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For Presentation Mon., Oct. 4, 2004, Afternoon, in Rm. 17, Session No. 13, Vancouver, B.C., Canada, at
Int’l Astronautical Federation 55th Congress; Paper No. IAC-04-J.1.08
GRAVITY-BASED STATIC LIQUID ORBITAL
ASTRONOMICAL MACRO-OPTICS
By John H. Bloomer
Pres., DISCRAFT Corp.
1990 SE 157th Dr., Portland, OR 97233, USA; Ph. (503)251-6914 ; FAX (503)252-2383
August 27, 2004
ABSTRACT
Inexpensive large space astronomical-quality (telescope) optics, and an optical method for
collecting/handling/transporting energy of nearby stellar sources (in arbitrary quantity for arbitrary
wavelength range) continuously in space for terminals in space and on nearby associated planetary
surfaces, are desired. For energy-transfer, radially and transversely moving or stationary terminals within
practical cost/time-frame/engineering limitations, would be necessary. For diameter-size of the precision
(giant) optics required for both astronomy and electromagnetic energy-handling, there is no substitute.
However, the cost of “diffraction-limited” (or geometrically optically unimprovable for given wavelengthand-diameter) optics even here on earth, goes up by the fourth power of the diameter. Most of the cost
is due to the expense of achieving by grinding and polishing a solid surface such that its normally-directed
overall deviations are less than the Rayleigh limit of /4 (or 5 × 10-6 inches for green light), and such that
its surface also is polished to exhibit an overall molecular-thick “Beilby” layer. But such a smooth “Beilby”
layer, one due to “surface tension”, is exhibited already by any non-reactive liquid surface in any
equilibrium gravity-field (e.g., such as water under 1G), and it is also a characteristic of any non-reactive
liquid in space zero-g occupying a precision cylindrical container, to exhibit a “perfectly” spherical
(precision) surface with radius-of-curvature determined by the contact angle and diameter of the
container. But a precisely (astronomically so) spherical surface (albeit on a liquid in space zero-g) is
geometrically very near a precise (astronomically so) paraboloid, for example. Too, the order-ofmagnitude of the force needed, at the precision needed, to turn such a spherical surface into such a
paraboloidal surface, can be shown to be lent by ordinary gravity of special but reasonably-sized,
bulkheads shaped & placed in space zero-g just for the limited purpose of statically (continuously)
“pulling” that spherical surface into that (“desired”) paraboloidal surface. Of course any desired
(geometrical) surface whatsoever can thereby as well be achieved simply by implementing the
appropriate gravitational normal potential function at every surface point, via a specific geometry of
bulkhead enclosing a “massive” liquid of a given specific gravity. Astronomical-quality orbital telescope
optics thus theoretically can be fashioned equally well for “passive” or “active” use, and as well one can
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thereby form laser-end-mirrors. Of course the cost of such a technology promises to confer orders-ofmagnitude savings over large ordinary solid astronomical space optics fabricated by any conceivable
system of segmentation, alignment, actuators, holograms, spinning, grinding or polishing, on earth or in
space. The diameter-size of such boundary-constrained precision static liquid astronomical surfaces in
orbit, can apparently be indefinitely extended by use of this inventor’s “calibration islands” concept for
control of a surface-tension “membrane” in zero-g, comparable in effect to NASA’s “New Generation
Space Telescope” concept for control of a solid membrane. The NASA system would consist of a
membrane-backing matrix of tiny turnbuckle “actuators” capable (point-by-point) of pulling a solid
(membrane) spherical space mirror into a precision paraboloid, but implemented of course (NASA-wise)
electro-mechanically rather than physico-chemically, with therefore a certain consequent unit-cost
penalty. Too, inventor’s physico-chemically self-functioning “calibration islands” in liquid space optics,
should be needed in orders-of-magnitude less areal density (therefore at proportionally less expense)
than NASA’s (similarly-normally-driven) “actuators” in solid (membrane) space optics. Reflective (mirror)
liquid-metal described here is gallium, whereas refractive (corrector) clear-plastic liquid described is DowCorning silicone No. 200. Development from Ta Li’s variational approach, of a fundamental
synthesis/analysis equation linking optical, gravitational & physico-chemical parameters, is given.
I.
STATIC ADAPTIVE LIQUID SPACE MACRO-OPTICS: INTRODUCTION
In 1965, I published via American Astronautical Society1, a treatise showing one should be able to
fabricate for space zero-gravity operation, a diffraction-limited, static-liquid-surfaced, mirror-andcorrector Maksutov (Fig. 1) catadioptric system, namely a 300-inch diameter orbital telescope (going the
Palomar 200-inch one better!) This instrument would rely on physical chemistry of liquids in zero-gravity
for precision optical sphericity, and on special masses emplaced in the spacecraft to literally and
permanently gravitationally deviate those spherical surfaces to “figure” them as required. The Maksutov
System at the given aperture requires very little figuring (retouching) indeed of its spherical-surfaced
mirror and spherical-surfaced refractive (lens) corrector. But literally any other optical system – especially
the various paraboloidal-primary ones – can as well be built to operate in space in this manner I showed.
[Cost – even on the ground – of major astronomical (SOLID) optics goes up by the fourth power of the
diameter; and, like football teams, “there is no substitute for size in major optics”!]
Later in ’65 I published a summary of the AAS conclusions in a paper for the SPIE 10 th Symposium2.
I claimed in a subsequent publication, that we could as a result build and operate far larger inverted
(propulsion) laser-beaming astronomical optical systems, far less expensively, in space than on the
ground3. This was ’66 when – fired by invention of the laser (of ’63) – I showed that rapid-transit
interplanetary – and even interstellar – missions could be powered by sun-driven orbital lasers mated to
(inverted-telescope) optical systems incorporating my giant new lightweight high-precision, “beamtightening” lowcost mirrors. I even suggested as an afterthought, that such a propulsion system aimed at
earth, could serve as a “solar power satellite” (I didn’t call it that), one that would even oversupply all our
Page 3
present energy needs globally (star-rocketry requires far more energy than that currently used for all
purposes on earth).
FIG. 1. Maksutov Catadioptric Wide-Angle System
R1/D = -0.6120.66
R2/D = - 0.6120.66 - 0.0565 - 0.007A
R3/D = - 2.1070.983
A
B
d1/D = 0.1
y
D
d2/D = 1.111.14
y
R1
nD = 1.5163
R2
= 64.1
n
fy
f
R3
Sy’
d1
S’
Sy’
d2
© 1966 BY J.H. BLOOMER
DWG. BY J.H. BLOOMER
ALL RIGHTS RESERVED
MAKSUTOV SYSTEM
(AFTER D.D. MAKSUTOV,
JOSA,
MAY,
1944)
FIGURE 1
But in those days I couldn’t seem to elicit any interest in any of this, so in ’69 I put it all aside to begin
designing aircraft.
Then in the ‘Nineties, I reviewed, revived and revised my astronomical, propulsion and power satellite
systems, as exhibited by various new publications referenced below4,5,6,7,8.
II.
TO BEGIN WITH, WHAT IS (STATIC) LIQUID SPACE OPTICS?
Clean, chemically nonreactive, reflective static liquids (metals for example), inside clean, precision
cylindrical, circular cylindrical, straight, toroidal or spherical boundaries in zero-g, will attain minimum
potential energy (i.e., spherical) surfaces, which will exhibit at every point on the boundary the precise
characteristic contact angle for that liquid on that solid (irrespective of the amount of gravitational or
equivalent force acting). See Figures 2, 3, 4.
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FIG. 2. Liquid Contact Angles @ 1g and 0g
LIQUID
CHARACTERIZED
BY:
0o CONTACT
ANGLE
45o CONTACT
ANGLE
(AFTER PETRASH & OTTO)
90o CONTACT
ANGLE
135o
(VAPOR)
(LIQUID)
1-G
CONTACT
ANGLE
(VAPOR)
180o CONTACT
ANGLE
(LIQUID)
CYLINDRICAL
TANK
ZERO-G
© 1966 BY J.H. BLOOMER
DWG. BY J.H. BLOOMER
ALL RIGHTS RESERVED
EXPERIMENTAL
CONFIGURATIONS OF
LIQUIDS IN 1-G & ZERO-G
FIGURE 2
FIG. 3. Liquid Contact Angles in Spherical Tanks
(AFTER PETRASH & OTTO)
 = 0o
(VAPOR)
 =45o
© 1966 BY J.H. BLOOMER
DWG. BY J.H. BLOOMER
ALL RIGHTS RESERVED
 = 180o
(LIQUID) 
 = 90o
SPHERICAL TANKS PARTFILLED WITH LIQUIDS OF
VARIOUS CONTACT ANGLES 
o
 = 135

CONFIGURATIONS
(EXPERIMENTAL) OF LIQUIDS IN
ZERO-G. FIGURE 3
Page 5
FIG. 4. Water Menisci in Tank @ Constant Contact Angles
N=1
N = 0.5
N = 0.05
N = 0.0
N = VERTICAL LOAD IN G’S
 = CONTACT ANGLE
INFINITELY LONG TROUGH
© 1966 BY J.H. BLOOMER
DWG. BY J.H. BLOOMER
ALL RIGHTS RESERVED
(WATER)


MENISCI FOR WATER FOR VARIOUS
LOAD FACTORS N (AFTER E.
BENEDIKT). FIGURE 4
N
Figure then of such a spherical “mirror,” will be purely and precisely spherical within the constraints
(controllable, as will be known) of the surface instantaneous axial section total circularity deviation,
where the latter is inherently due to a number of closely related influences as quantized by this author.
This quantization proceeded from analysis and application of results of a variational treatment by Ta Li of
liquid rocket fuel in zero-g and near-zero-g9.
Most useful applications of such a space-optical art, however of course will require parabolization of such
objects, introduced here by a totally new and revolutionary paradigm in space mechanics: It actually is
proposed here to figure these optics (with real-time control) by applying local gravity itself – of calibrated
liquid-metal masses and geometries – thereby achieving the tiny ultrahigh-precision axial modifications
required to change a (liquid) spherical large astronomical telescope mirror in space, into a paraboloidal
one (Fig’s 5, 6). In a sense, manufactured optics (in space) will then become infinitely “accommodating,”
much like the human eye.
FIG. 5. Error  Due to Axial Effective Gravity (Begin)
C
R3
F
D

N
= CHARACTERISTIC = /lv
= RADIUS OF MIRROR
= FOCAL LENGTH
= APERTURE
= F/D
= LOAD IN g0’S
10
3
D
1
 =  (c,R3,N,D) = ERROR =  1 +  2    Ng 0 C

2
Page 6
© 1966 BY J.H. BLOOMER
DWG. BY J.H. BLOOMER
ALL RIGHTS RESERVED
ERROR  DUE TO A FINITE
AXIAL LOAD Ng0. FIGURE 5
Ng0
FIG. 5. Error  Due to Axial Effective Gravity (Conclude)
FIG. 6. Practical Example of Gallium Paraboloid Due To Gravitational Deviation of Meniscus

20,000-lb 
Mercury
Figuring Mass
(“pancake”)


 ½”



25-ft dia.
 
paraboloidal

 mirror


2.5 ft
1/8” Deviation (Gravitational) or “error”
(Difference Between Limb & Vertex “Elevation” By 
Fundamental Analytic/Synthetic (“Error”) Equation)
 









Focus of

Paraboloid

0.57o Paraboloidal Field-of-View



Initial Spherical Surface (Radius = 3,695 ft,
Imposed by Contact Angle) Before

Gravitational Deviation
Solid Boundary Ring


(NOT TO SCALE)


o
88.2 Contact
Angle (of Natural Gallium Spherical Surface with Boundary)


Gallium Mirror

Mass
FIG. Date: 1999

Final F/100 Paraboloid
Surface After
Gravitational Deviation
(Gallium, free liquid)



2500 ft paraboloid FOCAL LENGTH

PRODUCTION OF F/100 ORBITAL PARABOLOID “FIRST-CUT” BY GRAVITATIONAL-FORCEFIGURING (“PANCAKE” SURFACE, SHAPE+POSITION SHOWN ARE ONLY SYMBOLIC). ZERO-G

Page 7
III.
SELF-FORMING (LIQUID) SPACE OPTICS
The self-forming, free liquid surface static orbital macro-optic paradigm introduced here, could ultimately
succeed in lowering cost – especially when commercial large-optics-fitted solar-power-satellites are
common – by whole orders of magnitude, even negating or reversing the sign of the cost (See Fig’s 7, 8).
To do so, however, intense R&D effort will evidently need to be directed along wholly new lines, i.e., in
physical chemistry, surface chemistry, space physics and mathematical physics of the gravitational
potential and of space liquid systems’ total energy.
FIG. 7. Adjustment & Operation of “Calibration Islands”
“HIGH” CASE I. MIRROR SURFACE “LEVEL”, “HIGH” AT CALIBRATION PT.
“CALIBRATION ISLAND”
ACTUAL LIQUID
“LEVEL” SURFACE
NON-WETTING CONE:
REPELS MENISCUS
NON-WETTING MENISCUS
DESIRED “LEVEL”
SURFACE
CAPILLARY
RESULTANT FORCE
“DOWN”
CALIBRATION ISLANDS IN LIQUID
SPACE OPTICS FIGURE. Date: Aug. ‘04
CALIBRATION POINT
WETTING CONE:
ATTRACTS MENISCUS
LIQUID GALLIUM
J. H. BURGE ET AL JOSA/SPIE VOL. 3356 3/98
PICOMOTOR ACTUATOR SETS CALIBRATION POINT
CASE II. MIRROR LIQUID SURFACE “LEVEL” “LOW” AT CALIBRATION PT.
DESIRED “LEVEL”
SURFACE
ACTUAL LIQUID
“LEVEL”
NON-WETTING CONE:
REPELS MENISCUS
“CALIBRATION ISLAND”
CALIBRATION POINT
WETTING MENISCUS
CAPILLARY
RESULTANT FORCE
“UP”
WETTING CONE:
ATTRACTS MENISCUS
LIQUID GALLIUM
Page 8
FIG. 8. Matrices of “picomotor actuators” recommended for NASA/AFRL & Bloomer “Membranes”, RESP.
PRINCIPLE BY J. H. BURGE ET AL, SPIE J.,
3/98: “NEXT GENERATION OF SPACE
TELESCOPES”
DESIGN
AFTER
“LIGHTWEIGHT MIRROR TECHNOLOGY
USING A THIN FACESHEET WITH ACTIVE
RIGID SUPPORT”
ADAPTED PRINCIPLE (GREATLY EXPANDED)
BY J. H. BLOOMER, INVENTOR, YR. 2000:
METHOD FOR INDEFINITELY EXPANDING
DIAMETER OF “LIQUID EPIHYDROSTATIC
CAPILLARY SPACE OPTICS” (LECSO)
8 M.
8 M.
AFRL/NASA 8-M.-DIA. MIRROR
PICOMOTOR ACTUATOR (BACKSIDE)
LAYOUT WITH 2500 ACTUATORS
INDIVIDUALLY CONTROLLED (50
ACTUATORS PER M2): EVERY
INTERSECTION AN ACTUATOR
ONE “PICOMOTOR ACTUATOR” (A
LA BURGE ET AL) PER LECSO
MIRROR 8-M.-DIA. AREA, TO FIX
PHYSICO-CHEMICAL
SELFADJUSTMENT
POINT
OR
“CALIBRATION ISLAND LEVEL” FOR
INSTANTANEOUS CORRECTION OF
“HEIGHT” OF MIRROR LIQUID
SURFACE
(SURFACE
TENSION)
“MEMBRANE”
BLOOMER
8-M.-DIA.
LECSO
DIFFRACTION-LTD. MIRROR (OR 8M.-DIA. AREA IN LARGER OPTIC)
FITTED W/ ONE ACTUATORCONTROLLED
“CALIBRATION
ISLAND”
COMPARISON OF NGST RIGID-MEMBRANE MIRROR W/ JHB LIQUID
MEMBRANE ONE FIGURE 8. DATE: AUG 27 ‘04
Page 9
IV.
LIQUID SPACE OPTICAL SUPPORT STRUCTURES NOT CRITICAL
A free-surface shallow liquid-optic “pool” in zero-g in space, incidentally constrained as it is only by
capillary forces and self-gravitation, does not care what minor strains and misalignments are suffered by
its support structure – especially when support structure itself has been formed in channels by first
capillary action then solidification. High precision and high rigidity of this structure are not required (See
Fig’s 9, 10, 11, 12).
FIG. 9. Why Liquids Can’t “Leak” in Liquid Space Optics
ACTUAL MENISCUS
HYPOTHETICAL DEVIATION
ANY MOTION OF THE LIQUID MASS OUT OF THE ENVELOPE
SHOWN BY THE SOLID-LINE MENISCUS REQUIRES A NONUNIFORM CHANGE IN CONTACT ANGLE AROUND THE
BOUNDARY AT CONSTANT TOTAL POTENTIAL ENERGY (HENCE
CONSTANT TOTAL SURFACE AREA).
© 1966 BY J.H. BLOOMER
DWG. BY J.H. BLOOMER
ALL RIGHTS RESERVED
WHY LIQUIDS CANNOT
“LEAK” OUT OF
BOUNDARIES. FIGURE 9
THIS IS IMPOSSIBLE.
FIG. 10. Non-Propagation of Liquid Space Optic Boundary Small Imperfections (Begin)

Fc

Fs
Page 10
FIG. 10. Non-Propagation of Liquid Space Optic Boundary Small Imperfections (Conclude)
AT P:
(1)
(2)
 1
1 

Fs = INWARD NORMAL FORCE DUE TO SURFACE TENSION = 2M =  
.
 R1 R 2 
Fc = OUTWARD (MOLECULAR, COHESIVE) NORMAL FORCE FROM PRESSURE OF LIQUID
 1
1 

INTERIOR =   
.
 R1 R2 
© 1966 BY J.H. BLOOMER
DWG. BY J.H. BLOOMER
ALL RIGHTS RESERVED
SURFACE TENSION, BOUNDARY
ERRORS, COHESIVE FORCE IN ZEROGRAVITY. FIGURE 10
FIG. 11. Non-Propagation of “Dimple” in Liquid Space Optic Boundary
11
Page 11
FIG. 12. Why Toroidal Channels Give Precision Circularity
12
V.
GRADED LIQUID SPACE OPTICAL EXPERIMENTS PLANNED
A logical consequence of the identification of vast liquid-space-optics potential, is an initially purely
scientific research and development program, one which might proceed somewhat along the lines of the
experimental apparatus described in Fig’s 13, 14, 15, 16, 17, 18, 19, 20, concluding in the ultimate, the
“Liquid Optics Test Satellite” (or LOTS) shown in Fig’s 16, 17, 18, 19, 20.
VI.
LI’S FOUNDATIONAL DIFFERENTIAL EQUATION
Total liquid orbital mirror axial-section circularity deviation is permitted convenient expression by the
“error” approximation derived from Li’s math model, which is a function of the net (paraxial) gravitational
attraction (or acceleration) as independent variable, itself a sum of the deliberately introduced gravityfiguring force plus (shown sufficiently minizable) perturbations such as extraneous self-gravitation (of
spacecraft parts), gravity gradient (of earth, for example), attitude control torques, etc. 10. This deviation
expression – based on our adoption of the first few terms of Li’s rapidly converging power series solution
Page 12
to his variational differential equation (which expresses the accelerated optical system’s resulting total
paraxial deviation from spherical) – is found to be directly proportional severally to (paraxial) net effective
gravitational force (independent variable), to the difference (parameter) between the density of the
liquid metal and its vapor, and to the cube of the mirror diameter (a second parameter); while on the
other hand deviation is at the same time shown proportional inversely to the liquid-mirror surface
tension (third parameter) and to the mirror focal ratio or f-number, (fourth parameter).
This approximate equation for paraxial overall surface deviation from circularity (or sphericity) is derived
from Li’s formulation as follows:
FIG. 13. Initial 1G Tests of Liquid Space Optics
J
F
A
C
Q
B
I
G
E
5”
R
D
H
K
A
F
1”
S
T
L
M
O
“1”  FORWARD
“0”  REVERSE
L
10
OPTICS ZERO-G SIMULATION VIA MATCHED LIQUID DENSITY
VII.
P
10
OPTICAL INTERPRETATION OF LI’S DEVELOPMENT OF ZERO-G EPIHYDROSTATICS
Final development of the ultimate general mathematical tool of “zero-gravity epihydrostatical optics,” is
due to a treatment by Dr. Ta Li9. The following development by Li, illustrated in Fig. 21, is characterized by
liquids in containers whose interiors are in the form of surfaces of revolution.
Page 13
14
FIG. 14. Zero-g Drop Test From a Tower
Page 14
15
FIG. 15. Zero-g Drop Test From Balloon
Page 15
FIG. 16. Liquid Optics Test Satellite Erection in Orbit
16
lv, ls and sv are respective surface tensions at the liquid-vapor, liquid-solid, and solid-vapor interfaces,
in dyne/cm; in the plane of Fig. 21, the “horizontal boundary” is the intersection of the surface (liquidvapor interface) with the toroidal annulus shown. u = u(x) is the vapor-annulus interface, v = v(x) is the
liquid-annulus interface, and z = z(x) is the liquid-vapor interface, where u=u(x) and v=v(x) are known and
z=z(x) is sought.
The total potential energy K of the system consists of the sum of surface energies and the effective (net)
central-force-field or contact-force potential energy, where
 x1
K  2 lv   1  z 2
 0


1/2
x1

xdx    1  u 

2 1/2
0
x1

xdx   ng 0  z 2  v 2 xdx ,

0


(1)
while thermodynamic equilibrium implies:
V

2
x1
where:
 u  z xdx ,
0
(2)
Page 16
 = ratio of vapor volume to container volume
V = Volume of container
g0 = Sea-level acceleration of gravity = 980.665 cm/sec2
=
 sv   ls
 lv
 = (liquid - vapor),
C=
 liquid  vapor
 lv


 lv
liquid = density of fluid
vapor = density of vapor above liquid
Above equations (1) and (2) define an isoperimetric problem in the calculus of variations with a mobile
upper limit. The total energy K is to be minimized and the resulting differential equation (the “solution”)
will itself be solved to provide a fundamental design formula, or expression. This expression will be critical
both in the analysis of systematic errors engendered by net residual acceleration (all sources combined),
and in the synthesis of diffraction-limited optical surface figures by designing masses (shapes, sizes and
positions) which will deviate the optical surfaces slightly by gravitational attraction.
Minimization of K can be facilitated by introduction of the “Bond Number” (dimensionless physicochemical constant characteristic of the system):
B0 
ng 0 R 2
 lv
,
(3)
where:
n = load factor = sea level g’s of acceleration operating on system,
R = radius of curvature of liquid-vapor interface
Applying standard techniques of the variational calculus, Li found the differential equation

d  d

d  d

2

 d  
1   

 d  





1 / 2 

  B0 ,


(4)
where:
 
1
R

2R 
 z  z 0  
,
B0 

(5)
and
  
2
x 
 ,
 2R 
(6)
Page 17
17
FIG. 17. LOTS Final Erection In Orbit
Page 18
18
FIG. 18. LOTS Interior Assembly Components
Page 19
FIG. 19. LOTS Interior Assembly Pictorial
FIG. 20. LOTS Rendering Showing Downlink Antenna
FIG. 21. Model for Li’s Treatment of Zero-gravity Epihydrostatics
Z-AXIS
LIQUID IN CONTAINER OF
REVOLUTION IN LOW GRAVITY FIELD
FIGURE 21
© 1966 BY J.H. BLOOMER
DWG. BY J.H. BLOOMER
ALL RIGHTS RESERVED
Page 20
Li’s solution for ordinate (height) z(x) of the meniscus surface at a lateral distance x (ABSCISSA) from the
vertex, was the infinite series,
2
4
6
1
1
20
1
 x 
 x 
 x 
z  2   4  B0     12 
B0  B0 2    ... .
2
3
3
6
 2R 
 2R 
 2R 
(7)
This equation is the required solution to the characteristic (Euler) differential equation of the given
variational problem.
From Fig’s 21 and 5, we define:

,
  z 0 x1   z 0 0   z B x1   z B 0 
0
0
(8)
where:
z0(x1) = ordinate of the (spherical) liquid surface at x=x1 when Bond No. B0=0  n=0 or “zero
gravity”; see Fig’s 21 and 5;
z0(0)=0= ordinate of the vertex (at x=0) of the (spherical) liquid surface when Bond No. B0=0  n=0
or “zero gravity”;
zB x1   ordinate of the (non-spherical) liquid surface at x=x1 when B0 ≠ 0  n ≠ 0 (“non-zero0
gravity”);
z B 0   ordinate of the (non-spherical) liquid surface at x=0 when B0 ≠ 0  n ≠ 0 or “non-zero0
gravity”.
We also define:
R0 = radius of curvature of unperturbed liquid surface,
x1 = radius of the meniscus “aperture”
 = f/number = focal ratio = F/D,
F = focal length,
D = aperture,
g = ng0 ,
RB = radius of curvature at the vertex of the (non-spherical) surface of revolution
0
obtained when n (and therefore B0) is finite.
Substituting various approximations in (8) and substituting for the Bond No., B0 from (3), reducing and
neglecting higher-order terms, we find
1  ng 0 x1
    
 2    lv R0
5
4

 .


(9)
If the liquid meniscus surface is considered an optical surface (reflector or refractor) we may redefine the
parameters of  as follows:
Page 21
R3 = R0
= radius of curvature of an optical mirror or lens,
F = R3/2
= focal length for paraxial rays striking an optic of curvature radius R3,
D = 2x1
= aperture of optical system,
 = f/no. = focal ratio = F/D =
1  R3 
.
4  x1 
Further substituting in (9) and reducing, we find, finally
10
D3
1
    ng 0 C
.

2
(10)
We may define (10) as “the fundamental expression of liquid space optics,” equally useful for analysis and
synthesis.
From (10) note that, as you would expect, the error  imposed by the net axial acceleration, is inversely
proportional to both liquid-vapor surface tension and f/no. , where telescopes with large f/no.’s already
have relatively flat primaries. Hence they are going to be, if liquid, less susceptible to net axial
acceleration, which is felt merely as a flattening influence anyway.
Note also from (10), the error  will be directly proportional to net axial acceleration, ng0 , and not
surprisingly will be directly proportional to the third power of the liquid space optical system’s primary
diameter, nominally.
From Fig. 5, note that flattening at the circumference of liquid optic under a net downward acceleration,
is reflected by positive quantity  1  z 0 x1   zB x1 , whereas concomitant flattening at the vertex of
0
the liquid optic under the same net downward acceleration, is reflected by positive quantity
 2  zB 0  z 0 0 .
0
VIII.
DEVIATION EQUATION FOR LIQUID-SPACE-OPTIC ANALYSIS AND ROUGH
SYNTHESIS (RECAP):
10
1
D3
    ng 0 C
,
2

 = total paraxial overall mirror (or lens) surface deviation from sphericity, measured as
the sum of flattening at both limb and vertex occasioned by a net gravitational
acceleration (see Fig. 5);
n = net paraxial load in g0’s;
g0 = 980.665 cm/sec2;
C=
 liquid  vapor
 lv
liquid = density of optical liquid;
Page 22
vapor = density of optical liquid vapor above its liquid mass;
lv = surface tension of liquid (reflective or refractive);
D = optic diameter;
 = f/no. (i.e., ratio of focal length to diameter of primary mirror).
IX.
FIRST ROUGH CUT AT PARABOLIZING A NATURALLY SPHERICAL LIQUID SPACE
(ZERO-G) OPTICAL SURFACE
When for example n at each point on the mirror surface is about 10-8 (earth gravities) on the zero-g liquid
gallium mirror illustrated in Fig. 6 (paraxially), and when in addition mirror diameter D is 25 ft as
illustrated (defined by the solid boundary ring shown), we pull the liquid spherical (minimum potential
energy) surface backward such as to create (roughly) a parabolic mirror with a focal ratio of  = f/100 (the
contact angle on the precision cylindrical boundary is exactly 88.2o). This can be achieved by the mercury
“pancake” of 25 ft. diameter and ½-inch thickness as shown, at a distance of 2.5 ft behind the mirror
(although other elements, masses, and positions can be selected, and can be optimized to reduce weight
and eliminate toxicity).
Field-of-view of this telescope would be 0.57o, which is perhaps about maximum for standard
astronomical paraboloid reflectors, hence presumed probably equally practical for both ordinary imaging
and laser-beaming. Focal length is 2500 ft. Deviation  to achieve this parabolization is (accomplished):
  1  2 
X.
1
inch.
8
NECESSITY OF “CALIBRATION ISLANDS” TO INDEFINITELY ENLARGE DIAMETERS OF
STATIC LIQUID SPACE OPTICAL PRIMARY MIRRORS
It is possible that the physico-chemical method shown in Fig. 7 may succeed in indefinitely extending
diameter of static liquid primaries, wherein we’ve taken a leaf from the NASA/AFRL “NGST” (Next
Generation Space Telescope) book11. AFRL/NASA evidently favor a matrix of individually controllable
“actuators” as shown in Fig. 8. to pull a solid membrane into diffraction-limited performance (primarymirror-wise). Quantity would be some 50 per square meter. In the static liquid “epihydrostatic capillary”
space optical method, however, capillary (surface tension) forces may actually drastically reduce the
necessity for such actuators, say from 2500 per 8-meter diameter mirror (as shown in Fig. 8) to only 1 per
8-meter-diameter mirror (also shown in Fig. 8). If indeed such devices can work (with appropriate physical
chemistry) to enlarge (diffraction-limited in visible light) operation of liquid space optics, then it appears
astronomy is in for an enormous boon: But it does appear such devices will be necessary to enlarge liquid
space optic diameters much above 25 ft, (if they’re to operate diffraction-ltd.), since LSO primaries rely on
their boundaries (and precision of same), to control their surfaces (and precision of same). But the ratio
of boundary units to area units of the surface, is
Page 23
2R
R
2

2
.
R
Thus, for a given f/no., liquid, and net equivalent gravity, the “error” grows with 1 st to 3rd pwr. of optic
diameter. Thus, some kind of “calibration islands” will be needed for very large (diffraction-limited) LSO
units, evidently.
XI.
REFERENCES
1) Bloomer, J.H., “The 300-Inch Diffraction-Limited ‘Orbiting Eye’,” American Astronautical Society
(Space Electronics Symposium Vol. 6), Ed. By C.M. Wong, Los Angeles, CA. (USA), 1965.
2) Bloomer, J.H., “Liquid Space Optic Optics,” Society of Photo-optical Instrumentation Engineers,
10th Technical Symposium, San Francisco, CA. (USA), 1965.
3) Bloomer, J.H., “The Alpha Centauri Probe,” International Astronautical Federation XVIIth
International Astronautical Congress Proceedings, Madrid, Spain (published Poland), 1966.
4) Bloomer, J.H., “Earthly Millennium Energy and Interstellar Shuttle Propulsion Potentials of Liquid
Space Optics,” 28th Intersociety Energy Conversion Engineering Conference, Atlanta, Georgia (USA,
1993.
5) Bloomer, J.H., “Liquid Space Optical Theory of Manned Starflight with Earthly Applications,” 23 rd
International Electric Propulsion Conference, Seattle, WA. (USA), 1993.
6) Bloomer, J.H., “Conversion of Solar Energy Via New Aerospace Technology,” 29th Intersociety
Energy Conversion Engineering Conference, Monterey, CA. (USA), 1994.
7) Bloomer, J.H., “Foundations of Liquid Space Optics for Astronomy, Solar Power Satellites &
Interplanetary Shuttles,” Space Power, Vol. 13, No’s 3 & 4, Phoenix, Arizona (USA), 1994.
8) Bloomer, J.H., “Foundations of Optics Technology for Mars Colonization, Paid Astronautics, Energy
Millennium, Interplanetary/Interstellar Shuttles,” 31st AIAA/ASME/SAE/ASEE Joint Propulsion
Conference and Exhibit, San Diego, CA. (USA), 1995.
9) Li, Ta, “Hydrostatics in Various Gravitational Fields,” General Dynamics Astronautics Space Physics
Group, Applied Research, San Diego, CA. (USA), 1960.
10) Oliver, Bernard, Personal Correspondence and Replies of 1/17/66 and 1/29/66.
11) Burge, J.H. et al, Lightweight Mirror Technology Using a Thin Facesheet with Active Rigid Support,
SPIE Journal, Vol. 3356, March, 1998.
12) Carreras, Dr. Rich, Air Force Research Lab, Albuquerque, N.M., personal communication, Aug.
1999.