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Date: Fri, 5 Mar 2010 20:04:14 -0600
To: Henno.Boomkamp@esa.int
From: "John C. Ries" <ries@csr.utexas.edu>
Cc: Eric Leuliette <Eric.Leuliette@noaa.gov>,
        Eelco Doornbos <e.n.doornbos@tudelft.nl>,
        "Mercier Flavien (contact)" <flavien.mercier@cnes.fr>,
        Frank Lemoine <Frank.G.Lemoine@nasa.gov>,
        "ids.analysis.forum@cls.fr" <ids.analysis.forum@cls.fr>,
        "jean-paul.berthias@cnes.fr" <jean-paul.berthias@cnes.fr>,
        Cerri Luca <luca.cerri@cnes.fr>,
        Marek Ziebart <marek.ziebart@ge.ucl.ac.uk>,
        Michiel Otten <michiel.otten@esa.int>,
        Nikita Zelensky <nzelensky@sgt-inc.com>,
        "remko@altimetrics.com" <remko@altimetrics.com>,
        Scott Luthcke <scott.b.luthcke@nasa.gov>,
        Steve Klosko <sklosko@sgt-inc.com>
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Subject: Re: [ids.analysis.forum] RE: surface forces

At 12:35 PM +0100 3/5/10, Henno.Boomkamp@esa.int wrote:
>Dear all,
>
>At the risk of introducing a bright red herring, I would like to mention
>a software problem that I ran into last year and which could have a
>signature similar to what you are all seeing (beta & two-beta signal).
>
>In multi-step numerical integrators you have to be careful with the
>zero transitions of the wing from the box-wing model, which typically
>do not occur for SRP but you can get them for IR, albedo or drag.
>When the panel rotates "through zero" in the direction of the Earth,
>the projected wing area for IR first decreases more or less linearly,
>can even be zero for a split second and immediately starts increasing
>again. This implies a discontinuity in the first and higher derivatives
>of any surface forces that are looking sideways at the solar panel.
>
>Multi-step integrators really don't like jumps in higher derivatives.
>The result can be non-negligible errors in the computed acceleration,
>which is the same as having an additional perturbation on the satellite.
>Arcs with two such occurrances per revolution (~20 per day...) will
>definitely come out worse than arcs that do not have any of these
>zero-crossings at all. Without having checked it in detail, I suppose
>that these movements are governed by the beta angle, and as it
>affects the absolute value of the projection (2-per-rev) there could
>well be a spectral peak at two beta as well.

Hi Henno,

It is clear that a multi-step (at least fixed-step) integrators will 
not precisely represent the integration when it steps across a 
discontinuity. We considered this some time ago because the problem 
is even larger than this when the satellite crosses into shadow. In 
many cases, the step goes directly from light to shadow (or from full 
SRP to zero) since the penumbra might be very thin. In fact, it was 
this effect that caused us to shorten the Lageos-2 integration step 
from 300 seconds to 250, because 300 was an unfortunate resonance 
with the shadow crossings for L2, and the shadow crossing was 
systematically 'mis-sampled'. John Lundberg, while he was here, even 
came up with a way to 'correct' the back difference table of the 
multi-step integrator to account for the discontinuity (I can 
probably provide that report if anyone is interested). Variable step 
integrators will sometimes detect these kinds of things and shorten 
the integration step, but we found they ended up spending a lot of 
time 'chattering' around each shadow crossing, without a substantial 
improvement in the accuracy.

In the end, we found that this problem for our fixed-step multi-step 
integrator had only a small effect for short arcs, and, barring an 
unfortunate resonance with the step size for long arcs, it tended to 
just average out. I would expect the contribution for the solar-array 
and the Earth's albedo/IR to be even more benign.

Best regards,
John