The Lageos measurement satellites

Two passive satellites are used for laser geodesy. These are 60 centimeter, 400 kilogram solid metal spheres covered with retroreflectors, used for precisely measuring the geometry of the Earth's surface and the Earth's gravitational field.

The m288 semimajor axis (12789km) is 458km above LAGEOS 2 apogee (12330.5 km). Server sky orbits should stay above LAGEOS! Server sky orbits are not quite circular, but elliptical to compensate for light pressure drag. If we want a 2x safety margin, the minimum perigee for server sky thinsats will be 12560km, and the maximum eccentricity will be 0.018 .

These satellites are interesting because their orbits are very precisely defined, and we can use them to verify both our orbital math and learn about drag at those extreme altitudes.

\large \mu = 3.98600448e14 {m^3}/{s^2} ~ \buildrel{?}\over{=} ~ \omega^2 a^3 = 3.988154435e14 = 1.000539376 \mu

The difference is probably due to J2 perturbation dragging forwards both perigee and mean motion. However, because LAGEOS 1 is in a highly inclined and slightly retrograde orbit, the equations I have (from Piscane, page 113) suggests it should get dragged backwards. Meanwhile, those same equations suggest that M288 must be about 3.4 kilometers higher to counteract the extra acceleration due to J_2

MoreLater

Drag, and implications for Server Sky

LAGEOS 1 is slowed down by drag, mostly drag against the high flux of radiation particles. It is dense enough to absorb essentially all the van Allen belt particles that pass through, and change momentum as a result. The "encounter rate" is not limited by ram speed, as it is with gas molecules in the dense atmosphere, but by the speed at which the radiation particles themselves move. If \rho_p is the mass density of particles in the van Allen belt in kg/m³, and v_p is their average speed in m/s, then LAGEOS sheds power into them proportional to 0.5 \rho_p v_p v^2 A , where v^2 = \mu / r and A is the effective cross section. The power reduces the orbital radius r by 0.5 M a_g dr / dt where a_g is gravity and M is the LAGEOS mass. Since dr \ dt = 1.3e-8 m/s, the mass flux per unit area is:

\large Flux = \rho v_p = { M \over { A ~ r } } { { d r } \over { d t } } = 5.1E-12 kg/m² s

That is the radiation flux from all directions. Since LAGEOS is in a highly inclined orbit, the radiation flux changes character in a complex way as a function of magnetic latitude. At high latitudes, the flux is lower.

Thinsats spend their entire orbit near zero magnetic latitude. The flux is higher, but the thinsats present half the cross section to isotropic radiation flux - 2 \pi r^2 (front and back) for a disk rather than 4 \pi r^2 for a sphere. The flux is not isotropic, as the particles are gyrating around field lines. Very complex - but as a crude approximation, assume all these factors cancel out within a factor of 2 or so.

The big question is the amount of the flux of momentum is actually absorbed. Since thinsats are so thin, much of the flux will pass right through without much momentum change. Is the fraction close to 1.0? 0.1? This needs calculation. Imagine a sandwich of thinsats thick enough to absorb all the particles. That whole sandwich will have a fraction close to 1.0, but the amount absorbed by each thinsat will be that fraction divided by the number of thinsats. So the fraction for an isolated, individual thinsat is likely to be small.

This analysis also assumes that the absorbed particles have zero average momentum. They protons actually drift towards the west, with a velocity of 700km/sec at 10MeV, 7 km/sec at 100keV, with the bulk of the particle flux at the lower energies. If most particle energies are below 10keV, there will be little effect from this.

If we assume complete absorption, a 50 micron thinsat will have a much larger area to mass ratio, and will slow down much faster than LAGEOS.

\large { { d r } \over { d t } } = { { A ~ r } \over M } Flux ~ = ~ { { 0.024 * 12789000 * 5.1E-12 } \over 0.003 } m/s ~ ~ = 0.5 mm/sec = 1.6 km/year

While this analysis is quite crude, it does show that radiation flux drag will eventually bring down thinsats, far more quickly than the neutral gas density might suggest.

MoreLater

Orbital elements from various sources

Two Line Elements

LAGEOS1 TLE:

• 1 08820U 76039A 11094.82079485 +.00000016 +00000-0 +10000-3 0 01141
• 2 08820 109.8462 077.3556 0044210 050.5232 309.9115 06.38664784558975

LAGEOS2 TLE:

• 1 22195U 92070B 11095.53472161 -.00000009 00000-0 10000-3 0 3192
• 2 22195 052.6462 175.4139 0138581 279.8295 078.6642 06.47294193436229

. . . 11094 = April 4, 2011 . . . 11095 = April 5, 2011

References:

• http://space.jpl.nasa.gov/msl/QuickLooks/lageosQL.htm

• http://www.n2yo.com/satellite/?s=8820

• http://www.n2yo.com/satellite/?s=22195

• http://nssdc.gsfc.nasa.gov/nmc/spacecraftOrbit.do?id=1976-039A

• http://www.springerlink.com/content/v316482316vt1735/

• http://www.innovateus.net/content/satellite-lageos