Light Pressure Modified Orbits

Light pressure effects modify server-sat array orbits. In the nominal orbit, server-sats 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 server-sat 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" server-sats relatively close to the earth, such as 7 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. The precession, expressed as the number complete precessions per year, is J \approx -3 J_2 ( Y / P ) ( R_e / R_s ) ^ 2 where J \equiv precessions per year caused by oblateness, J_2 \equiv spherical harmonic of gravity causing oblateness, Y \equiv year period in seconds, P \equiv orbit period in seconds, Y/P \equiv number of orbits per year, R_e \equiv earth equatorial radius = 6378km, R_s \equiv orbit equatorial radius.

Light Pressure

This page needs reworking - previously I was not using the correct math for fictious forces in a rotating frame. http://en.wikipedia.org/wiki/Rotating_frame has a good discussion, although they use \Omega where I use \omega_{m288} .

It turns out that for 3 gram 50 micron thick server satellites in m288 orbits, the orbital perturbations from light pressure are small, 360 meter elliptical oscillations along the line of the orbit. If the satellites get thinner, the perturbations increase proportional to the area to mass ratio. If the orbits move further out, the perturbations increase proportional to the cube of the orbit radius, because the orbit period grows, and the perturbations are proportional to the square of the orbit period. There is also a cumulative perturbation caused by the eclipse time, which is probably small enough to be corrected by optical maneuvering.

The fictitous forces are proportional to the radial and tangential position from mean perturbation center, and the tangential velocity relative to that center. Assume a circular orbit - toroidal orbits do not deviate much from that.

An m288 orbit has the following parameters (subject to verification, please help me check them):

In a circular orbit, the centrifugal acceleration balances the centripedal gravitational acceleration:

v^2 / r = \omega^2 r = a = \mu_{\oplus} / r^2 ~ ~ ~ ~ = 2.4371020 m/s^2 at m288

\omega^2 = \mu_{\oplus} / r^3

If x \equiv the tangential distance forward of orbit center, then for small x the triangle of tangential and radial accelerations is proportional to the tangential and radial distances, from congruent triangles:

\partial a_x / a = - \partial x / r

a_x = -( a / r ) x = \omega^2 x

With no perturbations, the vertical acceleration is:

a_r = \omega^2 ~ r - \mu_{\oplus} / r^2 = v^2 / r - \mu_{\oplus} / r^2 = 0

If the tangential velocity is perturbed, the radial acceleration is perturbed:

\partial a_r = 2 v / r \partial v_x = 2 \omega ~ \partial v_x

If the radial distance is perturbed, the radial acceleration is also perturbed:

\partial a_r = \omega^2 + 2 \mu_{\oplus} / r^3 \partial r = 3 \omega^2 \partial r

So the total radial acceleration, for small perturbations of y \equiv \Delta r and x is:

a_y = 3 \omega^2 ~ y + 2 \omega ~ v_x

The radial and tangential accelerations are caused by the light pressure from the Sun. That makes one rotation per 14400 seconds around the guiding center, and is interrupted when the satellite is eclipsed by the Earth. We will compute the effect of this later. If we approximate the rotation as the sidereal period instead, and assume we can make up for eclipse and rotation changes by maneuvering (dangerous assumption), then we can approximate the light pressure components as:

a_{{\lambda} ~y } = a_{\lambda} \sin( \omega t )

a_{{\lambda} ~ x } = a_{\lambda} \cos( \omega t )

Assume that a_{{\lambda} ~ x } = a_x , so that:

a_x = \ddot x = a_{\lambda} \cos( \omega t )

Integrating for x:

v_x = \dot x = ( { a_{\lambda} / \omega } ) \sin( \omega t )

Integrate again for x:

x = - ( a_{\lambda} / \omega^2 ) \cos ( \omega t )

That is the x position relative to orbit center.

Assume that a_{{\lambda} ~ y } = a_y so that

3 \omega^2 y + 2 \omega v_x = a_{\lambda} \sin( \omega t )

3 \omega^2 y + 2 a_{\lambda} \sin( \omega t ) = a_{\lambda} \sin( \omega t )

3 \omega^2 y = - a_{\lambda} \sin( \omega t )

y = -( { a_{\lambda} } / 3 \omega^2 ) \sin( { \omega t } )

Define \lambda as the light pressure distance:

\lambda \equiv a_{\lambda} / \omega^2 = 181.2 meters

MORE LATER, needs checking!

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Over a very long time, tumbling and completely out of control, random variations in up and down orbit acceleration would eventually drive it up and down, probably in something resembling a random walk. This process is likely to be very slow.

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