60,000 Miles up: Space elevator by 2035:

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If you work out the numbers, it turns out that launching from LEO takes less energy than launching from the Moon's surface if you want to get to Mars.

A mass launched from Earth LEO distance must be traveling at 11.7 km/sec in order to have enough velocity left over to reach Mars. A mass driver in LEO is already moving at 7.7 km/sec so the mass driver has to add 4 km/sec.

A mass launched from Moon orbit distance must be traveling at 3.4 km/sec to have that same velocity left over. Orbital speed at that distance is 1 km/sec. So a mass driver in orbit at lunar distance has to add 3.4 km/sec.

Now while this is less than needed at LEO, it is not that much less.

This is the velocity that would be have to be left over after escaping the Moon's gravity when using a Moon surface mounted mass driver. When you figure in how fast the object would have to be moving as it left the driver on the Moon, you end up with a result that is greater than the velocity needed for the LEO mass driver. The extra velocity needed to escape the moon tips the scales in favor of the LEO mass driver.

While a LEO mass driver would have to be 73 miles long at 7g to get a payload to Mars, and Moon surface mass driver would have to be 128 miles long to do the same job.

 

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I am always impressed by the extent of your knowledge in these subjects, so hesitate to make following comment:

I think the lightest weight mass launcher should be curved with constant radius of curvature about the gravitational attractions center, not radially away from it. Thus if in LEO about 4000 miles away from Earth's mass center, the launch package would have speed just sitting on the mass launcher of about 4000/1.5 =2667mph and need less added by the mass launcher to get to Mars.

A launch packages accelerating up a radial linear mass launcher would make strong force transvers to the line of the launcher*; so it must be much stiffer that one for which the reaction force is always along the launcher. It would also have that earth orbiting speed but it is perpendicular to the speed given by a radial mass launcher.

I'm too lazy to do even the simple calculation for LMO but bet LMO has much less "free" speed, but that speed is not quite "reusable" as the lauch reaction will send the mass launcher into an elliptical orbit. That could I think be made circular if the next launch was at perigee of the ellipse. If that is false extra energy will be required to make oribit circular again. Also the circular orbit needs to be higher than least LEO as will take some months to "reload" for next launch and don't want the perigee dipping into significant air drag.

A radial structure even "only" 73 miles long in LEO has considerable tension in it due to the gravity gradient. At JHU/APL where I worked for 30 years, we invented and were first to intentionally apply the gravity gradient concept to automatic passive stabilization of satellite attitude - keep the antennae pointing at the mass center of the earth, passively {no energy required).
I. e. the satellite consisted of two parts with an extendable "scissors boom" ** that deployed these say 100 meters apart once in orbit. There were a some high magnetic hysterous loop material used to damp out the oscillations about the stable radial position. [Perhaps some or all of the scissor links were made of that material - I forget it was more than 40 years ago.}

I said "intentionally" as it is very likely more than 500,000 gravity gradient stabilized satellites were launched a few years earlier. There were at least that number of fine wires about a meter long bound together by a moderate vapor pressure "glue" to form a cylinder that fit inside a Scout rocket as I recall. - Idea was the make a very low cost passive communication satellite that radio beams could reflect from for trans pacific communications ranges. The wire length was resonate for max back scatter of the RF frequency to be used. The glue was to release the individual wires with near zero relative velocity, but slowly they would form a earth circulating ring of random oriented RF reflectors.

Well the wires were never seen - most likely they got gravity gradient stabilized with only their ends pointing at earth.

*It must be constantly changing the orbital velocity with force applied by the launcher to the lunch package.

** That "scissor boom" may have been used only once - was soon replaced by long thin-walled tube {phosphor bronze as I recall) that was split / cut on one side its entire length; yet it could be compactly "rolled up" flat around a small drum and it essentially deployed itself - no joints, no motor etc.

By edit: If all the mass launcher length is a constant distance from the center of the earth then as the launch package is picking up speed relative to the launcher it "wants" to be in a higher orbit. Don't resist that. Let it pull that end of the mass launcher farther from the center of the earth.

Alternatively, intentionally have the "leave launcher end" slightly farther from the center of the earth so there is less strong pull on launcher transvers to its arc. It may even have small slow oscillation so that end "A" is first further then end "B" is further from mass center of the Earth.

Perhaps the next launch from the perigee can fix these changes in launcher atitude too (as well as return the elliptical orbit to a circular one.} This question is too complex for me - lets see if Janus 58 has any comment on my post, especially this question.

 

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All my calculations were done assuming a optimal launch vector by the mass driver. So for example, For the LEO driver, the payload leaves heading in the same direction as the driver orbits to take the most advantage of the orbital velocity of the mass driver. IOW, these are the minimum velocities needed under ideal conditions.

 

OK -we agree completely on that primary point. How about the secondary ones:

Because P^2 ~ a^3 and P ~ a/s where P is orbit period, a is semi major axis and s is some average of the speed we have, eliminating P, that a(s^2} is essentially constant; I think a straight launcher ~75 miles long would have the center 25 mile section closer to the Earth's mass center and orbital speed slower than a circular orbit needs at that lower than ends altitude. Likewise the ends of the launcher are going too fast for that higher altitude. They "want" to be lower and the central 25 miles "wants" to be higher. I. e, each "wants" to adjust its altitude to make that nearly constant a(s^2}.

This makes a bending that does lower the ends and raise the center, making their "desires" weaker, as if the structure were intelligent and knew if it could only get to be an arc of a circle with the circle's center at the Earth's mass center then all the bending stress would go away. Hence it cost more to put a long straight launcher into orbit than one slightly curved as it must be stronger.

Am I correct on this so far?

Now when the launch package is nearing the end of the launcher it is traveling much too fast for the orbit the rails of the launcher are holding it in. It is either going to rise up off the launcher (breaking free of the presumably EM drive force accelerating it}, or bend the end of the launcher upward or rotate the entire launcher.

The ends of a straight launcher were already too high for the a(s^2} constant, so now are much too high with a stronger "desire" to move lower. This probably would excite an attitude oscillation in the launcher. It is long so I am very uncertain how that develops or resolves.

In any case it may be good idea for the end the launcher the launch package leaves from to be not only curved down as suggested earlier, I. e. not straight, but excessively curved down so it like the central section "wants" to go higher as the leaving launch package will make it do.

I suspect this problem, optimum for least weight but adequate stiffness and strength can only be found by detailed simulation. That BTW, is the main reason the Indians have orbited Mars on the first try for only 11% of the cost the US spends to do that. - lots of simulations of everything instead of hard ware tests.

 

Ideally any linear accelerator is shaped such that the force on the load at any given time is entirely axial. This is the primary consideration based on design constraints of any such launcher. All that matters is the trajectory at the exit point is correct; it does not need to meet that criteria before exit, so an "orbitally shaped" launcher is not required.

Had the Indians had only simulations they would likely not have made it on their first attempt. They had simulation results - but more importantly they had the indirect experience of over a dozen such attempts by other countries.

 

I completely agree with that (except word "linear" does not imply the launcher is a straight line). "Linear" only implies that the force applied to the launch package is always parallel to the rails currently contacting the launch package.

I also explained why regardless of orientation that is not achieved by a straight mass launcher and noted a vertical (radial) orientation is worse than one horizontal, which in turn is worse than one curved to be an arc of a circle with center at the mass center of Earth, but even that, as I explain does not met the criteria we agree is best. Then I speculated on how to curve it to remove or at least reduce forces not accelerating the launch package along the rails of the launcher, based on recognitions of how those forces perpendicular to the rails arose.

I think the best mass launcher looks sort of like a ski jump - end from which the launch package leaves, turning ever more sharply up, but that upward bend is not why the launch package turns ever more sharply up. Reason for that is the fact that the launch package is being accelerated and moving to a higher orbit, I think, but this is "intuition" not analysis.

My "a(s^2) is essentially constant" analysis concluded that the end should actually, very-counter intuitively, turn down to lower "a" as "s" increases. This implies that the "start end" of the mass launcher is significantly tilted up so the launch end's downward turn does not direct the launch package into the atmosphere as it might if the start end were horizontal.

I hope Janus58 jumps in here soon as I have, as was said in APL's space department, the feeling I am in "deep yogurt" and confused.

 

For a launcher on Earth that might make sense since there is great benefit in leaving the atmosphere quickly.

However, in no case does an orbit around a single body "bend upwards." It always tends towards the body - just to greater or lesser degrees depending on speed, distance etc. So any "ideal" orbital launcher won't have the "ski jump" thing on the end, unless there is some reason to accelerate the load in a direction other than the velocity vector, and assuming the launcher can handle those loads.

 

The tendency would be to curve in the other direction. The ends, moving faster than orbital speed for their altitude will want to climb away, and the center, moving slower will want to sag. It is the basic tidal force effect. The gravitational differential along the length pulls upward on the higher parts and downward on the lower parts.

As for the payload "rising up" off the launcher, since we are in orbit there is no reason we need to place the payload on the "upper" side of the launcher. Just launch from the "underside" and forces will work for you in holding the payload in place. As far as the effect this has on the launcher itself, this is where we come up with an issue that needs to be addressed. Since the tidal forces acting on the launcher work in the opposite manner in which you proposed, the effect of the launcher of any such rotation would be for it to continue until it is is aligned vertically, which would be its natural stable orientation. This would be true whether you started out with a straight or curved launcher.

In fact, the horizontal launcher is only stable in the way a pin balanced on its point is stable. The slightest nudge will cause it to "fall" into the vertical alignment.

There are a couple of ways to combat this. One is gyroscopic; massive flywheels which can be sped up or slowed down to keep things lined up. The other is counter-masses. If you add arms that extend at a right angle to the launcher and put heavy enough masses at the ends, the tidal forces on these arms will be the overriding factor and keep your launcher in the right orientation. The best solution might be a mixture of the two methods. Counter-masses for overall stability and flywheels to nullify any recoil rotation caused by a launch.

The other issue is the effect the launch has on the orbit of the launcher itself. The mass ratio between launcher and payload would be high but each launch does nudge the launcher. A solution to this is to make the launcher two way and alternating your launches between the day lit and night sides of the Earth. This way you still launch in the direction the Earth's orbit, but in opposite direction with respect to the launcher's earth orbit. Of course this also means that when launching in one direction the launcher orbital velocity is aiding the launch and in the other it is retarding it. This means that the "muzzle" velocity would have to be greater in one direction than it is in the other to reach the same destination. Smaller payloads at a higher acceleration could do this while still giving the required adjustment nudge to the launcher. Such launches would be reserved for cargo that is not g-force sensitive.

The engineering of such a structure would be an ambitious undertaking. The loading factors on the counter-mass arms would be huge, and for rigidity I imagine suspension bridge like cables running from counter-mass arms to launcher. Then there is the solar panel array to power it. You have to wonder what level of Earth to Mars traffic would warrant the investment.

It would be something to see however. A 73 mile long launcher would have a ~.5 degree angular size as seen at the horizon to a 22 degree angular size when directly overhead as seen from the surface of the Earth. A ribbon of light that grows as it climbs in the sky and then rapidly shrinks to nothing as it passed into the Earth's shadow.

 

Thanks. I knew I could count on you to pull me out of my "deep yogurt" confusion.

That is the correct way to view it. My "a(s^2) wants to be a constant" probably is correct IF there were three 24 mile long unconnected straight, in-line sections almost touching end-to-end (this alignment due to tiny continuous thrusters that turn off at t=0) and have the same horizontal speed, S. I. e. the central piece's "a" is less than the "a" of the end sections, so to continue in the end-to-end but unconnected orbit the speed of that central section needs to be higher than the S it is, so it can not keep the altitude it has. - Not enough centripetal "force" to resist the pull of gravity, or as your easier to follow analysis shows it moves closer to Earth.

Or in my more complex approach, the central a(S^2) is greater than the essentially constant value of all orbits. So that section's "a" will reduce and with a then smaller circumference to orbit, it will "fly" under and ahead of the end pieces. Where I went wrong, I think, is ignoring the fact the three pieces ARE connected and thus have mutual interaction forces. The central section can just bend (sag as you say) letting there be an away from Earth force applied to it by the two end sections and when that "helping force" is just what will allow the centripetal "force" plus the "helping force" to equal gravity's force on it, the sag is "just right."

Thus I think a 75 mile long thin (very slight but not zero stiffness) metal ribbon can orbit the earth with constant curved shape, * concave side towards space, if up high enough to not have air drag. That ribbon could at launch be rolled up in small diameter roll, tossed out of LEO (say >100 minute orbit period) with very affordable cost and soon become your "A ribbon of light that grows as it climbs in the sky and then rapidly shrinks to nothing as it passed into the Earth's shadow." It might even have some economic justification as indicator of residual gas density - help predict when space junk (and other items) will soon burn up in the air.

Clever! and I thought I was good at "thinking outside the box." I was ~20 years in APL's space department but worked on implantable medical devices and several energy systems with rare exceptions, one being called in on a few "tiger teams" discussing how to help a "sick satellite." All knew I did not know much about space craft but that ignorance let me use my recognized ability to "think outside the box" and ask questions that some times got thinking along a new line going; usually, however, my questions just gave a needed humorous break to the meeting.

I must be getting too old to do that well anymore. - I even discussed gravity gradient for passive radial stabilization of satellites in my posts but did not connect that knowledge to the dynamics of a long horizontal rigid structure. The two weights you mention are over powering that to make "horizontal stability" of the mass launcher possible.

Thanks again.

* The ribbon will "want" to gravity gradient stabilize radially - that must be "over powered" by some rods with weights at their Earth end along the length of the ribbon, much like you suggested could be done with the mass launcher to keep it from "going radial."