General stationkeeping deltavee requirements?

I'm looking for general ways of calculating estimates of stationkeeping
deltavee requirements (e.g., m/(s y)) for satellites/stations in various
orbits (low, medium, high, and apostationary if relevnat) around
arbitrary bodies, and at each of the five Lagrange points between each
(sensible) combination of bodies.

Obviously this is something of a tall order, but for my purposes the
estimates need not be exact; within an order of magnitude or so is good
enough. I managed to find e-folding instability times for a few
specific situations, but obviously 1. that doesn't answer the question
generally and 2. there's more to stationkeeping than just pure e-folding
time; there's also perturbations for nearby objects.

Could someone suggest a relatively simple algorithm to compute such
things to some reasonable degree of accuracy for each of the scenarios
described above? Thanks.

--
__ Erik Max Francis && && http://www.alcyone.com/max/
/ \ San Jose, CA, USA && 37 20 N 121 53 W && &tSftDotIotE
\__/ For whatever reason / We don't see the seasons / Change again
-- India Arie

Erik Max Francis wrote in message ...

I'm looking for general ways of calculating estimates of stationkeeping
deltavee requirements (e.g., m/(s y)) for satellites/stations in various
orbits (low, medium, high, and apostationary if relevnat) around
arbitrary bodies, and at each of the five Lagrange points between each
(sensible) combination of bodies.

For the benefit of groundlings, would you mind explicating exactly
what "station keeping" and delta v means in this context?

In nautical terms, station keeping can be expressed as remaining
stationary in the rest frame of the body on which you are keeping
station. I assume this definition works in space, too, with
complication added by the fact that station keeping on a body in free
fall will in general _not_ place you in free fall: you will drift away
without constant course correction. True at sea also, even if the
physics is different!

Ok? Then what does "deltavee requirements" mean as a term of art?

Obviously this is something of a tall order, but for my purposes the
estimates need not be exact; within an order of magnitude or so is good
enough. I managed to find e-folding instability times for a few
specific situations, but obviously 1. that doesn't answer the question
generally and 2. there's more to stationkeeping than just pure e-folding
time; there's also perturbations for nearby objects.

e-folding time:

"The time required for the amplitude of an oscillation to increase or
decrease by a factor of e."

http://euromet.meteo.fr/demos/courses/glossary/efolding.htm

Hmm... further search reveals that this is a much-beloved term of art
in orbital mechanics, but doesn't immediately clarify. Since you have
chosen to cast your net to sci.physics, will you perhaps also favor
the general audience with a brief explanation in context?

...

"Erik Max Francis" wrote in message
...

Could someone suggest a relatively simple algorithm to compute such
things to some reasonable degree of accuracy for each of the scenarios
described above? Thanks.

Would a numerical simulation method suffice? It's relatively
easy to set up a simulation of a few bodies and a "test
particle". Run the thing until the station-keeping limit is
reached (probably a distance discrepancy?) and then look at
the particle's velocity and the time elapsed.

There are analytical treatments for the motion of bodies
at the Lagrange points in most texts on celestial mechanics.

Ed Green wrote:

| Erik Max Francis wrote in message ...
|
| I'm looking for general ways of calculating estimates of stationkeeping
| deltavee requirements (e.g., m/(s y)) for satellites/stations in various
| orbits (low, medium, high, and apostationary if relevnat) around
| arbitrary bodies, and at each of the five Lagrange points between each
| (sensible) combination of bodies.
|
| For the benefit of groundlings, would you mind explicating exactly
| what "station keeping" and delta v means in this context?
|
| In nautical terms, station keeping can be expressed as remaining
| stationary in the rest frame of the body on which you are keeping
| station. I assume this definition works in space, too, with
| complication added by the fact that station keeping on a body in free
| fall will in general _not_ place you in free fall: you will drift away
| without constant course correction. True at sea also, even if the
| physics is different!

For L points it is the energy cost of remaining within a given distance
(i.e., tolerance) of the particular point. For example, Earth-Sun L1 is
not an equilibrium point, but the energy cost of staying there is low
enough that you can put a satellite there for several years.

| Ok? Then what does "deltavee requirements" mean as a term of art?

"station keeping" is the act of staying at the point

"delta vee" is the energy cost, usually in terms of a single maneuver,
but in this case you have to take into account the necessary frequency
of maneuvers

The calculations Erik is asking for are not simple and I'm not sure
there is a simple scaling with masses and orbital parameters, because it
would also depend on the nature of the perturbations making station
keeping more (or less) necessary from case to case.

| e-folding time:

in the context he is using, it is the growth time of an instability,
following a simple linear equation like dy/dt = gamma y, where the
e-folding time is 1/gamma

--
cu,
Bruce

drift wave turbulence: http://www.rzg.mpg.de/~bds/

In article ,
Edward Green wrote:
For the benefit of groundlings, would you mind explicating exactly
what "station keeping" and delta v means in this context?

Station keeping (or stationkeeping) is maintaining a particular orbit
despite perturbations which tend to mess it up. For example,
geostationary comsats burn fuel regularly to maintain orbital positions
despite disturbances from (mostly) the Moon and the Sun.

Delta-v is velocity change -- it's what you get from a rocket burn. The
propulsion requirements for most kinds of space maneuvers, including
stationkeeping, are conveniently expressed as how much velocity change
they require. Given a particular spacecraft mass and rocket technology,
this translates directly into fuel requirements. For example, a comsat
which must not drift more than half a degree north or south from GSO will
need a rocket burn of about 50m/s about once a year, and with rockets
burning hydrazine as a monopropellant, that will require about 25kg of
hydrazine per ton of spacecraft.

"The time required for the amplitude of an oscillation to increase or
decrease by a factor of e."
Hmm... further search reveals that this is a much-beloved term of art
in orbital mechanics, but doesn't immediately clarify.

Many of the perturbations of interest in some stationkeeping problems show
up as gradually increasing oscillations around the desired position. Some
modest amplitude will typically be tolerable, and clearly the more quickly
the oscillations grow, the more often maneuvers will be needed to keep
them from growing too far. But there are other variables in the problem,
I fear; *how big* those corrections have to be is a separate issue.
--
MOST launched 30 June; science observations running | Henry Spencer
since Oct; first surprises seen; papers pending. |

Erik Max Francis wrote in message ...

I'm looking for general ways of calculating estimates of stationkeeping
deltavee requirements (e.g., m/(s y)) for satellites/stations in various
orbits (low, medium, high, and apostationary if relevnat) around
arbitrary bodies, and at each of the five Lagrange points between each
(sensible) combination of bodies.

Obviously this is something of a tall order, but for my purposes the
estimates need not be exact; within an order of magnitude or so is good
enough. I managed to find e-folding instability times for a few
specific situations, but obviously 1. that doesn't answer the question
generally and 2. there's more to stationkeeping than just pure e-folding
time; there's also perturbations for nearby objects.

Could someone suggest a relatively simple algorithm to compute such
things to some reasonable degree of accuracy for each of the scenarios
described above? Thanks.

Since Bruce Scott and Henry Spencer were kind enough to translate your
request, I've thought about your interesting question in the
interstices of a day.

You have a very complicated problem. ;-)

I'd restate things as follows: A body is in free fall in trajectory
governed by the gravitation fields of one or more large objects. This
is the intended trajectory. The body is perturbed by the
gravitational fields of objects not included in defining the intended
trajectory, and other unknowns. The body is maintained in its
intended trajctory by the actions of a control system running a
control algorithm and consuming a resource (deltavee). You want an
estimate of resource consumption.

(This puts the Lagrange points and the orbits on the same page).

In general, four considerations control resource consumption:

* spectrum of the perturbations

* goodness of the algorithm

* tolerance to deviations

* shape of the potential

By spectrum we consider if the perturbations are modeled by white
noise or the known influence of identifiable bodies. The spectrum is a
given, as is the potential. By goodness we consider our cleverness in
adopting a control scheme to the remaining factors in achieving our
objectives with maximum economy. We must also take into account the
rate, lag and accuracy of our navigational fixes.

The local environment (potential shape), the spectrum of perturbations
and our tolerance of deviation all interact to determine the optimal
control scheme; there is no one-size-fits-all. For example ...

If a body in flat space is subject to white noise impulses it will
execute a random walk. It is reasonable to expect that the resource
requirement will decrease the greater our tolerance. But if the body
is at a point of unstable equilibrium greater tolerance -- in
ignorance of the shape of the potential -- may actually lead to
greater resource expenditu it's cheaper to control small tickles at
the hilltop than to allow them to grow as the ball rolls down. We may
also take advantage of known behavior following perturbation
(oscillations) in minimizing expenditure; the way a hobo falling off a
freight waits for it to come back his way instead of running after the
train! But in another case the optimum "control algorithm" is
particularly simple: if we are subject to a known push which we are
required to deal with in real time (no waiting for the train to come
back), deltavee is simply equal to the deltavee supplied, no question.

Any problem is subject to complications, but in yours, I'd say they
are not even first order, they are zero order: on the order of the
size of the problem itself. You will simply have no choice but to
consider a variety of special cases: there is no useful general case.

Erik Max Francis wrote:

I'm looking for general ways of calculating estimates of stationkeeping
deltavee requirements (e.g., m/(s y)) for satellites/stations in various
orbits (low, medium, high, and apostationary if relevnat) around
arbitrary bodies, and at each of the five Lagrange points between each
(sensible) combination of bodies.

Obviously this is something of a tall order, but for my purposes the
estimates need not be exact; within an order of magnitude or so is good
enough. I managed to find e-folding instability times for a few
specific situations, but obviously 1. that doesn't answer the question
generally and 2. there's more to stationkeeping than just pure e-folding
time; there's also perturbations for nearby objects.

Could someone suggest a relatively simple algorithm to compute such
things to some reasonable degree of accuracy for each of the scenarios
described above? Thanks.

Begin with "three body problem" on google, 13,900 hits.