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| Summary: while 50 micron thinsats are probably manufacturable, they may be a bit too thin, perhaps by a factor of 2. Light-pressure-related eccentricity gets larger as the thinsat area to mass ratio gets larger. THIS NEEDS FURTHER STUDY. | 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, sunwards-perigee 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. |
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| 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 ( -1.082626683E-03 , 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 ~ ~ $ where ||$ N_{pr} $ ||precessions per year caused by oblateness || ||$ J_2 $ ||spherical harmonic of gravity causing oblateness || ||$ Y $ ||year period in seconds, || ||$ T $ ||orbit period in seconds || ||$ Y/T $ ||number of orbits per year || ||$ R_e $ ||earth equatorial radius = 6378 km || ||$ A $ ||orbit semimajor axis (radius if circular) || 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 uses $ a ( 1 - e^2 ) $ instead of $ A $ and has a correction for inclination, but both inclination and ellipsicity are small for these near-equatorial near-circular orbits, so we can use the above approximation to a few percent accuracy. || orbit || radius (RE) || orbits/year ||$N_{pr}$|| || LEO || 1.047 || 5758.5 || 17.061 || || m288 || 2.005 || 2191.5 || 1.771 || || m360 || 2.264 || 1826.2 || 1.157 || || m480 || 2.627 || 1461.0 || 0.688 || || m720 || 3.182 || 1095.7 || 0.351 || || m1440 || 4.168 || 730.5 || 0.137 || || GEO || 6.611 || 365.2 || 0.027 || |
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 ( -1.082626683E-03 , 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 $ || -1.082626683E-03 ||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 near-equatorial near-circular 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 || |
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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. |
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| || 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) = { { 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 $ \Delta \vec{ e } = { { 2 \Delta V } \over V } \left( \matrix{ cos~s_b \\ sin~s_b } \right) $ . . . Soup pg 54 ---- $ \lambda $ = 3.65e-5 s^-2^ is the light pressure acceleration, which is the solar pressure ( 4.56e-6 N/m^2^ ) times the area to mass ratio, 8 m^2^ = ( 0.024 m^2^ / 0.003 kg ) for a 50 μ m 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 $ { { d\vec{ e } } \over dt } = { 2 \over V } \left( \matrix{ cos~s \\ sin~s } \right) { { d V_t } \over dt } + { 1 \over V } \left( \matrix{ sin~s \\ -cos~s } \right) { { d V_r } \over dt } $ . . . Soop pg 93 $ { { \partial \vec{ e } } \over { \partial t } } = { \lambda \over { 2 \pi V } } \int_0^{2\pi} \left[ 2 \left( \matrix{ cos~s \\ sin~s } \right) sin( s-s_s ) - \left( \matrix{ sin~s \\ -cos~s } \right) cos(s-s_s) \right] ds $ . . . Soop pg 93 $ { { \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 ) = { { 3 P \sigma Y } \over { 4 \pi V } } \left( \matrix{ cos~s_s\\sin~s_s } \right) $ . . . Soop page 95 Without the J2 modifications, a light-pressure-modified orbit(with $ \vec e $ and perigee pointing towards the sun ) for a 3 gram, 240cm^2^ thinsat ( \sigma = 8 m^-1^s^-2> would have a light pressure eccentricity of: $ e_{\lambda} = { { 3 \lambda Y } \over { 4 \pi V } } = { { 3\times3.65e-5\times3.156e7 } \over { 4\times\pi\times5582.74 } } $ = 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^ |
|| 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 ~~~$ = 3.65e-5 s^-2^ is the light pressure acceleration, which is the solar pressure $ P $ ( 4.56e-6 N/m^2^ ) times the area to mass ratio $ \sigma $, 8 m^2^ = ( 0.024 m^2^ / 0.003 kg ) for a 50 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(s-s_s) - \left( \matrix{ ~\sin(s) \\ -\cos(s) } \right) \cos(s-s_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*s-s_s) \\ -\cos(2*s-s_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(2s-s_s) \\ -\cos(2s-s_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(2s-s_s) \\ -\cos(2s-s_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 light-pressure-modified 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 ~ {3.65e-5} ~ \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^ MoreLater ---- |
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| $ \lambda = \lambda_0 ~ cos ~ \delta_\odot $ | $ \lambda = \lambda_0 ~ \cos(\delta_\odot) ~ ~ ~$ |
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| $ \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 } $ $ \langle \lambda \rangle \approx 0.9592 ~ \lambda_0 $ so $ e_{\lambda} $ is reduced somewhat, to 0.047 . ---- === Modifications for earth shadow eclipse time === |
$ \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 . ---- === Modifications for earth shadow eclipse time and night pollution minimization === |
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| 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, $ \lambda $ is a function of $ s_s $ and so we need to integrate: $ { \Large { { \partial \vec{ e_{\lambda} } } \over { \partial t } } = { 1 \over { 2 \pi V } } \int_0^{2\pi} } \lambda( s - s_s ) \left[ 1.5 \left( \matrix{ -\sin(s_s) \\ ~\cos(s_s) } \right) + 0.5 \left( \matrix{ ~ ~ \sin(2*s-s_s) \\ -\cos(2*s-s_s) } \right) \right] ds ~ ~ ~ $ ||<-2> $ \theta $ || $ \lambda( \theta ) $ || || from || to || for m288 orbit || || $ -\pi/2 $ || $ \pi/2 $ || \lambda_0 || || $ \pi/2 $ || $ 2\pi/3 $ || function here || || $ 2\pi/3 $ || $ -2\pi/3 $ || 0 (eclipse) || || $ -2\pi/3 $ || $ -\pi/2 $ || function here || MoreLater |
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| 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 50μm 8 m^2^/kg thinsat, $ e $ = 0.045 / 0.771 or 0.058, or perhaps a little larger because of the twice-annual variation in tangential light acceleration. Assume that the distance to orbits near M288 (for example, LAGEOS) should be at least 0.06*12789 km, so there should be no valuable equatorial crossers between 12000 and 13300 km . FIXME ... the LAGEOS satellites cross the equator at around 12230km - WE MAY NEED TO MAKE THE 50μm THINSAT HEAVIER!!! Or maybe our rising and descending nodes can be made to match LAGEOS in some manner, so we always stay many kilometers away. Needs more analysis!!! MORE LATER |
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 50μm 8 m^2^/kg thinsat, $ e $ = 0.045 / 0.771 or 0.058, or perhaps a little larger because of the twice-annual variation in tangential light acceleration. Assume that the distance to orbits near M288 (for example, LAGEOS) should be at least 0.06*12789 km, so there should be no valuable equatorial crossers between 12000 and 13300 km . MoreLater ... the LAGEOS satellites cross the equator at around 12230km - to avoid their orbit WE MAY NEED TO MAKE THE 50μm THINSAT HEAVIER!!! Or maybe our rising and descending nodes can be made to match LAGEOS in some manner, so we always stay many kilometers away. Needs more analysis!!! MoreLater |
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| The M480 orbit ( 16756km ) is slower so $ e_{\lambda} is larger: $ e_{\lambda488} = { { 3 \lambda Y } \over { 4 \pi V } } = { { 3\times3.65e-5\times3.156e7 } \over { 4\times\pi\times 4877.51 } } $ = 0.056, reduced to perhaps 0.55 with eclipse effects. 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 50μm 8 m^2^/kg thinsat, $ e $ = 0.055 / 0.312 or 0.176 . Assume that the distance to orbits near M480 should be at least 0.18*16756 km, so there should be no valuable equatorial crossers between 13700 and 19800 km . This is a very wide band; we probably must make m480 thinsats heavier. |
The M480 orbit ( 16756km ) is slower so $ e_{\lambda} $ is larger: $ e_{\lambda488} = { { 3 \lambda Y } \over { 4 \pi V } } = { { 3\times3.65e-5\times3.156e7 } \over { 4\times\pi\times 4877.51 } } ~~~$ = 0.056, reduced to perhaps 0.55 with eclipse effects. 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 50μm 8 m^2^/kg thinsat, $ e ~~~$ = 0.055 / 0.312 or 0.176 . Assume that the distance to orbits near M480 should be at least 0.18*16756 km, so there should be no valuable equatorial crossers between 13700 and 19800 km . This is a very wide band; we probably must make m480 thinsats heavier. |
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| Because of the eccentricity, we will incline the orbit, so that the central orbit can be set aside for heavier thinsats. 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 $ = 0.058 ) 1°, then the orbit passes above and below the 12789 km central orbit by 223 km, compared to the 740 km sideways distance. The circular orbital speed at M288 is 5583 m/s. The relative velocity of the two orbits at crossing ( r = A, $ cos( \theta ) = -e $ = 0.058, or $ \theta $ = 93.3° ) is: || tangential || $ e V cos( \theta ) = e^2 V $ || 18.8 m/s || || radial || $ e V sin( \theta ) $ || 323.3 m/s || || total || $ e V $ || 323.8 m/s || The "velocity shear" across the orbit is 1.45 m/s per km or 1.45e-3 / second. If we imagine a series of increasing-weight thinsats around the central orbit, lighter at the edges and denser near the central orbit, then M288 orbits that differ in velocity by 1.45 m/s will always more than 1 km apart. |
Because of the eccentricity, we will incline the orbit, so that the central orbit can be set aside for heavier thinsats. 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 ~~~$ = 0.058 ) 1°, then the orbit passes above and below the 12789 km central orbit by 223 km, compared to the 740 km sideways distance. The circular orbital speed at M288 is 5583 m/s. The relative velocity of the two orbits at crossing ( r = A, $ \cos( \theta ) = -e ~~~$ = 0.058, or $ \theta ~~~$ = 93.3° ) is: || tangential || $ e V \cos( \theta ) = e^2 V $ || 18.8 m/s || || radial || $ e V \sin( \theta ) $ || 323.3 m/s || || total || $ e V $ || 323.8 m/s || The "velocity shear" across the orbit is 1.45 m/s per km or 1.45e-3 / second. If we imagine a series of increasing-weight thinsats around the central orbit, lighter at the edges and denser near the central orbit, then M288 orbits that differ in velocity by 1.45 m/s will always more than 1 km apart. |
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| MORE LATER ---- === A note about thinsats 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 much higher power per kilogram, which increases the forces needed for sunlight station keeping. MORE LATER --------------- Old (incorrect) analysis removed. |
MoreLater ---- === 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_{\lambda-GEO} = { { 3 \lambda Y } \over { 4 \pi V } } = { { 3 \times 2e-6 \times 3.156e7 } \over { 4\times\pi\times 3074.6 } } $ = 5e-3 . 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, were 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. MoreLater |
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, sunwards-perigee 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 half-thrust 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 black-body 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 ( -1.082626683E-03 , 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 |
-1.082626683E-03 |
spherical harmonic of gravity causing oblateness |
\mu |
398,600.4418 km3/s2 |
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 near-equatorial near-circular 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/m2 |
|
Speed of Light |
2.998e+8 |
m/s |
|
Light pressure |
4.56e-6 |
kg/m-s2 |
|
Thinsat parameters |
|||
mass |
3e-3 |
kg |
|
thickness |
50 |
μm |
|
density |
2.5e+3 |
kg/m3 |
|
volume |
1.2e-6 |
m3 |
|
area |
2.4e-2 |
m2 |
|
length |
18.5e-2 |
m |
rounded thruster top |
force |
1.0944e-7 |
kg-m/s2 |
|
acceleration |
3.648e-5 |
m/s2 |
|
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 J2. 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 quasi-inertial; very accurate analyses will require full numerical solutions. The X-Y 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 ~~~ MXXX 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 ~~~ = 3.65e-5 s-2 is the light pressure acceleration, which is the solar pressure P ( 4.56e-6 N/m2 ) times the area to mass ratio \sigma , 8 m2 = ( 0.024 m2 / 0.003 kg ) for a 50 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(s-s_s) - \left( \matrix{ ~\sin(s) \\ -\cos(s) } \right) \cos(s-s_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*s-s_s) \\ -\cos(2*s-s_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(2s-s_s) \\ -\cos(2s-s_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(2s-s_s) \\ -\cos(2s-s_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 light-pressure-modified orbit (with \vec e and perigee pointing towards the sun) for a 3 gram, 240cm2 thinsat ( σ = 8 m2kg-1 ) would have a light pressure eccentricity of:
e_{\lambda} = { \Large { { 3 \lambda Y } \over { 4 \pi V } } = { { 3 \times ~ {3.65e-5} ~ \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 T1/3
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 eccentricity-modifying 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 .
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 sin2 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, \lambda is a function of s_s and so we need to integrate:
{ \Large { { \partial \vec{ e_{\lambda} } } \over { \partial t } } = { 1 \over { 2 \pi V } } \int_0^{2\pi} } \lambda( s - s_s ) \left[ 1.5 \left( \matrix{ -\sin(s_s) \\ ~\cos(s_s) } \right) + 0.5 \left( \matrix{ ~ ~ \sin(2*s-s_s) \\ -\cos(2*s-s_s) } \right) \right] ds ~ ~ ~
\theta |
\lambda( \theta ) |
|
from |
to |
for m288 orbit |
-\pi/2 |
\pi/2 |
\lambda_0 |
\pi/2 |
2\pi/3 |
function here |
2\pi/3 |
-2\pi/3 |
0 (eclipse) |
-2\pi/3 |
-\pi/2 |
function here |
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 50μm 8 m2/kg thinsat, e = 0.045 / 0.771 or 0.058, or perhaps a little larger because of the twice-annual variation in tangential light acceleration. Assume that the distance to orbits near M288 (for example, LAGEOS) should be at least 0.06*12789 km, so there should be no valuable equatorial crossers between 12000 and 13300 km .
MoreLater ... the LAGEOS satellites cross the equator at around 12230km - to avoid their orbit WE MAY NEED TO MAKE THE 50μm THINSAT HEAVIER!!! Or maybe our rising and descending nodes can be made to match LAGEOS in some manner, so we always stay many kilometers away. Needs more analysis!!!
M360 orbits 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 \lambda Y } \over { 4 \pi V } } = { { 3\times3.65e-5\times3.156e7 } \over { 4\times\pi\times 4877.51 } } ~~~ = 0.056, reduced to perhaps 0.55 with eclipse effects.
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 50μm 8 m2/kg thinsat, e ~~~ = 0.055 / 0.312 or 0.176 . Assume that the distance to orbits near M480 should be at least 0.18*16756 km, so there should be no valuable equatorial crossers between 13700 and 19800 km . This is a very wide band; we probably must make m480 thinsats heavier.
Inclination - Flattened Toroidal Orbit
Because of the eccentricity, we will incline the orbit, so that the central orbit can be set aside for heavier thinsats. 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 ~~~ = 0.058 ) 1°, then the orbit passes above and below the 12789 km central orbit by 223 km, compared to the 740 km sideways distance. The circular orbital speed at M288 is 5583 m/s. The relative velocity of the two orbits at crossing ( r = A, \cos( \theta ) = -e ~~~ = 0.058, or \theta ~~~ = 93.3° ) is:
tangential |
e V \cos( \theta ) = e^2 V |
18.8 m/s |
radial |
e V \sin( \theta ) |
323.3 m/s |
total |
e V |
323.8 m/s |
The "velocity shear" across the orbit is 1.45 m/s per km or 1.45e-3 / second. If we imagine a series of increasing-weight thinsats around the central orbit, lighter at the edges and denser near the central orbit, then M288 orbits that differ in velocity by 1.45 m/s will always more than 1 km apart.
The locus of all these orbits will be a "double Möbius" 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 J2 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/s2. 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/m2 WAG) these 40m2 satellites are subject to 180 μN of sunlight force, or 0.12 μm/s2 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/s2, or 110 m/s/year .
If we do not correct the sunlight forces, then e_{\lambda-GEO} = { { 3 \lambda Y } \over { 4 \pi V } } = { { 3 \times 2e-6 \times 3.156e7 } \over { 4\times\pi\times 3074.6 } } = 5e-3 . 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, were 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.