Geostationary Space Based Solar Power

The geostationary orbit is a seemingly attractive place to put Space Based Solar Power arrays. It is also already full of valuable satellites, whose missions could be severely compromised by SSPS arrays as currently proposed. Further, the orbital dynamics are more complicated than a naive two body Kepler orbit without light pressure modification and tidal forces, invalidating naive assumptions about orbital placement and stability.

The most important thing to know is that international law and common decency require that the hundreds of GEO satellites up there stick to their assigned angular slot, and must not deviate from their assigned position more than 0.1° or so, north-south or east-west. They also must not create radio interference with those other satellites.

Earth antennas for GEO communication services are usually fixed, and have narrow angular apertures, but they are not perfect and will receive off-axis sidelobe power, creating interference. SBSP satellites will produce millions to billions of times more power than a communication satellite, so even tiny amounts of sidelobe leakage may swamp ground receivers, even receivers intended for other frequency bands.

If an SBSP can interfere with existing services, it will not be allowed to launch. If it is launched and does not perform within licensed limits, it will be shut down, disabled, or destroyed, whatever it takes to get it off the air and out of the way of other licensed satellites. If your hypothetical solar power satellite cannot operate within these constraints, go back to the drawing board and figure out something that will.

I will show that naive ultralight designs ( 10m²/kg ) require significantly eccentric orbits, with a large tangential angular variation. That can interfere with the mission of existing comsats, and possibly threaten collisions.

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For this analysis, I will ignore the effects of eclipse, the Earth's axial tilt, higher order gravitational harmonics, special relativity, and perturbations from Jupiter and other planets. These effects are not trivial, but they require a book-length treatment (such as Soop and Piscane and Wertz, referenced below).

I will also not reference closed-source computer programs such as AGI's Satellite Toolkit. These programs can be useful, but they are opaque. You can't see the math behind the pretty GUI, so you don't know when you've taken the models beyond their range of applicability. Before using any computer-aided design tool, you must learn the underlying math and physics, or you will almost certainly misapply it. CAD tools automate understanding, they do not replace it.

Effects Discussed

We will assume that the orbit is near circular, but will be slightly elliptical. It will be perturbed away from that orbit, which will require appropriate choice of eccentricity (zero eccentricity is far too expensive) and quite a bit of \Delta V ~~~ thrust to keep the inclination near zero, in spite of out of plane perturbations by lunar and solar tides. A related analysis is on this wiki, for lower and more stable MXXX orbits.

• Precession of the argument of periapsis by light pressure.

• Precession of the argument of periapsis by J₂ earth oblateness.

• Inclination perturbation of the north-south alignment by the Moon and the Sun.

• Radio sidelobe interference caused by off-axis power.

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Simplified Orbits

For this discussion, we do not need to dwell on all six parameters of a classic elliptical Kepler orbit, or the high precision modifications of it. To start with, assume our solar satellite orbit has no inclination (and thus no "ascending node"). I will show that the orbit must be elliptical, so assume a Kepler orbit with an ellipticity e ~~. It is geosynchronous (though as we will see, not exactly geostationary), and that means that the orbital period is a synoptic 86,400 seconds and a sidereal 86,164+ seconds. The orbit turns in the same direction as the earth turns around the sun, "eastbound", so it makes one extra "sidereal day" every year.

To first order, orbits with the same period have the same semimajor axis, half the end-to-end distance from perigee to apogee on the orbital ellipse. The semimajor axis for a circular geostationary orbit is the same as the radius, 42,164 kilometers. The altitude above the Earth's equatorial bulge (6,378 kilometers) is 35,786 kilometers, though that is not too significant in our discussion.

Since the semimajor axis is the same for an elliptical orbit, that means that if perigee is X kilometers closer than the circular radius, then apogee is X kilometers further away. There are some slight differences because of the equatorial bulge (resulting in the J₂ cube law modification to the gravitational field). We will deal with those later. More importantly, the relative position of a body in an elliptical orbit traces out a smaller ellipse (approximately) in the equatorial plane, relative to coordinates of a body in a circular orbit. For modest sized eccentricities, that smaller ellipse is twice as wide as it is tall (see Soop for details), so the ellipse is twice as wide (east to west) as it is radially (up and down compared to the earth.

Looking down from the north, you will see the earth moving counterclockwise in its orbit, and rotating counterclockwise on its axis. A geostationary orbit will rotate counterclockwise around the earth, and a slightly elliptical geostationary orbit will move counterclockwise around a rotating 2:1 ellipse, centered on an imaginary, orbiting geostationary point.

Someday I will create an animation showing that.

Here are the relevant parameters of a geosynchronous orbit (rounded to the nearest second and kilometer, and an eccentric example (arbitrarily chosen):

(it is an easy-to-remember coincidence that the last three digits of those numbers is "164"; no deep significance)

Note that even with that very small eccentricity, we have exceeded a ±0.1° east-west deviation requirement by more than a factor of 10. Legal geostationary satellites must be positioned very precisely.

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Precession of the argument of periapsis by light pressure

Light pressure accelerates our satellite - if it has a big "lightsail" area, it accelerates a lot. The light pressure acceleration depends on the optical thrust multiplier \alpha ( 1.0 + albedo, between 1.0 and 2.0, assume 1.2):

a ~ = ~ \Large { \alpha \times { { P_{solar} } \over c } \times { { Area } \over { mass } } }

P = ~ 1367 W/m², c = ~ 3e8 m/s . Some SBSP papers have proposed an area to mass ratio of 10 meters² per kilogram; Version 4 server sky thinsats assume 0.5 m²/kg at the much lower M288 orbit.

We will define the optical ballistic parameter \sigma \equiv \alpha \times { { Area } / { mass } } , so our acceleration formula becomes:

a ~ = ~ \Large { \sigma \times { { P_{solar} } \over c } }

Using the 10m²/kg number, and an \alpha of 1.2, the acceleration is 55 μm/s² , or 1725 m/s of delta V per year. If we tried to hold a satellite in exact position against that light pressure with some kind of rocket engine, we would need to ship up enough fuel and provide enough energy to generate a LOT of delta V. That is three times the delta V provided by the ion engines of a modern comsat over a 10 year mission. That means either a very short-lived SBSP satellite, or frequent resupply of lots of ion engine fuel.

Comsats have area to mass ratios around 0.01kg/m², and \alpha \approx 1.5, about three orders of magnitude less. They are also perturbed by light pressure, but do not compensate for it. Over the course of a year, the radial light acceleration integrates into to a circle, and the circle is small enough to stay within the position error box for these relatively heavy satellites.

The simplest, naive model is to assume that the satellite follows a one year circle around its intended position. The orbit can't actually do that, but just for fun, let's pretend it can. The radius of our imaginary circle would be:

naive radius(wrong): r = a \left( { Y \over { 2 \pi } } \right)^2 ~~~ where Y ~~ = 31556926 seconds, one year.

For the σ = 0.015m²/kg comsat, the acceleration is 6.9e-8 m/s², so the radius would be 1730 kilometers. Oopsie, we fell right out of our ±0.1° or ±74km box. Fortunately, that is not the way it actually works.

Real comsats are actually in a slightly elliptical orbit, which precesses once per year to stay lined up with the sun. The actual effect of light pressure is to add a retrograde increment of ellipticity to the ellipticity vector \vec e . If \vec e is pointed away from the sun (that is, an elliptical orbit with the argument of perigee pointed towards the sun), the ellipse will rotate around the sun once a year.

The increment can be estimated by thinking of the effect of light pressure - to speed up the orbit outbound from the sun, and slow it down inbound. If the orbit started out circular, and the apparent position of the sun in the sky was stationary, the effect would be to make the orbit more and more elliptical, with perigee east of noon and apogee west of noon. Eventually, the eastward perigee would intersect with the earth, bye-bye orbit.

However, the sun appears to move around the sky, once per year relative to the stars as viewed from the earth, and that means our eccentricity increment follows a circle, too. That will make the eccentricity vector trace a circle. There will always be eccentricity, but we can minimize it if we design it to be the integrated increment over 2π tracing a circle around the zero eccentricity point once a year.

Slight quibble: This is modified by the inclination of the earth's equatorial plane, which reduces the eccentricity increment by 4.1% (0.5×(1-cos(23.44°)), and by oblateness (next section), which adds 2.7% additional "turning" of the orbit. Overall, this analysis is overly pessimistic by 1.4%, but our estimate of α may be off by more than that.

So, you say impatiently, what is the eccentricity increment?

I won't derive it here; you can see an incomplete derivation at MXXX orbits or look in Soop. After all the deriving, we learn the eccentricity we must add to our orbit to follow that circle around:

e_{\lambda} \Large{ = {{ 3 P \sigma Y } \over { 4 c \pi V }} = {{ 3 P \alpha A Y \sqrt{r} } \over { 4 c \pi m \sqrt{\mu} }} }

For constant eccentricity, m / A \propto \sqrt{ r } . For constant radial distance range = 2 e \times r , e must be inversely proportional to r , so m / A \propto r^{3/2} .

For our σ = 0.015 m²/kg comsat, with an orbital velocity of 3075 meters per second, this works out to

e_{\lambda} = ~~~ ( 3 × 1367 × 0.015 × 31556926 ) / ( 4 × 3e8 × π × 3075 ) = 1.67 e-4

The spread along the orbit is ± 2 e radians, or ±360°×e/π degrees, or about ±0.02°, 20% of our control box. Note that the control box will have to contain other peturbations, but at least the light pressure variation is tolerable. We can reduce weight by a factor of 5 before we fall outside it.

But that is a big problem for SBSP satellites in geosynchronous orbit. If we make them any lighter than around 20kg/m², they will no longer stay in their assigned ± 0.1° box. If we made them 0.1 kg/m² with \alpha = 1.2, they will sweep out an orbital angle of ± 24 degrees over the course of a year - for each SSPS region (containing multiple satellites, hopefully), we must convince 48 other satellite operators to give up their orbital slot. If we choose only 4 SSPS regions around GEO, we will be taking up more than half of the assignable slots in GEO.

That might be worth it in the long run, but whoever does this is going to need hundreds of billions of dollars to pay incumbent operators and users to find some other way to communicate.

Also, remember that the same night light pollution concerns that apply to server sky in M288 orbit will apply to GEO. GEO is about 5 to 6 times further away, so we can increase our population by 30x over the 1TW server sky estimated soft limit, but we cannot accomodate the kind of global power increases likely to occur over the next century.

Mitigations?

At first glance it might seem to be possible to incline the SSPS orbit relative to the Earth's orbital plane so that it is above or below other satellites at their orbital angle. However, when you think about the evolution of orbits around the year, and the fact that every geosynchronous orbit crosses the equatorial plane twice per day, it is evident that those crossings will intersect all the satellites within the "eccentricity skew" band, twice a day for four lengthy periods per year. The satellites will need to be turned off.

Adding ballast to the SSPS satellites, perhaps dumb lunar rock, can raise their mass to area ratio so that their eccentricity can be smaller. However, this is more mass subject to inclination perturbation, requiring north/south station-keeping Δ V.

Using lower server-sky region orbits increases orbital velocity and reduces the light pressure perturbation some. The lower orbits are also easier to launch to. From temperate latitudes, MEO SBSP satellites will be at different inclinations than GEO comsats in higher orbit. However, the night light pollution concerns are worse, earth tidal orientation effects are worse, and of course the satellites are sweeping through the skies at a couple of degrees per minute, meaning that a distributed set of sources will need to sweep past many ground rectennas.

Much higher SBSP satellites, at lunar radius, will orbit slowly, and even more subject to light pressure perturbation. But they will also be off-angle to GEO, easier to orient in earth tides, and produce very little light pollution. These high orbits (perhaps near lunar L2, L4, and L5) need more sophisticated analysis, but there may be solutions not available at lower orbits. Antenna sizes will be gigantic, given the long distance and refraction limits, but it may prove possible to "re-lens" microwave power with gossamer structures in lower orbits, focusing the power on smaller ground rectennas.

Most of these problems are because the GEO orbit is already in use. When visionaries wrote about Space Solar Power in the early 1970's, there were relatively few GEO satellites and few users, with few books and analytical tools describing the details of GEO orbit design. 40 years later, GEO comsats are an integral part of the world economy. Visplacing communication satellites with another technology would be terribly damaging.

This leads to a bold suggestion: Replace GEO comsats with MEO constellation systems like Server Sky. MEO thinsats are lighter, easier to launch, 50 times easier to communicate with, and close enough for short ping times and low-delay telephone communication. Without the station-keeping and radio interference issues, we can populate "generation 2 GEO" with SBSP, thinsat arrays in skewed elliptical orbits, and other lightweight systems designed for harmonious cooperation.

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Precession of the argument of periapsis by J₂ earth oblateness.

See LightOrbit for a derivation. MoreLater

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Inclination perturbation of the north-south alignment by the Moon and the Sun

The moon and sun create tidal forces on satellites, more force for higher orbit satellites. They are usually inclined relative to the equatorial plane, so there is a north-south component to that force, pushing the satellite north and south. If left unchecked, this can eventually push the satellite up to inclinations exceeding the axial tilt, so comsats thrust about 14 meters per second per year delta V to compensate for solar perturbations, and an average of 37 m/s-y to compensate for lunar perturbations. The lunar perturbation varies over the 18.6 year lunar orbit precession cycle, from 31 m/s-y to 43 m/s-y. So a 10 year mission over the time that the moon is at maximum relative inclination may require as much as 540 m/s delta V. See page 157 in Wertz.

If we make the SBSP satellite heavier, we can cut down the eccentricity, but we add to the mass, and the thrust needed for North/South stationkeeping. Although we may be able to tolerate more north/south variation with an agile rectenna, the inclination can accumulate to quite large values over many years - we must keep it in bounds, so a small target box is not much more costly than a big one.

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Radio Sidelobe interference caused by off-axis power

I will analyze off-axis sidelobe power (perhaps a billion times higher than comsats) and radio receiver front end selectivity, intermodulation, and saturation to estimate the effects of a gigawatt SSPS satellite on communications. MoreLater

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References

E. M. Soop, "Handbook of Geostationary Orbits", Springer; 2010 Softcover reprint of hardcover 1st ed. 1994 edition, ISBN-10: 9048144531

V. L. Pisacane, "The Space Environment and Its Effects on Space Systems", AIAA Education Series, Hardcover, 2008 | ISBN-10: 1563479265

W. L. Larson & J. R.Wertz, "Space Mission Analysis and Design", Microcosm Press, Kluwer Academic Publishers, 3rd ed, 1999 , ISBN-10: 1881883108