Solar-powered space flight
7c. Achieving the desired level of
optical precision: Examples
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7.5 For mirror
layouts as per Figures 9 and 10 then 
are as per Table 6
for a range of well spread out values of 
. We note that in the
situation where 
and 
is constant for all 
(for
a given 
), i.e. 
and 
can be ignored, the
following apply:
(a) For small , the terms in 
and 
predominate, and as
we might expect have the desired signs, i.e. they involve the wires pulling the
mirror along in the direction of the negative x-axis and radially
inwards.
(b) For any given , the terms in and still predominate
for sufficiently large 
. This means that
sufficient tension to achieve a smooth reflective surface can always in
principle be achieved as long as is large enough.
Table 6: Values of 
arising from 
, 
and 
for various 
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0.999
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-1.43
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-1.05
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-1.51
|
1.05
|
-1.05
|
0.31
|
-0.95
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0.99
|
0.04
|
-1.05
|
-1.05
|
1.06
|
-1.04
|
0.31
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-0.95
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0.95
|
0.48
|
-1.05
|
-0.95
|
1.11
|
-1.00
|
0.31
|
-0.95
|
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0.90
|
0.57
|
-1.05
|
-0.99
|
1.17
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-0.94
|
0.30
|
-0.95
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0.81
|
0.62
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-1.04
|
-1.11
|
1.29
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-0.84
|
0.28
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-0.96
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0.70
|
0.66
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-1.03
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-1.31
|
1.47
|
-0.72
|
0.24
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-0.97
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0.62
|
0.68
|
-1.02
|
-1.51
|
1.66
|
-0.63
|
0.22
|
-0.98
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0.52
|
0.69
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-1.02
|
-1.81
|
1.94
|
-0.53
|
0.19
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-0.98
|
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0.4
|
0.70
|
-1.01
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-2.31
|
2.42
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-0.42
|
0.15
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-0.99
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0.30
|
0.72
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-1.01
|
-3.26
|
3.34
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-0.30
|
0.11
|
-0.99
|
If, say, 
= 
kg m-3, 
= 1 MPa (so is roughly 8 – 14
MPa during flight), 
= 0.01 and = 200 m then 
, 
and 
(and if there is a
single stanchion per mirror area element, 
) are as per Table 7
for various values of . The above values of
, etc. have been
chosen to be akin to those that might arise when lifting a reasonably modest
payload into orbit. We note that 
and 
have the desired
signs as per (i) for all relevant . As long as the
stanchions are sufficiently close together the tension will vary only a little
in the regions between stanchions, making it possible in principle to achieve
the desired level of optical accuracy. Some cross buttressing might also be
desirable, to limit vibration of the mirror during flight, unless a
particularly smooth acceleration is delivered by the engine.
Table 7: 
, 
and hence 
as multiples of 
(and hence angle,
, that a single
cross-stanchion should make to the x-axis) needed in order to achieve
uniform tension in larger mirror for a hypothetical mirror example
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( )
|
()
|
()
|
( )
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0.999
|
-17.6
|
-23.3
|
29.2
|
52.9
|
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0.99
|
-10.3
|
-20.8
|
23.2
|
63.6
|
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0.95
|
-8.1
|
-19.2
|
20.8
|
67.1
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0.90
|
-7.6
|
-17.9
|
19.4
|
66.9
|
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0.81
|
-7.3
|
-15.9
|
17.5
|
65.4
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0.70
|
-7.0
|
-13.6
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15.3
|
62.7
|
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0.62
|
-6.9
|
-11.9
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13.8
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60.1
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0.52
|
-6.7
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-10.0
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12.1
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56.1
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0.41
|
-6.6
|
-7.9
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10.3
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50.2
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0.30
|
-6.5
|
-5.7
|
8.6
|
41.2
|
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