how much would better optics be worth?  

At 2584 stars per dollar, 5.3E7 stars would be worth $20,500. 
Subtracting the base cost of the telescope, we see that we would
break even at a cost of $3500 per lens.  We hope to pay a lot less, so 
we will continue to explore this design.

There are a bunch of intangibles. 

In favor of the Mak:

1) It is a solution in hand.
2) We can buy the Maks one at a time as we need them. Low financial 
risk.  
3) This is a relatively high production item, so we can expect them to 
be uniform in quality.  But possibly not great quality.  
4) The Mak has a built in focus mechanism, and it would be easy to 
motorize it.
5) It is a sealed design, which makes for a practical installation.

Mak problems:

1) Field uniformity.  We don't know what it is yet.  But it does not 
promise to be too good.

If favor of the home built design:

1) Somewhat larger aperature.
2) More uniform field (by design).
3) Better quality control as we are contracting to build the lenses. 

Home built problems:

1) Most Serious Problem.  We have to buy a bunch at one time to keep 
the cost down.  This is then a high risk venture.  
2) We have to design a focus scheme.

All things considered, I am in favor of pursuing our own design.  But 
it is not such a clear choice.  The commercial Mak would not be so bad 
if the special design does not work out.  

For variable star fans, it looks like the equipment costs for finding 
new variable stars might work out to $5 - $20 per star.  Cheap! 
Assuming that we measure 100,000 stars with the Mark IIIs and that 
they cost me about $30,000 and that 1% of the stars measured will be 
variable, then we should get $30 variables with the Mark III program.
Almost a fixed cost per star found.  I have been doing these estimates 
in various forms since I started.  The numbers have not changed much.
They get even more constant if you include other costs.  

This just to show that engineers think differently from most 
scientists.  Cost is always a part of engineering decisions.   

[1]Paczynski, B. 12th IAP colloqium: Variable Stars and the 
Astrophysical Returns of Microlensing Surveys, July 12, 1996. 

[2]Allen, C. W. 1973, Astrophysical Whantities, p-243 (The Athlone 
Press, University of London)

Tom Droege