b
Ç
[
cos
2
x
2a
h
+
a tan x
t
where
is the allowed tolerance on
. For h = 0,
= 45 deg,
= 25
µ
as, a = 30 cm, and
=
we have
and
. Changes in the temperature gradients
2 $ 10
8
K
-
1
,
b
È
[ 5.1 mK/m
b
Ç
[ 5.1 mK/m
within the CMA of this order or larger on timescales of the order of the spacecraft spin period or
less will require direct means of either controlling or measuring the basic angle fluctuations.
(Static gradients
2
are of little or no concern in this respect, since the basic angle is a solution
parameter in the data reduction process.) Figure 2 is a contour plot of the temperature gradient
tolerance in mK/m as a function of
in
µ
as and the CTE in units of 10
-8
K
-1
. The CTE of the
ultra-low expansion glass Zerodur is at most 5x10
-8
K
-1
in the temperature range 0-50 C. For
smaller individual pieces (less than ~600 lb.) it is not unreasonable to expect values that are bet
-
ter than this. I use 2x10
-8
K
-1
in this memo for illustrative purposes. For comparison, the CTE of
silicon carbide is 4.3x10
-6
K
-1
.
Initial thermal work performed at JPL for the FAME project indicates that the thermal environ
-
ment may in fact be stable to better than ~7 mK changes over one spin period.
3
If this is the case,
and if further analysis supports the results found here, then perhaps relatively inexpensive further
attention to the thermal environment of the CMA in particular could replace the more costly,
complex, and untested (in space) laser metrology system currently proposed.
Additionally, it should be pointed out that, since the CMA lies in collimated light beams, the
additional longitudinal shifts from non-ideal placement of the CMA attachment points to the
optical bench have no significant effect for longitudinal gradients. However, for transverse gra
-
dients the deformation of the glass between the supports and the mirrors contributes to the rota
-
tion of each mirror in opposite directions. Therefore, it is important that
h be as close as possible
to zero (Figures 3 and 6).
Finally, there is a potential problem concerning beam divergence due to curling of the mirrors
from static gradients.
4
Changes in the temperature gradients are likely to be too small to have a
noticeable effect. Wavefront errors on the order of 18
picometers would result from gradient
changes of 5 mK/m. However, static thermal gradients, according to a JPL study,
5
are likely to
be ~5 K/m. This would produce a noticeable warpage of the mirrors, with associated wavefront
error of order ~30 nm, or ~
/18 at
= 550 nm.
Basic Angle Temperature Gradient Sensitivity
FTM-USNO-95-01
3
5
Results are shown in the Step 1 proposal (see Figure 2.2-8 on page 19).
4
The effects of gravity unloading and non-uniform CTE should also be looked at in this respect.
3
The ~7 mK/m fluctuation shown in the Step 1 proposal is actually a worst-case scenario which
was driven by Earthlight at perigee (J. McGuire, private communication).
2
I.e., gradients that do not change on timescales smaller than the spacecraft spin period.
