Sloan Digital Sky Survey Telescope Technical Note 19971020
Steve Kent and Walter Siegmund
The 2.5 m optics have the capability for piston motion of both the primary and secondary mirrors. Thus it is possible to move the primary and secondary simultaneously such that focus is maintained, but the scale factor at the focal plane is changed. This allows aluminum plug-plates, that have a much different coefficient of thermal expansion than the borosilicate optics, to be predrilled and used over a range of ambient temperatures.
What are the equations relating the mirror motions to each other and to the scale factor? How much mirror motion is necessary to compensate for the ambient temperature range over three months?
Define the following quantities:
Z = Displacement of primary along the optical axis from some nominal position in mm. S = Displacement of secondary along the optical axis from some nominal position in mm. DF = fraction change in scale factor M = magnification factor of the secondary D2 = nominal distance of secondary mirror vertex from the prime focus
For the 2.5 m optical design, D2 = 1980.47 mm and M = 2.22.
The primary and secondary motions are related by the equation:
S = Z * M^2 / (1. + M^2) = 0.83*Z
The sense is that the primary and secondary both move towards or away from the focal plane. The fractional change in scale is given by
DF = ((M - 1) / (1. + M^2)) * (Z/D2) = Z / 9614
The sense is such that the scale factor increases (more arcsec/mm) if the primary moves closer to the focal plane.
To good accuracy, these equations should work equally well for both imaging and spectroscopy. They were tested by running ray traces on the spectroscopic design.
The plug-plates will be drilled for the mean temperature of the month that they are expected to be used. Records are available of the extreme maximum and minimum temperature of each month for Sunspot, NM for a 20 year period from 1954 to 1974 ("A Twenty-Year Summary of Sacramento Peak Weather: August 1954 Through July 1974", Sloan Digital Sky Survey Telescope Technical Note 19970822). The mean temperature of each month is taken to be the mean of the extreme high and extreme low for that month. Note: this is not the normal definition of the mean temperature, but it is appropriate since it minimizes the maximum departure that must be accommodated. Subsequently, the average monthly mean was calculated by averaging the monthly means month by month over the 20 year period and plotted (Figure 1). It is this temperature, adjusted for the daytime temperature swing, that will be used to drill plates.
The maximum positive and negative departures from the monthly mean were calculated for each month over the 20 year period. Subsequently, the 20-year extremes were found month by month by taking the most extreme values of the monthly departures over the 20 year period. These were plotted as well (Figure 1).
The results show that the most extreme departure from the monthly mean (as defined above) occurs in January. It is due to the -23 °F record low in 1962. With this event removed from the data, the largest range is 74 °F and occurs in April. For intervals longer than a month, it is necessary to consider the time rate of change of the monthly mean temperature (Figure 2). April and May have gradients near 9 °F/month while October has a gradient of just less than -9 °F/month. The largest three-month range occurs in February-April and is 86 °F. The range for all year, still removing the January 1962 record low, is 103 °F.
Figure 1: Mean temperatures and maximum departures. The means of the extreme maximum and minimum temperatures for each month are calculated. These are averaged month by month over 20 years and the result plotted. The maximum positive and negative departures from the monthly mean for each month are also plotted. The negative departure for January would be -38 °F if the record 1962 low temperature were deleted from the data.
Figure 1
: Mean temperatures and maximum departures. The means of the extreme maximum and minimum temperatures for each month are calculated. These are averaged month by month over 20 years and the result plotted. The maximum positive and negative departures from the monthly mean for each month are also plotted. The negative departure for January would be -38 °F if the record 1962 low temperature were deleted from the data.
Figure 2: Rate of change of the mean temperature per month.
Figure 2
: Rate of change of the mean temperature per month.
Since the plug-plates are not used during the daytime, the maximum nighttime temperature is clearly more relevant than the maximum temperature. Extreme maximum temperatures invariably occur during clear days with strong solar heating. Consequently, it is quite plausible that the maximum nighttime temperatures are 15 °F less than the daytime high temperatures. Also, the mean temperature should be reduced by about 8°F for plug-plate drilling (Table 1). With this adjustment, the largest three-month range occurs in February-April and is 71 °F. The range for all year, still removing the January 1962 record low, is 88 °F.
The expansion coefficient for aluminum alloy 6061 is 24.3 microns/m-°C while that for borosilicate glass E6 is 2.8 microns/m-°C. The expansion coefficient difference is 21.5 microns/m-°C. Combining this number with the scale factor equation from the previous section gives a coefficient of 0.207 mm/°C for primary motion. The primary motion range needed for three months is 8.2 mm and for all year is 10.1 mm. The secondary motion range needed for three months is 6.8 mm and for all year is 8.4 mm.
Table 1: Estimated mean monthly nighttime temperatures. These are from the mean curve of Figure 1 reduced by 8 °F to adjust for the daytime temperature swing. Plug-plates intended for use in the given month should be drilled according to this table.
Table 1:
Estimated mean monthly nighttime temperatures. These are from the mean curve of Figure 1 reduced by 8 °F to adjust for the daytime temperature swing. Plug-plates intended for use in the given month should be drilled according to this table.
month °F °C Jan 20.3 -6.5 Feb 21.7 -5.7 Mar 26.3 -3.2 Apr 35.1 1.7 May 43.9 6.6 Jun 53.1 11.7 Jul 55.1 12.8 Aug 52.4 11.3 Sep 46.5 8 Oct 37.8 3.2 Nov 27.8 -2.3 Dec 23.9 -4.5
month
°F
°C
Jan
20.3
-6.5
Feb
21.7
-5.7
Mar
26.3
-3.2
Apr
35.1
1.7
May
43.9
6.6
Jun
53.1
11.7
Jul
55.1
12.8
Aug
52.4
11.3
Sep
46.5
8
Oct
37.8
3.2
Nov
27.8
-2.3
Dec
23.9
-4.5
For the primary mirror, the axial motion range needed to adjust the focal surface scale to compensate for differential thermal expansion between the optics and the plug-plates is 8.2 mm for the worst case 3 month temperature range. It is 10.1 mm for the worst case range over the entire year. The secondary motion range needed for three months is 6.8 mm and for all year is 8.4 mm.
Additional motion range of the secondary is necessary to compensate for telescope flexure (about 0.5 mm), differential expansion of the truss and the optics (about 0.8 mm), and departures from the nominal spacing during assembly (about 1 mm). The nominal design range of both the primary and secondary is 13 mm and appears to be more than adequate. Plug-plates intended for use in a particular month should be drilled for the temperature given in Table 1.
Date created: 10/20/1997 Last modified: 10/20/1997
