Space Solar Power System
Design Formulae
Transmitted radiation – Size of the resolution spot on the ground, x
The Power Star can match an arbitrary power distribution on the ground – but within the limits
of the spatial resolution, or smallest “spot size”, x . To keep the rectenna sizes to the minimum,
the power Star would distribute its total power output into several beams, each of them having
width given by:
z
x 
DA
  Operational wavelength
(1)
z  transmit distance (35, 786 km for GEO)
DA  Diameter of the phased array
Average power density on the ground,  :
At each of (probably several) power reception stations, the maximum power density is:
2
4D 
   A  Pstation
  z 
Pstation  Power delivered to an individual rectenna
(2)
 Some fraction of Pt , where:
Pt  Total transmitted power from the entire satellite
Safe power level: Assume a safe level is <10% of the solar insolation at the Earth’s surface, and
at zenith (approximately 1050 W/m2). To ensure that the rectenna dimension, x , is no larger
than necessary,  should equal this maximum tolerable level:
2
 max 
4  DA 
Pstation  105W m 2
   z 
(3)
Note that combining (1) and (3) and solving for x , we get:
x 
4 Pstation
max
(4)
Thus, when the safety constraint is imposed, the receiving station size is independent of altitude.
Therefore the two important constraints are Equation (4) and Equation (5) below.
1
  z 
Pstation  105W m  

4  DA 
2
2
(5)
This allows us to calculate the number of rectennas that can be serviced.
Sun- Synchronous Orbit Specifications:
Such orbits use the J 2 -induced precession of the argument of perigee to maintain the orbit plane
at a fixed orientation with respect to the sun (obviously in this case at maximum angle of
incidence). The orbit inclination, I, is given by:
cos i    a 12352km 
72
   P 3.795 Hrs.
73
a  orbit radius (circular orbit assumed) in km
(6.a-c)
P  orbit period in hours
For example, for an orbit altitude of ~ 2200km , we have 11 orbits per day and a maximum
latitude of ~ 740 .
Some quantitative estimates of the reduction in necessary aperture size when a formation of
several, N s , satellites are used to beam power to the same location at distance z can be obtained
by using Equation (5) and assuming that the power density on the ground adds incoherently
(because the satellites are certainly many wavelengths distant from one another), and imposing
the requirement that the total power density be in the safe range ( max  105.0W m2  N S ):
DA   z
4 105
 Pt N S
(7)
Patch Antenna Dimensions:
A patch antenna consists of a metal “patch” mounted on a grounded, dielectric substrate as
shown:
The dielectric provides a resonant cavity to amplify the transmitted signal. Since L is the
resonant dimension, we must have:
2
L 2
(8)
W is usually chosen as 1.5L to get higher bandwidth, but we shall assume W  L   2 here.
Power (angular) Distribution:
We assume a randomized distribution of patch antennae (with W  L   2 ) having an average
distance of s between their centers. Randomized distribution eliminates highly concentrated side
lobes to produce a single main lobe. The angular power distribution is approximately modeled
as:
2
  D 2
 DA  1
A
P    
 
 2 exp   

2
 2s  


2




(9)
Total Power Delivered:

Pt   d  P      Psa
 2s 
Psa  Total power collected by the solar arrays
2
2
(10)
 
Note that the maximum value of s is  2 , so that the factor   reflects the sparse aperture
 2s 
theorem.
2
 
For a given s the fraction of the frontal area occupied by the solar arrays is 1    , therefore:
 2s 
2
   2   2
Psa  1     DA eff Qs
  2s   4


eff  Solar cell efficiency
(11)
QS  Solar insolation  1367 W m 2
Summary of Power Delivered:
 
Pt   
 2s 
2
   2   2
1     DA eff Qs
  2s   4
(12)
Optimal Spacing and Maximum Power:
The optimal spacing is:
3
soptimal  
2
(13)
for which the maximum power is:
1   2

  DA eff Qs 
 4 4

Pt max  
(14)
This assumes that the solar cells and patch antennas must not overlap. If they could completely
1
overlap, the factor   in the above would become unity.
4
Stowed Dimension and Mass:
Assume the balloon is initially folded into a spherical capsule having diameter Dstowed . We
assume the maximum possible packing efficiency, which simply preserves the total volume of
material. The relationship is:
1 3
Dstowed
6
w  Thickness of the "skin"
DA2 w 
(15)
And the total mass is:
M   DA2 w skin
 skin  volumetric mass
(16)
density of the "skin"
Orbit Lifetime - Aerodynamics
For lifetimes greater than 10 years, the lifetime as a function of the initial orbit radius of a
circular orbit is nearly independent of the launch time relative to the solar maxima or minima
(see SMAD, 3rd Edition, Figure 8.4). Thus the orbit lifetime can be estimated using the average
atmospheric density as a function of altitude. Further we may assume small drag forces such that
the decaying orbit takes the form of a tight spiral with a slowly varying “instantaneous” orbit
radius. Then we have approximately that the orbit life time, t f , is:
tf  
ai
R
E
da
1
 Atm  a   a
ai  Initial orbit radius
RE  Radius of the Earth
(17)
  Gravitational constant (GM) of the earth
 Atm  a   Atmospheric density at orbit radius a
4
where  is the ballistic coefficient:

M
CD A
M  Power star mass   DA2 w skin
(18)
A  Frontal area  4 DA2
Thus, for the Power Star and assuming free molecular flow ( CD  2 ), we get:
  2wskin
(19)
which is just twice the areal density of the “skin” and independent of diameter.
The atmospheric density function assumed here combines the tabulated values in SMAD and the
US Standard Atmosphere. A spline interpolation of these values up to 1000km altitude is shown
on the following chart:
As an example, suppose that an object has a ballistic coefficient of 65kg m2 . The next figure
shows the orbit lifetime as a function of initial orbit altitude obtained by numerical computation
of the above integral. The result is in good agreement with that of SMAD, 3rd Edition, Figure 8.4
for the average atmospheric density function.
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Solar Pressure
The acceleration due to solar pressure, denoted by asol , for the spherical Power Star is given by
(see Reference ):
 DA2  0  a 
2
4

 1  9 1     ref  pˆ
4M c  r 
 0  Solar intensity in space at the Eath's orbit
asol  
 1367 W m 2
c  Speed of light  2.998  108 m s
a  One AU distance  1.496  1011 m
(20)
r  Distance of the Power Star from the Sun
 ref  Reflection coefficient of the Power Star surface
  Coefficient of specular reflection
1, purely specular reflection

0, purely diffuse reflection
pˆ  Unit vector pointing from the Power Star center to the Sun
Using Equation (18.b) for the mass and setting r  a , we get:
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asol   0 pˆ
0 
0
1  4 1     ref 
4cw skin  9
(21)
Since the Power Star semi-major axis in its Earth orbit is much smaller than an AU, the unit
vector, p̂ , is approximately along the line from the center of the Earth to the Sun. Let the x-axis
be aligned with this vector, and the x-axis together with the y-axis defines the orbit plane.
Introduce polar coordinates  r ,  , where  is the angle from the Earth-Sun line as shown in the
figure below. We also neglect the rotation of this reference frame.
y
Power
Star
r

Earth
x
With these assumptions, the dynamical equations reduce to two: The energy integral and the
expression for the rate of change of angular momentum due to the solar pressure:
2
 2 0 r cos   2 E
r
r  2r   0 sin 
r 2  r 2 2 
(22)
where E is the total initial energy. In general the solution is periodic. For the Power Star we can
assume that the nominal motion when the solar pressure is neglected is a circular orbit at orbit
radius denoted by r0 , and that initially, r  r0 , r  0, and   0 with   0. . Then the leading
terms in a Fourier series expansion are:
 1

r  r0 1   2  cos  2mt   1  O   3  
 8

  mt   sin  mt   O   2 
m  Mean motion in a circular orbit   r03

(23)
0
r02  0
1  94 1     ref 

2

r0 m
4  cw skin 
7
Performance Calculations for a Specific Example:
First, use Equation (14) to determine the total power that can be produced as a function of
the Power Star diameter, D A , and for various values for the solar array efficiency, eff . Results
are shown in the Figure below. Presently, the solar array printing technologies capable of mass
production have efficiency in the 1-2% range. Thus we plot the case eff  0.01 . However,
several manufacturers are confident that this can be raised to 5% with modest near-term effort.
Thus the plot for eff  0.05 represents the likely current capability. For a 1 to 5% solar array
efficiency, the Figure shows that a one kilometer power Star could deliver power in the range 2.6
to ~12 Mega-Watts (MW). If advanced development produces large quantities of 20% efficiency
solar array, then the I km Power Star could deliver ~50MW.
The deployable size of the system depends on the mass capability of the launch vehicle
and its fairing dimensions. To address this we consider a specific case: The Zenit-2SLB capable
of putting 13MT in MEO orbit. The fairing is a mono-conic with a cone height, B, of ~4m, an
end-to-end length, L, of 13.65m and a cylinder diameter, D, of 3.9m. Thus, the fairing volume is:

1

D 2  B  L   178.9 m3
4
3

(20)
8
At this point, we need to determine if 13MT of folded balloon material can be fit into this
volume. A reasonable choice for the “skin” material might be Mylar or Kapton. Their volumetric
densities are similar, so let us choose the value for Mylar. Then:
 skin  1390 kg m3
(21)
Thus, if the entire fairing volume were filled with the folded Power Star, the total mass would be
178.9 m3 1390 kg m3  248.8 MT - which far exceeds the launch capability. Thus, with any sort
of reasonable packing efficiency, the folded, 13MT Power Star can easily be accommodated
within the Zenit fairing.
The next question becomes: How big can a Zenit-launched Power Star be? Let us use
Equation (16) and solve for DA :
DA 
M
2.98

 wskin
w
(22)
The Mylar skin of the Echo 2 satellite had a thickness of 0.013mm. It is reasonable to assume
that we can presently reduce this by approximately a factor of two - to w  0.006 mm . With this
value, we obtain:
DA  704.4 m
(23)
With this value, we can use Equation (15) to approximately determine the diameter of a spherical
container enclosing the folded Power Star in the fairing of the Zenit launch vehicle:
Dstowed  2.61m
(24)
Which easily fits the fairing. With an operational diameter of 0.7km, and a reasonable 5% solar
array efficiency, the delivered power graph shows the system capable of ~7 MW output.
Regarding orbit life time as a function of the altitude of the initial orbit, we use Equations (17)(19) to obtain the following graph. With w  0.006 mm , we have:
  2wskin  2 1390 0.006mm  0.01668
(25)
Note that the de-orbit time function is independent of the diameter of the system and directly
proportional to  , which is twice the areal density of the skin. The results indicate that a long
lifetime is ensured by placing the Power Star at roughly 2000km or above. Thus a MEO orbit or
above is suitable for a long-term system.
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