Not so fast! First of all, I want to say that there was no attempt at obfuscation. If my post didn't make sense, that is only because I am a lousy explainer. For those that have had their fill of orbital mechanics, skip this post.
What started all of this lifetime, fuel orbital mechanics discussion was that I merely wished to point out that there are two categories of life limiting "things". Mechanical wear out (including batteries and solar cells) and fuel. During the design cycle these are coordinated to be the same expected value. But when the rubber meets the road, sometimes Providence smiles and you get more fuel than expected because you launch on a different rocket than what you had designed for, and you are blessed with more life giving fuel. If you are double lucky and your satellite beats the odds against old age, you have the fuel to support it. I then pointed out that in G* case, the Delta gave a lot more fuel reserves because of the higher injection.
But the concept that there is "a lot of fuel" or that orbit keeping fuel requirement is "small" is just plain wrong. Once upon a time I made a post where I referred to a "plane change" and really meant "inclination change" and ccryder caught me. He made a very good point. The two are not the same. I knew that, but was being sloppy when I referred to it in the wrong way. For example, G* satellites are in 8 planes of 52 degrees inclination. While it is "expensive" to change from one plane to another, it is possible - and it was discussed in this very thread (OK, maybe it was the LOR thread) last summer, I think. This plane change is done without changing the inclination. Changing the inclination is prohibitively expensive. Executive summary: Change inclination - no. Change plane - hopefully not.
I mention this because now I want to talk about the specifics of a in-plane, no inclination change delta-v. An analogy to delta-v can be the distance between two points. Say, in round numbers 3,000 miles from LA to NY. To get from one orbit to another takes so much delta-v, or change in velocity. If I want to get from one orbit to another, I go faster, and I go "up" to the higher orbit., so I need so much delta-v. If I want to go between two points, it takes so many miles. That's the analogy I would like to try and make in the following discussion.
Using a Soyuz injection of 900 km, and assuming a circular orbit, and also a target orbit of 1414 km, again circular, it is really easy to calculate the delta-v it takes to get from the 900 km orbit to the 1414 km orbit. Assuming a hohmann transfer (explained below) that number is 241.7 meters/sec to get from 900 km to 1414 km. The calculation used to obtain this figure is very easy to make, the most difficult math being a square root. I can post the formula, as well as the details of the calculation, if anyone is curious, but the formula can be found on page 164 in Bate, Meuller & White's "Fundamentals of Astrodynamics". No self respecting rocket scientist wouldn't have a copy on his or her desk. You can probably type in "hohmann" into Yahoo! and get a gazillion hits, too. Executive summary: 900 km circular orbit to 1414km circular orbit is 241.7 meters per sec delta-v (assuming a hohmann transfer).
Using the mileage analogy, a hohmann transfer is the shortest distance between two orbits. NY to LA may be 3,000 miles in a straight line, but if you drove, depending on your route, you may drive a lot further. But, with fuel being a scarce resource and with no roads to limit us in space, the hohmann transfer is a pretty good gauge to what is achieved in practice. For those familiar with thermodynamics, the hohmann transfer is the Carnot cycle of orbital mechanics.
CSM quoted 15 to 75 meters/sec per year for orbit keeping in LEO. That's about right. But again, using the mileage analogy, "your results may vary", depending on how tight the requirements are that you are trying to maintain, your orbit (the lower the orbit, the more delta-v due to atmospheric drag, some inclinations have higher requirements, etc.) If you use Maurice's higgidlypiggidly orbit control, you may only require 15 m/s, or even lower. But, orbit keeping can top 100 m/s per year in some cases.
Now then, to get from delta-v (meters per second) to kg of fuel requires two bits of information. The mass of the satellite (how heavy it is) and the specific impulse (Isp) of the thrusters it uses. Two satellites can both require 15 m/s per year for orbit keeping, but one uses only 1 kg of fuel, while the other uses 100 kg. Back to the mileage analogy - Isp is like the mileage on your car. My car gets 50 miles per gallon, and yours gets 10. We both drive from LA to NY, I use 80 gallons of gas, and you use 400. But two satellites can also use the same thrusters, with the same Isp, and require different amounts of fuel because the satellites have different mass (one is heavier). If I put a 4-cylnder engine in a Toyota, it will get a certain mileage, and if I put the same engine in a big 18-wheeler semi, it will get an altogether different mileage. And then Isp will change over life. Does your old Ford get the same mileage it used to? Thought not. The efficiency (Isp) of a thruster can change from 10% to 100% over the useful design life, depending on it's design.
But, all this does little more to support the original point: Fuel is a precious, life limiting resource, and sometimes it runs out before your satellite wears out, and other times, your satellite wears out before your fuel runs out. Good design practice attempts to coordinate the two life-limiting items.
One more thing about Teledesic and satellite in-orbit collisions and conservation of momentum. If two objects collide in space, their respective center of mass will not change from before the collision to after. So, if the two satellites are in the same plane before the collision, it stands that the center of mass of the two objects is also in that plane. That means after the collision all the little bits, when summed together, their center of mass will still be in the plane. So, as a whole, the shrapnel is "in the plane", but some pieces may no longer be "in the plane". If the two objects weren't in the plane before the collision, then the visualization becomes a bit more complicated, but the result the same --the center of mass of the two bodies does not change after the collision. That's "conservation of momentum".
One last thing before I shut up. When a satellite is "de-orbited", its orbit is usually raised higher, not lowered. It would only be put lower if there was sufficient atmosphere to pull it in rather quickly to burn up on re-entry. That's because when you stop maintaining the orbit, eccentricity (due to solar pressure) starts to change the shape of the orbit, but the semi-major axis remains roughly constant. So, if you lowered the orbit, eccentricity would eventually bring the "satellite graveyard" back up to the operational altitude, and the orbits would intersect, opening the possibility of a collision. To explain that with any detail would probably test the patience of most readers. And I think I've shot my wad with this post.
Regards,
Mr A
PS. I have another long-winded post in me about the Strom Thurmond Nationalist Protection act. That'll have to wait until another day, however. This post did me in for now. |
