Light Pressure Modified Orbits
Summary: Light pressure adds an "eccentricity vector" of an orbit perpendicular to the light pressure. If this is added to the vector of am equatorial, sunwardsperigee elliptical orbit with large J_2 effects, the orbit's eccentricity vector will precess to follow the apparent annual movement of the sun relative to the earth. If the solar sail ratio (the ratio of thinsat area to mass) is large, the eccentricity must be large to compensate.
Light pressure effects modify thinsat array orbits. In the nominal orbit, thinsats are at "half thrust", with each thruster half mirror and half transparent. Variations to full or zero reflectivity, and full or zero thrust, allow each thinsat to maneuver in relation to the array, or for the array as a whole to maneuver around its assigned centerpoint. The following is an analysis of two effects, earth oblateness and nominal halfthrust light pressure, on the orbit. We will assume that the arrays maintain a constant, slightly elliptical orbit that precesses once per year in the equatorial plane. We will assume continuous illumination tangential to the equatorial plane, and zero light pressure effects from Earth albedo or blackbody radiation, and no solar or lunar tides. These assumptions are somewhat crude approximations to get us into the ballpark of a solution. Precise solutions will probably demand accurate numerical solutions simulating many years of orbital evolution.
Earth Oblateness
For "heavy" thinsats relatively close to the earth, such as 3 gram satellites at m288, the dominant deviation from a perfect Kepler orbit is caused by the J_2 spherical harmonic of the gravity field, in turn caused by the oblateness of the spinning Earth. For small eccentricities, the eastward precession of the perigee of one elliptical equatorial orbit is proportional to the J_2 term of the WGS84 model ( 1.082626683E03 , see Pisacane 2008 ) and the inverse of the orbit radius squared. For small eccentricity and inclination, the precession, expressed as the number complete precessions per year, is N_{pr} \approx 3 J_2 ( Y / T ) ( R_e / a )^2 \approx \left(  3 J_2 Y \mu^{1/2} / ( 2 \pi {R_E}^{3/2} ) \right) / ( {a_{RE}}^{7/2} ( 1  e^2)^2 ) ) \approx 20.21822 / ( {a_{RE}}^{7/2} ( 1  e^2)^2 ) ) ~ ~ ~ where
N_{pr} 

precessions per year caused by oblateness 
J_2 
1.082626683E03 
spherical harmonic of gravity causing oblateness 
\mu 
398,600.4418 km^{3}/s^{2} 
Earth gravitational constant 
Y 
31556926 sec 
year period in seconds 
T 

orbit period in seconds 
Y/T 

number of orbits per year 
e 

eccentricity 
R_E 
6378.137 km 
earth equatorial radius 
a 

orbit semimajor axis (radius if circular) 
a_{RE} 
a / R_E 
orbit semimajor axis in earth units 
Note: Pisacane's formula (5.47b) is for d \omega / dt ~ ~ in radians per second, and is in terms of n = 2 \pi / T ~ ~ . Dividing both sides by 2 \pi ~ and multiplying by Y ~ seconds per year gives the number of complete 360° (=2π radian) precessions per year. His more accurate formula has a correction for inclination, but is small for these nearequatorial nearcircular orbits, so we can use the above approximation to a few percent accuracy.
N_{pr} is unity  the orbit precesses accurately without light pressure  when a = ( (1.5 J_2 Y ( R_E )^2 / \pi )^{2/7} \mu^{1/7}
 = 15,058 km radius.
orbit 
a_{RE}/radius 
e 
orbits/year 
N_{pr} 

LEO 
1.047 

5758.5 
17.061 

m288 
2.005 

2191.5 
1.771 
apogee towards sun 
m288 
2.005 
0.049 
2191.5 
1.780 
with significant eccentricity 
m360 
2.264 

1826.2 
1.157 
more sensitive to light pressure 
m390x 
2.361 

1715.1 
1.0000 
"infinite" light pressure sensitivity 
m394x 
2.374 
0.1 
1701.4 
1.0000 
"infinite" light pressure sensitivity 
m407x 
2.417 
0.2 
1657.1 
1.0000 
"infinite" light pressure sensitivity 
m480 
2.627 

1461.0 
0.688 
perigee towards sun 
m720 
3.182 

1095.7 
0.351 

m1440 
4.168 

730.5 
0.137 

GEO 
6.611 

365.2 
0.027 
To keep a thinsat orbit in the same orbit and orientation to the sun, the light pressure must retard the precession at m288 and m360, and advance it at m480 and m720, so that the total precessions per year is one. More complicated orbit evolution may allow different ratios, however, eccentricity should not accumulate over many years or the apogee and perigee will eventually intersect other orbital bands, causing collisions.
When N_{pr} is one, the light pressure generated eccentricity accumulates, pushing the the apogee towards the west until the orbit stabilizes. Higher eccentricity increases N_{pr} . This needs more study, but moving the orbit towards m390 may be a way to let light pressure drive the orbit to high eccentricity, bringing perigee down to the atmosphere and reentering the thinsats.
V2.0 thinsat characteristics
Light pressure parameters 

Light Power 
1367 
W/m^{2} 

Speed of Light 
2.998e+8 
m/s 

Light pressure 
4.56e6 
kg/ms^{2} 

IR adjustment 
0.45 


IR reduced L.P. 
2.05e6 
kg/ms^{2} 

Thinsat parameters 

mass 
3e3 
kg 

thickness 
44 
μm 

density 
2.7e+3 
kg/m^{3} 

area 
2.5e2 
m^{2} 

volume 
1.1e6 
m^{3} 

length 
18.5e2 
m 
rounded thruster top 
force 
5.12e8 
kgm/s^{2} 

acceleration 
3.71e5 
m/s^{2} 
Light Pressure
A useful starting analysis is in E. M. Soop, "Handbook of Geostationary Orbits". Soop's analysis is for geostationary orbits, which are rarely in eclipse and much less subject to J_{2.} perturbations. However, Soop's analysis is a good starting point. The book is practical, focused on satellite operation, the math is moderate, and the references are rather skimpy.
Soop analyzes the geostationary orbit in the cartesian MEGSD (Mean Equatorial Geocentric System of Date) coordinate system (pg. 15). This system is approximate and quasiinertial; very accurate analyses will require full numerical solutions. The XY plane of MEGSD is the Earth's equatorial plane (which slowly precesses 0.014° per year), with the x direction oriented sidereally, towards the Vernal Equinox or First Point of Aries, where the equatorial plane and the ecliptic plane intersect. The Z direction is north.
Soop describes the various orbital parameters as both scalars (pg 21) and vectors. Unlike Soop, I will use \vec{ x } instead of \overline{ x } .

scalar 
vector 

semimajor axis 
a 

eccentricity 
e 
\vec{ e } 

inclination 
i 
\vec{ I } 

right ascension of the ascending node 
\Omega 

argument of perigee 
\omega 

true anomaly 
\nu 

unperturbed orbit angle 
s 

radius 
r 
\vec{ r } 

velocity 
dr / dt 
d\vec{ r }/dt 

unperturbed orbit velocity 
V = \sqrt{ \mu / a } 

similar to Soop page 39 
apogee 
r_a 

perigee 
r_p 

period 
T = 2 \pi \sqrt{ a^3/\mu } 

radial velocity 
V_r = V (e_x~\sin(s)~~e_y~\cos(s) 

similar to Soop page 39 
tangential velocity 
V_t = 2 V (e_x~\cos(s)~+~e_y~\sin(s) 

similar to Soop page 39 
orthogonal (North) velocity 
V_o = V (i_x~\sin(s)~~i_y~\cos(s) 

similar to Soop page 39 
\vec{ I } = \left( \matrix{ \sin(i)~\sin(\Omega) \\ \sin(i)~\cos(\Omega) \\ \cos(i) } \right)
\vec{ r } = \left( \matrix{ x \\ y \\ z } \right) = { \Large { { a( 1  e^2 ) } \over { 1 + 3 \cos(\nu) } } } \left( \matrix { \cos(\Omega)~\cos( \omega+\nu )  \sin(\Omega)\sin(\omega+\nu )~\cos(i) \\ \sin( \Omega ) ~\cos( \omega+\nu )  \cos(\Omega) \sin( \omega+\nu )~\cos(i) \\ \sin( \omega+\nu )~\sin(i) } \right) ~~~ . . . Soop pg 25
a \equiv ~~~ M_{XXX} unperturbed orbit radius . . . similar to Soop page 26
\vec{ i } = ( i~\sin( \Omega ), i~\cos( \Omega ) ) ~~~ . . . Soop pg 27
\vec{ e } = ( e~\cos( \Omega + \omega ), e~\sin( \Omega +\omega ) ) ~~~ . . . Soop pg 27
r \approx A + \delta a~~A e \cos(\nu) ~~~ . . . Soop pg 29
s_b ~~~ is angle at thrust . . . Soop pg 53
{ \large { \Delta \vec{ e } } = { \Large { { 2 \Delta V } \over V } }} \left( \matrix{ \cos(s_b) \\ \sin(s_b) } \right) ~~~ . . . Soup pg 54
\lambda ~~~ = 3.65e5 s^{2} is the light pressure acceleration, which is the solar pressure P ( 4.56e6 N/m^{2} ) times the area to mass ratio \sigma , 8.333 m^{2} = ( 0.025 m^{2} / 0.003 kg ) for a 44 μm thinsat. . . similar to Soop pg 93
s ~~~ is sidereal angle of orbit . . . Soop pg 93 text
s_s ~~~ is sidereal angle of Sun . . . Soop pg 93 text
{ \Large { { d\vec{ e_{\lambda} } } \over dt } = { 2 \over V } } \left( \matrix{ \cos(s) \\ \sin(s) } \right) { \Large { { d V_t } \over dt } + { 1 \over V } } \left( \matrix{ \sin(s) \\ \cos(s) } \right) { \Large { { d V_r } \over dt } } ~ ~ ~ . . . Soop pg 93
{ \Large { { \partial \vec{ e_{\lambda} } } \over { \partial t } } = { \lambda \over { 2 \pi V } } \int_0^{2\pi} } \left[ 2 \left( \matrix{ \cos(s) \\ \sin(s) } \right) \sin(ss_s)  \left( \matrix{ ~\sin(s) \\ \cos(s) } \right) \cos(ss_s) \right] ds ~ ~ ~ . . . Soop pg 93
{ \Large { { \partial \vec{ e_{\lambda} } } \over { \partial t } } = { \lambda \over { 2 \pi V } }\int_0^{2\pi} } \left[ 1.5 \left( \matrix{ \sin(s_s) \\ ~\cos(s_s) } \right) + 0.5 \left( \matrix{ ~ ~ \sin(2*ss_s) \\ \cos(2*ss_s) } \right) \right] ds ~ ~ ~ . . . This is the equation if \lambda is constant
{ \Large { { \partial \vec{ e_{\lambda} } } \over { \partial t } } = { \lambda \over { 2 \pi V } } } \left[ 1.5 \times 2 \pi \left( \matrix{ \sin(s_s) \\ ~\cos(s_s) } \right) + 0.5 { \Large { \int_0^{2\pi} } } \left( \matrix{ ~ ~ \sin(2ss_s) \\ \cos(2ss_s) } \right) ds \right]
{ \Large { { \partial \vec{ e_{\lambda} } } \over { \partial t } } = { { 3 \lambda } \over { 2 V } } } \left( \matrix{ \sin(s_s) \\ ~\cos(s_s) } \right) + { \Large { \lambda \over { 4 \pi V } } { \int_0^{2\pi} } } \left( \matrix{ ~ ~ \sin(2ss_s) \\ \cos(2ss_s) } \right) ds ~ ~ ~ . . . the integral is zero for constant illumination
{ \Large { { \partial \vec{ e_{\lambda} } } \over { \partial t } } = { { 3 \lambda } \over { 2 V } } } \left( \matrix{  \sin(s_s) \\ ~ \cos(s_s) } \right) ~ ~ ~ perpendicular to the Sun, perturbing the eccentricity vector eastward . . . Soop page 95
Y ~~~ = one year . . . Soop page 95
{ { \vec{ e_{\lambda} } ( t ) = } { \Large { { 3 \lambda Y } \over { 4 \pi V } } } } \left( \matrix{ \cos(s_s) \\ \sin(s_s) } \right) ~~~ . . . Soop page 95
For a prograde orbit, this vector points eastward along the orbit, and has a magnitude of
e_{\lambda} = { \Large { { 3 \lambda Y } \over { 4 \pi V } } } = { \Large { { 3 P \sigma Y } \over { 4 \pi V } } }
Without the J2 modifications, a lightpressuremodified orbit (with \vec e and perigee pointing towards the sun) for a 3 gram, 240cm^{2} thinsat ( σ = 8 m^{2}kg^{1} ) would have a light pressure eccentricity of:
e_{\lambda} = { \Large { { 3 \lambda Y } \over { 4 \pi V } } = { { 3 \times ~ {3.65e5} ~ \times ~ 3.156e7 } \over { 4 \pi ~\times ~5582.74 } } } ~ ~ ~ = 0.049
The eccentricity of the light modified orbit is around 0.05 . If the thinsat is thinner, that gets larger. It also gets larger for more distant orbits ( M360, M480, etc.). Proportional to period T^{1/3}
Inclination to the Ecliptic
Because of the inclination of the orbit relative to the ecliptic, some of the light pressure is towards the south during summer and the north during winter. The eccentricitymodifying light pressure is a function of the sun's declination \delta_\odot :
\lambda = \lambda_0 ~ \cos(\delta_\odot) ~ ~ ~
where:
\delta_\odot = \arcsin \left [ \sin \left ( 23.44^\circ \right ) \cdot \cos \left ({{360^{\circ}\times(day+10)}\over{365}} \right ) \right ] ~~~ where day is the day of the year starting with Jan 1 = 0
thus
\lambda = \lambda_0 \sqrt{ 1  \left [ \sin \left ( 23.44^\circ \right ) \cdot \cos \left ({{360^{\circ}\times(day+10)}\over{365}} \right ) \right ]^2 } ~ ~ ~
averaging over an orbit
\bar { \lambda } \approx 0.9592 ~ \lambda_0 ~~~ so e_{\lambda} ~~~ is reduced somewhat, to 0.047 .
NOTE ADDED: This needs refinement. When the orbit is inclined, less is in eclipse, so the thrust increases due to that.
Modifications for earth shadow eclipse time and night pollution minimization
The eclipse time has a small effect, because we are only subtracting an acceleration approximately equal to sin^{2} of the +/ 30° angle behind the earth. Integrating from 150° to 150° subtracts approximately 1/24 of the effect, so e_{\lambda} is reduced to about 0.045 .
A larger effect on light pressure is the intentional tilting of thinsats to minimize night time light pollution. Thinsats will be turned towards the terminator as they pass into the back half of the orbit. For M288 and lower, the orbits will have an apogee in the back, and orbit slower and spend longer in this reduced light (and light pressure) condition.
Soop's integral is intended for nearly circular GEO orbits, and assumes both ds / dt and \sigma (and thus \lambda ) are approximately constant. With the night light pollution minimization manuever, multiply \lambda by F( s  s_s )$.
Integrate:
{ \Large { { \partial \vec{ e_{\lambda} } } \over { \partial t } } = { 1 \over { 2 \pi V } } \int_0^{2\pi} } \lambda F( s  s_s ) \left[ 1.5 \left( \matrix{ \sin(s_s) \\ ~\cos(s_s) } \right) + 0.5 \left( \matrix{ ~ ~ \sin(2*ss_s) \\ \cos(2*ss_s) } \right) \right] ds ~ ~ ~
We can simplify and use sun angle zero, and integrate over the orbit:
{ \Large { { \partial \vec{ e_{\lambda} } } \over { \partial t } } = { \lambda \over { 2 \pi V } } \int_0^{2\pi} } F( s ) \left[ 1.5 \left( \matrix{ 0 \\ 1 } \right) + 0.5 \left( \matrix{ ~ ~ \sin(2*s) \\ \cos(2*s) } \right) \right] ds ~ ~ ~
s 
F( s ) 

from 
to 
for m288 orbit 
~ \pi/2 
~ ~ \pi/2 
1 (maximum) 
~ ~ \pi/2 
~ 5\pi/6 
\Large{{~2\sin(s)1}\over \sqrt{54\sin(s)}} 
~ 5\pi/6 
5\pi/6 
0 (eclipse) 
5\pi/6 
~ \pi/2 
\Large{{2\sin(s)1}\over \sqrt{5+4\sin(s)}} 
If we integrate function F( s ) times the sin() and cos() terms (with this program:sunint.c), we get a vector of \left( \matrix{ 0 \\ 0.732 } \right) , slightly higher magnitude than the power multiplier, 0.689 .
If we also assume a multiplier of 0.45 due to front side infrared filtering (0.333 ideal), the sun thrust power is multiplied by M = 0.33. Assuming a sail ratio \sigma = 8.333 m^{2}/kg (A 3 gram, 250cm^{2} thinsat),
e_{\lambda} = { \Large { { 3 M P \sigma Y } \over { 4 \pi V } } = { { 3 \times 0.33 \times ~ {4.56e6} ~ \times ~ 8.333 ~ \times ~ 3.156e7 } \over { 4 \pi ~\times ~5582.74 } } } ~ ~ ~ = 0.0169
Combining with the J2 Modification
The simplest case for M288 is an eccentricity vector diagram like so:
For a bounded elliptical M288 orbit, e_{J2} = 1.771 \times e and e_{ year } = e , so with apogee towards the sun, e_{ \lambda } = 0.771 \times e . That means that for a 44μm 8.333 m^{2}/kg thinsat, e = 0.0169 / 0.771 or 0.022, or perhaps a little larger because of the twiceannual variation in tangential light acceleration. Assume that the distance to orbits near M288 (for example, LAGEOS at 12331 km apogee) should be at least 0.022*12789 km = 280km, so there should be no valuable equatorial crossers between 12509 and 13069 km .
M360 orbits (The O3b satellites) have a J2 excess of 0.157 instead of 0.771, which means that the eccentricity to mass ratio will be almost 5 times larger than M288. M360 does not look like a practical orbit for low mass thinsats. Cheap launch and ballast will change this.
The M480 orbit ( 16756km ) is slower so e_{\lambda} is larger:
e_{\lambda488} = { { 3 M P \sigma Y } \over { 4 \pi V } } = { { 3 \times 0.33 \times 4.65E6 \times 8.333 \times 3.156e7 } \over { 4 \pi \times 4877.51 } } ~ ~ ~ = 0.0198 .
For a bounded elliptical M480 orbit, e_{J2} = 0.688 \times e and e_{ year } = e ~ ~ ~ , so with perigee towards the sun, e_{ \lambda } = 0.312 \times e ~ ~ ~ . That means that for a 44μm 8.333 m^{2}/kg thinsat, e ~ = 0.0198 / 0.312 or 0.0633 . Assume that the distance to orbits near M480 should be at least 0.0633*16756 km = 871 km, so there should be no valuable equatorial crossers between 15885 and 17627 km .
Inclination  Flattened Toroidal Orbit
Because of the eccentricity, we will incline the orbit, so that the central orbit can be set aside for heavier objects. We should consider a flattened torus so that this orbit does not go too far south or north. If we make the inclination for the M288 orbit ( e ~~~ = 0.0 ) 0.5°, then the orbit passes above and below the 12789 km central orbit by 112 km, compared to the 280 km sideways distance. The circular orbital speed at M288 is V = 5583 m/s. The relative velocity of the two orbits at crossing ( r = A, \cos( \theta ) =  e ~ ~ = 0.022, or \theta ~ ~ = 91.26° ) is:
tangential 
e V \cos( \theta ) = e^2 V 
2.7 m/s 
radial 
e V \sin( \theta ) 
122.5 m/s 
total 
e V 
122.5 m/s 
The "velocity shear" across the orbit is ( 122.5 m/s / 112km ) 1.09 m/s per km or 1.09e3 / second. If we imagine a series of increasingweight thinsats around the central orbit, lighter at the edges and denser near the central orbit, then M288 orbits that differ in velocity by 1 m/s will always more than 910 meters apart.
The locus of all these orbits will be a "double MÃ¶bius" strip (one full turn), with one edge along the circular central orbit and the other edge wrapping around it toroidally.
A note about thinsats and Space Based Solar Power in GEO
The J_{2} oblateness effect is almost nonexistent in GEO. The discussion by Soop above is focused on GEO satellites, which must be well positioned with very small errors in North/South and East/West position. The North/South position is strongly perturbed by tidal effects from the Sun and Moon, while the East/West position is perturbed by higher order nonuniformities in the longitudinal gravity field. These nonuniformities want to drag satellites towards 75°E and 105°W, while light pressure effects want to drive the orbit elliptical as above.
GEO is prime real estate; almost every orbital slot has an expensive satellite in it, and satellites are typically constrained to stay in an east/west dead band of about 0.1° so they do not encroach on their neighbors. Further, GEO comsats usually talk to fixed high gain dishes on the ground, with narrow beam angles, so they are constrained to 0.1° of drift in all directions. Server sky ground antennas are assumed to be steerable, so they are free to drift somewhat to the north and south. But they must be constrained east/west .
The total station keeping thruster budget for a GEO satellite averages 50 m/s per year or 1.6 μm/s^{2}. With a conventional satellite dry weight of 1.5 tonnes, the station keeping requires an average of 2.4 mN. Traditional satellites use solar panels massing about 500kg/10kW (WAG), and assuming 20% efficiency ( 270w/m^{2} WAG) these 40m^{2} satellites are subject to 180 μN of sunlight force, or 0.12 μm/s^{2} of acceleration, smaller than the station keeping force.
Solar power satellites are intended to produce 1 kW per kilogram, which increases the forces needed for sunlight station keeping to perhaps 2.0 μm/s^{2}, or 110 m/s/year .
If we do not correct the sunlight forces, then e_{\lambdaGEO} = { { 3 \lambda Y } \over { 4 \pi V } } = { { 3 \times 2e6 \times 3.156e7 } \over { 4\times\pi\times 3074.6 } } = 5e3 . The daily east/west drift would be about ±0.01 radians or 0.57°  clearly unacceptable in the tightly constrained GEO orbit. The mass must be increased to 6kg/kW, and thinsats would require similar mass to area ratios.
Although SSPS in GEO is "always on", it may be as much as 6 times heavier and 13.3 times harder to launch than thinsats to M480  even with thinsats at 30% utilization to one ground rectenna, the launch cost per Watt is 4x cheaper. M480 thinsats are also 2.5x closer, and antenna sizes can be 6x smaller. If the rectennas are 6x cheaper, then we can deploy 6 for the cost of 1 GEO rectenna, and sell all the power from the thinsats, all around the world, bringing utilization closer to 100%.
Note that in the real world of grid power, demand is not constant over a day, and peaking sources are expensive. If SBSP can supply peak load, by switching the beam to rectennas in other time zones, the power becomes much more valuable. Overall, SBSP supplied with MEO thinsats can be up to 20x more profitable than power supplied from GEO satellites. In addition, because the arrays can be in inclined orbits, we the rectenna elevation angles can be reduced, and the power beams are less likely to interfere with signal beams from GEO communication satellites.
Lagrange Orbits and Light Pressure
Because of the biological effects of nocturnal illumination, both MEO and GEO SBSP deployment will be limited anyway. It is better to deploy large scale SBSP at the Lunar Lagrange points, where lunar perturbations and solar perturbations can be balanced with much lighter thinsats in very large arrays, perhaps with significant north/south inclination to the lunar orbital plane.