NASASpaceFlight.com Forum
General Discussion => Q&A Section => Topic started by: Archibald on 06/05/2013 03:29 pm
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I have difficulties grasping how suborbital trajectories actually work.
Grondilu gently gave me some details in this post
http://forum.nasaspaceflight.com/index.php?topic=19541.msg1050451#msg1050451
I have hard time understanding why (for example) ICBM and shuttle abort modes (TAL and AOA) have such different flight profiles and trajectories.
Let's suppose a rocket boost to mach 20 - either a shuttle abort or an ICBM. It is followed by a coast phase before re-entry. How many time is available during the coast phase ?
Why is an ICBM topping at 500 km when the shuttle never go past 100 km ?
Most importantly I wonder what trajectory give the longer coast phase - the shuttle "flattened" trajectory, or the ICBM "pointy" one ?
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You are ignoring the flight path angle and time while powered. Each has major effect. That is why comparing shuttle to ICBM is not valid.
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Not sure what you're asking, but for one thing, ICBMs don't glide very well, and the abort trajectory to the Azores or wherever isn't the one you'd use if you'd planned the hop from the beginning, as with a missile launch. And shuttle aborts can be from anywhere, so you can only figure coast phase if you specify the exact time of the abort.
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ICBMs take the high apogee route because they can and it's the most energy efficient at maximum ranges. Manned vehicles cannot use it because reentry angle would be too steep and their destination orbits are usually lower anyway.
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A weapon is generally denser and can take more gees than a crewed vehicle.
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I have difficulties grasping how suborbital trajectories actually work.
Grondilu gently gave me some details in this post
http://forum.nasaspaceflight.com/index.php?topic=19541.msg1050451#msg1050451
I have hard time understanding why (for example) ICBM and shuttle abort modes (TAL and AOA) have such different flight profiles and trajectories.
Let's suppose a rocket boost to mach 20 - either a shuttle abort or an ICBM. It is followed by a coast phase before re-entry. How many time is available during the coast phase ?
Why is an ICBM topping at 500 km when the shuttle never go past 100 km ?
Most importantly I wonder what trajectory give the longer coast phase - the shuttle "flattened" trajectory, or the ICBM "pointy" one ?
The first thing to note is, the difference between "suborbital" and "orbital" is a pragmatic one. A suborbital trajectory is the same thing as an orbital one--as long as the orbiting object is above the ground and sufficiently clear of the atmosphere so drag is negligible. A suborbital trajectory is just an orbit where part of its path (usually most of it) is under the ground. But until it hits the atmosphere and then the ground, a craft on a suborbital trajectory is orbiting just the same as one whose orbit, closed or open, happens not to ever collide.
Indeed, everything is orbiting, all the time. You and are orbiting right now. The difference is, objects on Earth, or even in its atmosphere, are being accelerated into a different orbit constantly. A suborbital trajectory means that the object will enjoy free fall and thus its orbital parameters will remain unaltered--briefly. Then it will run into the same solid body that is continually changing our orbital parameters in the experience we call "having weight."
Now then--a suborbital vehicle is generally chosen because it is not important for the object in question to do anything but get to its destination elsewhere on Earth, and since rocketry is rather difficult and expensive, to get there with the minimum cost of reaction mass. Therefore what we want is the minimum energy orbit; there are in fact an infinite number of elliptical orbits that connect two points on the Earth's surface. (And more, including hyperbolic ones, that qualify except they'd have to tunnel through the Earth somehow; we are talking just about the orbits that avoid colliding with Earth until they get where they are going). Of all the possible orbits that connect two points though, just one has the minimum energy and therefore is going the minimum velocity at the launch point. I've figured out that the orbit which has the second focus of its elliptical path at the point where a straight line connecting the launch point and destination point intersects the major axis (at a right angle) is the minimum-energy orbit. Such an orbit departs the origin/launch point at an angle just under 45 degrees--to be exact, elevation above the horizon 45 degrees minus one quarter the distance, in angular terms, between origin and destination.
If you look at such ellipses, it will become evident how different they are from circular orbital trajectories. If they cover any really significant distance, they rise up to apogee heights considerably higher than any normal launch that aims for orbit--even launches of geosynch satellites usually, I believe, launch first to a low more or less circular orbit and then boost their payload to a very eccentric transfer orbit, with a final circularization burn once GEO altitude is attained. Since Apollo (which also used a low LEO parking orbit before translunar injection) all manned flights have been to LEO, no higher than 500 km altitude.
Say we wanted to launch a suborbital craft to just barely reach the altitude of 500 km with minimal velocity, and let's ignore air drag and assume we launch it instantaneously into its trajectory, as if we fired it out of a cannon. I figure the fraction of Earth's circumference it would cover before crashing down again would be just 19.85 degrees, which is 1191 nautical miles range. The semimajor axis of the complete ellipse would be 3689.3 km, so the orbital energy would be -54 million joules/kg. At the Earth's surface the gravitational potential energy is -62.5 million, so the kinetic energy there would be 8.535 million and the speed therefore 4132 m/sec, which is a whole lot lower than LEO orbital speeds. Reaching apogee at 500 km up, the kinetic energy margin is reduced to just 4 megajoules so it is then orbiting, tangent to Earth's surface and parallel to a circular orbit, at just 2828.42 m/sec. Whereas a satellite in circular orbit there would be going in the same direction at 7612.61 m/sec, a nearly 5 km/sec difference!
This should illustrate how it is that although suborbital minimum energy paths can easily reach much greater altitudes than we generally aim for in low Earth orbits, the latter are nevertheless more energetic, and therefore require more delta-V to reach. If our purpose in launching a ballistic missile were to reach a target 90 degrees away on the surface of the Earth, its minimum energy track would rise to about 1320 km altitude, getting into the inner boundary of the inner Van Allen belt (depending on latitude; at high latitudes the belt is higher and above IIRC 60 degrees or so it doesn't exist at all). Such a projectile would have to be fired at 6051.6 m/sec from a surface cannon. By the time it rose to 500 km altitude, passing our reference satellite there, it would be going somewhat slower--but note that a good part of its velocity is radial, outward climbing toward its high apogee. If we aim to go beyond 90 degrees, the apogee of the minimal energy orbit starts to come down again. With a 160 degree range, the apogee is again 500 km, only now the semimajor axis of the ellipse is nearly equal to Earth's radius and launch velocity is 7875! That's in excess of orbital speed at 500 km, and not by accident nearly the orbital speed at Earth's surface, if that were possible to sustain. Again, the thing is rising at launch because we have to incline it to match the trajectory--but note that the trajectory is now raised just 5 degrees above the horizon--the rule is, 45 degrees minus one quarter the range (in degrees of circumference) you wish to hit. Thus we see that for very short ranges it converges on the familiar artilleryman's rule of 45 degrees for maximum range.
If we wished to hit a target exactly 180 degrees away in fact, we would have to launch it into a flat circular orbit!
Well, we'd have other options--this is for the minimum energy trajectory; by going to higher delta-Vs and aiming higher, we can send it in a very long, slow and high ellipse that will eventually come down from a high angle. (Such as the aborted space probe mentioned in your post on the other thread, for instance).
So you see, the difference between suborbital and orbital paths is one of energy and inclination of launch.
In fact of course since circular orbits (and sustainable, therefore orbital and not suborbital, elliptical ones) must have minimum altitudes above the drag of the atmosphere, there is no way by using this imaginary cannon method to put something in such an orbit in one shot. First we must rise up above the atmosphere, and then circularize the orbit there.
And of course real rockets do not instantly jump to these ideal, simple elliptical trajectories--this is impossible both because we can't achieve instant velocity changes (and coming close to that, as with a cannon, involves crushing accelerations--I'm interested in human space flight so those are ruled out anyway). And because of air drag; we really don't want to be going at Mach 20 plus in the lower atmosphere! ICBM warheads are of course going nearly that fast coming down to be sure, but that's an extreme case.
Real rockets actually need to launch pretty much straight up at first, to get clear of the launch system and then to get into thinner air before building up really high speeds. So the practical difference is a matter of how fast the rocket rotates to start building up tangential, orbit-wise speeds, and of course at what speed it burns out and starts coasting. Suborbital rockets will generally want less tangential speed but have more radial, rising speed--this as it were buys time for the slower angular motion to cover distance. A rocket to orbit doesn't want to attain the same altitudes but does want more tangential speed as soon as it can get it.
Regarding your last, "most important" question about which gives the longer coast phase--well, the launch that was meant to be suborbital, hence lower energy, will have aimed for more climb and less tangential velocity, therefore it will be rising faster and take longer to be checked at apogee, and hence will have more time to fall to the surface. An aborted orbital launch will have invested more in sideways speed and less into climb, and will therefore have less time before it hits thick atmosphere.
Which one covers more distance is a trickier question because of course the latter will be speeding along faster in angular terms. But, if it is given that the cutoff velocity is the same, I'd say that clearly the suborbital one goes farther too, despite its lower tangential velocity, because presumably it was launched in the minimum-energy trajectory to achieve a given distance, so another rocket burning out at the same speed but a differently inclined trajectory must therefore fall short.
This might not always be true if it is not always true a suborbital craft is launched on minimum-energy trajectory. ICBMs for instance are not optimized for individual delivery missions--one has a standard warhead on a standard, nowadays almost certainly solid-fuel rocket, that has a given burning time and thus a given "mission delta-V". Obviously different missiles would be launched from different silos (or submarines) and have to reach a whole range of targets--the ranges are all over the map, while there is just one range, the maximum a given missile can achieve, that uses the minimum-energy trajectory. To hit closer targets, the missiles must be programmed to use up their excess delta-V somehow, either by lobbing themselves up into higher trajectories that will take a longer time to come down to their eventual targets, or by a faster, lower trajectory that is terminated with a second burn (or rather, in the case of a solid rocket, a turn in the final phase of burn) that diverts it toward the target.
Since such missiles are not going on optimum paths, they'd have shorter ranges than they ideally could, and perhaps the low fast trajectories might even have less time above the atmosphere than the aborted orbiter would.
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Page-11 (Page-9 actual) of this document:
http://ota-cdn.fas.org/reports/8410.pdf
Has a nice illustration of a couple of different ICBM trajectories (including reentry angles, apogees, and flight times) while Page-9 (Page-7 actual) shows the difference between a "10,000km" missile flight (12,000km apogee) compared to the "nominal" 250km Shuttle orbit. It also includes the booster burn-out altitude and time for a "nominal" launch. (200km and 180-seconds after launch respectivly)
(Some good stuff here also: http://space.au.af.mil/au-18-2009/index.htm )
The big thing to remember is that ICBMs are designed to do a very different "job" than the Shuttle or any other "space" booster. We can use "converted" ICBMs (stages mostly but some actual "decommisioned" ICBMs, the Russians mostly :) ) but you have to do alot of "reprogramming" and in some cases re-manufacturing to switch from one to the other. ICBMs are designed and built to get the majority of their motor burn(s) out of the way as soon as possible and to "coast" for most of their flight. (Note the "3-minute" burn for the example ICBM above compared to the Shuttle's almost 10 minute long flight) So the majority of ICBMs don't even make it to orbital speed before burn out. (Typically around Mach-20)
Anyway hope those links help some with the questions.
Randy
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Page 9 is a very interesting page, Thanks
One subtle takeaway is ICBM's fly lofted trajectories when they have to burn all the propellant (solids) verses a nominal trajectory when thrust can be terminated (liquids, Minuteman).
Lofted trajectories might seem like a disadvantage, but provide the ability to overfly some defense systems. The Aegis SM-3 comes to mind. It is very hard to reach 1200km when you are limited to a 21 inch missile.
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thank you all folks !
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One subtle takeaway is ICBM's fly lofted trajectories when they have to burn all the propellant (solids) verses a nominal trajectory when thrust can be terminated (liquids, Minuteman).
Very good observation! But also the reason why there's the depressed trajectory too. The trajectory math works out the way that gives two angles if range to target is less than maximum. And it's not even very complex, see attached the relevant equation and an example I just scribbled on paper (did not dare to scan) from book Fundamentals of Astrodynamics (http://www.amazon.com/Fundamentals-Astrodynamics-Dover-Aeronautical-Engineering/dp/0486600610/ref=sr_1_1?s=books&ie=UTF8&qid=1371654020&sr=1-1&keywords=fundamentals+of+astrodynamics). I've recommended this before and will recommend again, costs peanuts and is like the Rocket Propulsion Elements equivalents to orbit math. The book's language provides down-to-earth examples for instance in this very matter;
The fact that there are two trajectories to the target should not surprise you since even very short-range ballistic trajectories exhibit this property. A familiar illustration of this result is the behavior of water discharged from a garden hose. With constant water pressure and nozzle setting, the speed of the water leaving the nozzle is fixed. If a target well within the maximum range of the hose is selected, the target can be hit by a flat or lofted trajectory.
PS Q = 0.9 in the example at 100km burnout altitude translates to about 7.45 km/s velocity, only 400m/s shy of circular orbital speed at that altitude! The value is set by the capability of your launcher.