A good altitude for Ortillery? Math and astrophysics guys help out

Given:
1) Geosynch is roughly 22,000 km (if I'm not way off base)
2) Given geosynch is stationary wrt a point on the earth
3) Given a lower orbit will be (of necessity) faster? I think
4) Given the lower the orbit, the faster the orbit - so the further
off geosynch, the less time you'll be able to see a given point on the
earth before going out of arc

Can some of the math whizzes calculate an altitude where a ship
wishing to supply ortillery support can see a given point on the earth
(we'll call it the battlefield) for a period of at least an hour -
we'll call this the support-fire window. I figure an hour is a few DS2
turns, and at least 12 SG2 turns. Probably long enough to lay down
enough ortillery to matter.

So what altitude would give us an orbit slow enough to maintain
visibility on a point for an hour before going over the horizon? I
realize topography affects this, so an answer that covers a range of
altitude would be acceptable.

I'm just curious: Are we talking about 22,000 km (geosynch)? 400 km
(Hubble)? etc.

Depending on what this window is, and the ortillery window WOULD have
to coincide with fairly well time-wise with planned attacks or no
support would be available (or at least, it might take a lot longer to
arrive). I think I figured out that a shell travelling mach 10 on
average would take about 110 mins to arrive from geosynch... a little
long for a support request. From 400 miles, the answer is something
like 171 seconds. Now, Mach 10 may not be even a reasonable guestimate
for ballistic entry projectiles - I have not modelled terminal
velocity or acceleration due to gravity to guess how long it would
really take... I just picked Mach 10 out of the ether.

If your ortillery is beam, this might be moot except that you'd want
to reduce the amount of atmosphere you fire through to ameliorate
diffusion and defraction and refraction (bending and scattering
essentially). You'd still want to be in close. But your fire missions
would arrive rapidfast after the administrivia of getting them tasked.

But if you are using missile or beam, travel time is an issue. It can
be quite sizeable.

So.... what altitude would be reasonable and give us at least an hour
as a support-window? If someone had someone is gutsy enough to bash up
two formulas, one for the support window

delta-t(support window)= ? (some f(altitude))

and

t (inbound transit) = ? (some f(altitude, probably launch v and acc))

then we could get some ideas for the relationship between ortillery
ship altitude, the time the round takes (and hence how long the
support fire should take to arrive) and the time the ship is "on-call"
(the duration of the support window caused by the orbit altitude).

This might be instructive for all of us - it'll give the FTers a
feeling for how close to the surface you have to get and how long you
are useful at various heights when doing ortillery support. For DS and
SG players, it'd give some idea of the range or response times one can
expect (we'll still have to think about the delays in comms and
organizational levels but that's not hard) from ortillery and the
length of time ortillery support could be available.

It may be we can evolve some simple rules which let players choose a
lower altitude support mission (with a smaller window of availability)
as a trade off for faster response on the fire missions, or vice
versa - trade delay time for a longer ortillery support mission.

So, orbital mechanics brains.... and math whizzes.... do your
worst.... :)

Thomas Barclay
Software UberMensch
xwave solutions
(613) 831-2018 x 3008