So what would happen if.......

[Not Myself]

I'm just interested in the orbital mechanics that 'force' the craft into a large circle. Is it a greater gravitational force to its starboard side?

If I'm understanding the question correctly, you'd continually be at the "top" (that is, the highest latitude) of an orbit, in this sense - if you cut the engines at any point, the craft would follow an orbit that stayed within 28 N and 28 S.  You'd need to be continually be thrusting towards the north, or your trajectory will immediately start to "fall" to a lower latitude.

[Not Myself]

OK, here's another way to think about it.

If you are travelling in a circle directly about 28 N, you are constantly accelerating around a point on the earth's axis, which is r*sin(28) (argument taken in degrees, make the appropriate conversion for radians) north of the centre of the earth.  Earth's gravity is constantly accelerating you towards the centre of the earth.

So to maintain this "orbit", you need to be constantly applying acceleration which is equal to the vector pointing towards the axis north of the earth's centre, minus the vector pointing to the centre (i.e., the difference between the acceleration you need and the acceleration you get for free from the earth).  This difference needs to point north, but it could also point "inward" towards the earth, or "outward", depending on how fast you are going.

I think the minimum acceleration needed to maintain this trajectory is probably parallel to the surface of the earth, i.e., along a great circle pointing from your present position towards the north pole.  However, I haven't done the calculations to support that yet, so it might be wrong.

[Not Myself]

Some calculations.

It's easiest (for me at least) to use x/y/z coordinates where z=0 is the plane containing the equator.  Then the desired orbit is

x = r*cos(w*t)*cos(alpha)
y = r*sin(w*t)*cos(alpha)
z = r*sin(alpha)

where r is the distance from the centre of the earth, and alpha is the desired constant latitude.  w is related to orbital speed, and t is the time.  The acceleration needed to maintain this is

x'' = -r*w*w*cos(w*t)*cos(alpha)
y'' = -r*w*w*sin(w*t)*cos(alpha)
z'' = 0

The acceleration you get from gravity is

x'' = -r*beta*cos(w*t)*cos(alpha)
y'' = -r*beta*sin(w*t)*cos(alpha)
z'' = -r*beta*sin(alpha)

where beta depends on the earth's mass, the gravitational constant, stuff like that.  What you need is the difference,

x'' = r*(beta-w*w)*cos(w*t)*cos(alpha)
y'' = r*(beta-w*w)*sin(w*t)*cos(alpha)
z'' = r*beta*sin(alpha)

Assuming you have a fixed altitude in mind, then r would be fixed, so the acceleration "north", parallel to the earth's axis, is fixed.  However, the acceleration which is perpendicular to the earth's axis can be controlled by adjusting your speed (which is related to w).  The total acceleration needed is the smallest in magnitude if you choose w to be the square root of beta, and x'' and y'' are zero then.

So my initial intuition was wrong - the minimum acceleration needed, when the speed is just right, is parallel to the earth's axis, not parallel to the surface just beneath you.

[Not Myself]

One more thought - the acceleration needed to hover straight above the north pole would be r*beta.  To keep travelling in a circle just above 28 N, you're going to need at least 46.9% of that amount of acceleration, if you get the speed just right (and more if you don't).

So maintaining this type of orbit is going to be pretty tough.  If you get far away from earth, then beta will get a lot smaller, and it will become easier, although we have to think about what "above 28 N" means in this case - in the same plane the circle of latitude lies in, or "above" from the perspective of a person standing on the surface of the earth.  The latter case is the one that would require more fuel.

[ka9q]

So my initial intuition was wrong - the minimum acceleration needed, when the speed is just right, is parallel to the earth's axis, not parallel to the surface just beneath you.

This makes sense, as you're forcing the orbital plane to continually precess around the earth's axis, i.e., to constantly change the RAAN. To make a gyroscope precess around a given axis, you apply a force parallel to that axis. Think of a toy gyroscope spinning on a stand in gravity.

I seem to recall some serious theoretical proposals to do just this, to make it possible to have geostationary satellites over some latitude other than the equator. Obviously this can't be done with conventional reaction engines, but it might be possible with a solar sail. Probably a really big one.

[Echnaton]

I am thinking of it this way.  Is this more or less right.

A hypothetical launches due east from a point 5 degrees south of the north pole.  It is relatively easy to visualize that the track that stays on a line east from the launch site is not going to stay on the rather tight curve of the 85th parallel, which will be be moving away to the north with the curve of the earth as the rocket climbs.  As rises along the same line it will ultimately end up in a orbit with an 85 degree inclination.

[Count Zero]

I used Google Earth to illustrate:  The line starts at Pad 39A and goes due east on heading 090.



Looking straight down on the line, we can see that it is straight; yet on the globe we can see that it follows the great circle route and goes south of the equator.

Hope this helps.

[Not Myself]

I get the impression the OP is OK with the idea of a great circle, and is aware that the proposed trajectory isn't one.  I interpreted it to mean, what kind of thrust would you have to apply to maintain this circular but non-great-circle trajectory.

[Not Myself]

I seem to recall some serious theoretical proposals to do just this, to make it possible to have geostationary satellites over some latitude other than the equator. Obviously this can't be done with conventional reaction engines, but it might be possible with a solar sail. Probably a really big one.

Yes, it was a bit of an eye-opener for me to realise that it is not some small course correction which is needed, but something quite powerful.  Unless you deviate from the equator only slightly.

Do we have satellites which stay over a fixed line of longitude, but which oscillate back and forth between north and south?  A few of those at almost the same longitude would have the effect of keeping a satellite somewhat close to overhead (from a fixed point on the surface) all of the time.  Just not the same satellite at all times.

[Chew]

Do we have satellites which stay over a fixed line of longitude, but which oscillate back and forth between north and south? 

Some orbits can hover over an area but they make figure 8s; they can't stay over a longitude. See Tundra orbit

[ka9q]

Do we have satellites which stay over a fixed line of longitude, but which oscillate back and forth between north and south?

Yes, but as Chew points out they actually follow a figure-8 pattern around the longitude line. This is usually close enough.

This happens automatically when geostationary satellites run out of stationkeeping fuel. Perturbations from the sun and moon cause the inclination to slowly increase to a maximum and then decrease again. They also tend to migrate east or west to one of several "gravity wells" at specific longitudes due to the earth's lumpy gravity field.

These old satellites are often available cheap to those who can use them because they're pretty much useless for direct broadcasting or other applications with large numbers of mechanically fixed antennas. Broadcasters doing point-to-point feeds usually have antennas that can track, so they can use them. I believe Qualcomm's Omnitracs system makes use of such satellites because their truck antenna patterns are broad enough in elevation to cover the entire orbital path.

Another use of old satellites in inclined orbits is to provide polar coverage. The geostationary arc can't be seen at all from the poles, but a satellite in an inclined orbit may rise above the horizon for a few hours every day. This is how the South Pole Station has gotten its Internet connection for quite a few years now; for a long time they used TDRSS-1, the original one launched on STS-6 I think,  plus a couple of old Air Force experimental satellites. They didn't have 24 hour coverage, nor was it terribly fast, but it was better than nothing. When the Internet was young, before it was walled off by firewalls, I often demonstrated pinging spole.gov to visitors, most of whom had never seen the Internet before.

[ka9q]

See Tundra orbit

I actually chose Sirius because of this; I like to patronize companies with clever new ideas.

Sirius chose the tundra orbit so that its active satellites could be seen at much higher elevation angles over its service area, the continental United States, than conventional communications satellites. Their thinking was that this would better illuminate urban canyons without an expensive terrestrial repeater network filling in lots of holes. It didn't quite work out this way, but it does help.

[Glom]

For information, if you path has constant track, it's called a rhumb line.

[Not Myself]

Some orbits can hover over an area but they make figure 8s; they can't stay over a longitude. See Tundra orbit

Yes, I guess they couldn't stay over the same longitude.  The beginning of the idea, but I failed to think it all the way through 

This happens automatically when geostationary satellites run out of stationkeeping fuel. Perturbations from the sun and moon cause the inclination to slowly increase to a maximum and then decrease again. They also tend to migrate east or west to one of several "gravity wells" at specific longitudes due to the earth's lumpy gravity field.

These old satellites are often available cheap to those who can use them because they're pretty much useless for direct broadcasting or other applications with large numbers of mechanically fixed antennas. Broadcasters doing point-to-point feeds usually have antennas that can track, so they can use them. I believe Qualcomm's Omnitracs system makes use of such satellites because their truck antenna patterns are broad enough in elevation to cover the entire orbital path.

Would these be omnidirectional?  Or is the system for keeping them oriented separate from the system for keeping them in a particular orbit?

Another use of old satellites in inclined orbits is to provide polar coverage. The geostationary arc can't be seen at all from the poles, but a satellite in an inclined orbit may rise above the horizon for a few hours every day. This is how the South Pole Station has gotten its Internet connection for quite a few years now; for a long time they used TDRSS-1, the original one launched on STS-6 I think,  plus a couple of old Air Force experimental satellites. They didn't have 24 hour coverage, nor was it terribly fast, but it was better than nothing. When the Internet was young, before it was walled off by firewalls, I often demonstrated pinging spole.gov to visitors, most of whom had never seen the Internet before.

So you used their limited bandwidth on a ping   Pretty cool though.  But maybe not quite as cool as the other end of the ping.

[Not Myself]

I actually chose Sirius because of this; I like to patronize companies with clever new ideas.

Bugger, I can't remember what satellite radio service I had in my car when I lived in North America (and when I had a car).  I seem to recall that there were two main services - was this one of them?  (Maybe there are more than two now?)