Launching an object into orbit

[guss]

This is obviously a highly complex question, but I'm really interested in it, and I still don't know where to start after looking around a while.

If you had an object that was .019 kilograms, how fast and at what angle would you have to shoot the object at to get it to orbit at 200000 meters, accounting for air resistance and the deduction of gravity at the constantly increasing altitudes? Also, would the mass of the object you launch change the speed or angle at which you had to launch it at?

Any insight on how to solve this would be appreciated. Thanks!

 

[D H]

If you just shoot the object from the surface of the Earth such as from a cannon, one of two things will happen to the object. Give it enough energy and it will go on an escape trajectory. Fail to give it enough energy and it will come back to Earth. In either case it will not go into orbit. Two impulsive applications of force are needed to get something into orbit. You need to "shoot" from the surface of the Earth and then shoot again once the object is at orbital altitude.

Using a cannon is not a good idea. Even if you have a cannon with a very long barrel, the acceleration needed to accomplish the first step is immense. To keep the payload from turning into dust or mashed potatoes the force needs to be applied gradually, over the course of quite a few minutes. With that in mind, you might as well work toward both ends at once (raising the altitude by a few hundred miles and raising the velocity by 17,000 miles per hour.

 

[pallidin]

From: http://en.wikipedia.org/wiki/Space_gun

A space gun is a method of launching an object into outer space using a large gun, or cannon. It provides a method of non-rocket spacelaunch‎.

In the HARP Project a U.S. Navy 16 inch (406 mm) 100 caliber gun (40 m) was used to fire a 180 kilogram slug at 3,600 meters per second, reaching an apogee of 180 kilometers, hence performing a suborbital spaceflight.

However, a space gun has never been successfully used to launch an object into orbit.

 

[MattRob]

Sorry Guss. I'm looking at the N-Prize myself and doubt I'll get it first, but you'll need a rocket. No easy cheats exist to getting something into orbit :P

 

[D H]

However, a space gun has never been successfully used to launch an object into orbit.

The article explains why:

A space gun, by itself, is not capable of placing objects into stable orbit. The laws of gravitation make it impossible to reach a stable orbit without an active payload which performs orbital correction burns to change the shape of its orbit after launch. The orbit is a parabolic orbit, a hyperbolic orbit, or part of an elliptic orbit which ends at the planet's surface at the point of launch or another point. This means that an uncorrected ballistic payload will always strike the planet within its first orbit unless the velocity was so high as to reach or exceed escape velocity.​

 

[MattRob]

Interesting physics learned from Orbiter : One can only raise the apoapsis when at the periapsis, and vice versa. It's impossible to change an object's current point of the orbit by thrusting in any way. It will only change some other part of the orbit.

In other words, in order to make a perfect orbit in a single burn, one must accelerate straight up, and reach orbital velocity at an apoapsis high enough to be safe. Whatever altitude the rocket is at when it finishes the orbital burn, it's periapsis will stay that high unless actively changed.

Just thought I'd add my $0.02 ...

 

[guss]

Haha, thanks guys. I was going to change my argument to have the object just be able to orbit Earth non-sustainably - say 10 times. But then I saw it could only orbit once doing this.

But, surely there has to be some way to do this almost entirely with a space gun, with only minor corrections made during flight. Maybe having it drop of a mass (although this probably wouldn't work due to the laws of gravity, but this could be different?), or maybe a small booster on one side of the spacecraft . Also, couldn't the slight air resistance, although not so slight at the velocity we would be talking about, be enough to change the trajectory of the object so it could make a sustainable orbit, for at least a short period of time?

Also, if anyone wants to take the time, how would you figure out the angle and initial velocity required to launch a .019 kg object into an orbit at 200,000 km, I'm just curious?

Thanks.

 

[MattRob]

Unless the barrel is unimaginably large, you'll be launching mush into space, not cargo :p . Anyways, a small thruster should be enough to get it into a stable orbit once up.
The speed a space engine is capable of is determined by two things: the fuel used, and the ratio of fuel to vehicle. In a vaccum, it doesn't matter how massive the vehicle is, it's vehicle-to-fuel ratio that's important. So a small engine should be more than enough. But, mush issue, the engine would break on launch.

It might be possible to use some sort of re-entry control system to "skip" off the atmosphere to change your orbit, but you'd lose a lot of speed in the process, making the requirements for said cannon much higher, it is far more feasible to stick a small rocket motor on it.
But then you'd also need a guidance system and a small computer so it knows which direction to fire the motor...

 

[guss]

Unless the barrel is unimaginably large, you'll be launching mush into space, not cargo :p . Anyways, a small thruster should be enough to get it into a stable orbit once up.
The speed a space engine is capable of is determined by two things: the fuel used, and the ratio of fuel to vehicle. In a vaccum, it doesn't matter how massive the vehicle is, it's vehicle-to-fuel ratio that's important. So a small engine should be more than enough. But, mush issue, the engine would break on launch.

It might be possible to use some sort of re-entry control system to "skip" off the atmosphere to change your orbit, but you'd lose a lot of speed in the process, making the requirements for said cannon much higher, it is far more feasible to stick a small rocket motor on it.
But then you'd also need a guidance system and a small computer so it knows which direction to fire the motor...

Well, I'm talking more theoretically here, not worrying much about the object breaking. The motor is obviously the best way to do this - at least for a single object.

I suppose one way you could do it is to launch it at the moon in the right trajectory, then curve it back to Earth in the right direction for it to be in a sustainable orbit :P. Speaking of that, how would you go about calculating the trajectory and speed?

 

[DaveC426913]

The article explains why:

A space gun, by itself, is not capable of placing objects into stable orbit. The laws of gravitation make it impossible to reach a stable orbit without an active payload which performs orbital correction burns to change the shape of its orbit after launch. The orbit is a parabolic orbit, a hyperbolic orbit, or part of an elliptic orbit which ends at the planet's surface at the point of launch or another point. This means that an uncorrected ballistic payload will always strike the planet within its first orbit unless the velocity was so high as to reach or exceed escape velocity.​

Never realized this before, but of course, in retrospect it's obvious.

 

[guss]

Still trying to figure out what would be necessary to calculate the speed and trajectory to launch an object into orbit around Earth using the moon as a gravitational slingshot, if it's even possible. Any clues, anyone?

 

[rcgldr]

Wouldn't the fact that the Earth is orbiting the sun allow a single impulse on an object from the Earth's surface allow an orbital path (pretend there's no atmosphere)? For example, if the object is fired in the opposite direction of the Earth's orbit, the reduced speed relative to the sun will result in that object being drawn towards the sun, resulting in the equivalent of a second correction, but I'm not sure as the cycle completes if this all cancels out.

 

[pallidin]

No.
It really boils down to this: Very specific trajectory changes MUST be made after achieving orbit in order for sustainable orbiting to occur.

 

[cjameshuff]

You an use a third body to buy some time, achieving more than one orbit, but the resulting orbit will not be stable in the long term. Fire a payload into a trajectory that chases the moon at apogee, and perigee might be lifted enough to miss the atmosphere, but there will be more close encounters with the moon at apogee leading to major changes in the orbit over time. Fire into an elliptical solar orbit, and you're better off due to the larger distances and periods involved, but the resulting orbit will intersect Earth's orbit and occasionally be perturbed by a close pass.

 

[guss]

You an use a third body to buy some time, achieving more than one orbit, but the resulting orbit will not be stable in the long term. Fire a payload into a trajectory that chases the moon at apogee, and perigee might be lifted enough to miss the atmosphere, but there will be more close encounters with the moon at apogee leading to major changes in the orbit over time. Fire into an elliptical solar orbit, and you're better off due to the larger distances and periods involved, but the resulting orbit will intersect Earth's orbit and occasionally be perturbed by a close pass.

Thanks.

Can anyone actually show the calculations behind this? I'm really interested.

 

[D H]

Thanks.

Can anyone actually show the calculations behind this? I'm really interested.

There are no closed-form solutions to the issue raised by rcgldr in post #12 and cjameshuff in post #14. This is venturing into the realm of the N-body problem, a problem which does not have a general closed form solution.

That said, there are a handful of special cases of the three body problem that do have an exact closed form solution, and we do know how to solve the N-body problem in general using numerical techniques. Countless journal articles and even entire books have been written on the N-body problem. I'll give a few references at the end of this post. Google phrases such as "three body problem", "Lagrange point", "N-body problem", and "horseshoe orbit" to learn more.

We also know how to address the issue of a single fly-by such as the close encounter with the Moon discussed in post #14. This again is the subject of many journal articles and books. NASA has been using such close encounters, aka gravity assists or gravity slingshots, for quite some time to make its planetary probe missions possible. NASA now has five satellites on an escape trajectory from the solar system: Pioneer 10 & 11, Voyager 1 & 2, and New Horizons. None of these spacecraft were launched from Earth with anything close to solar system escape velocity. These vehicles instead attained escape velocity by using gravity assists.


Animated GIFs of a couple of objects with rather interesting orbits:
Near Earth Object J002E3: http://neo.jpl.nasa.gov/j002e3/j002e3d.gif
This object is rather apropos to the topic at hand. It was the upper stage of the Apollo 12 mission launched in 1969.

Near Earth Object 2002 AA29: http://neo.jpl.nasa.gov/2002aa29/2002aa29a.gif
This object is in a 1:1 resonance with the Earth and has a very cool horseshoe orbit.


Some references:
"Porkchop" is the First Menu Item on a Trip to Mars
http://mars.jpl.nasa.gov/spotlight/porkchopAll.html
NASA uses porkchop plots to plan missions to other planets. A linked set of such plots are needed to plan a sequence of encounters such as with the current New Horizons mission.

Slingshots and Space Shots by Bill Casselman
http://www.ams.org/samplings/feature-column/fcarc-slingshot
This featured AMS column discusses gravity assists and also discusses a rather interesting configuration of the three body problem, Burrau's problem, aka the Pythagorean problem.

Victor Szebehely, Burrau's problem of three bodies
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC335596/pdf/pnas00677-0079.pdf
This 1967 PNAS paper discusses Burrau's problem from the perspective of Newtonian mechanics.

M.J. Valtonen, S. Mikkola, S., & H. Pietila, Burrau's three-body problem in the post-Newtonian approximation
http://adsabs.harvard.edu/abs/1995MNRAS.273..751V
This more recent MNRAS paper discusses Burrau's problem from the perspective of general relativity.

The three-body problem by Mauri J. Valtonen and Hannu Karttunen
http://books.google.com/books?id=dvIXkeS17bAC
As I told you above, entire books have been written on this subject. This is one of them.

 

[guss]

Great post, thanks a lot.

 

[D H]

Fire a payload into a trajectory that chases the moon at apogee, and perigee might be lifted enough to miss the atmosphere, but there will be more close encounters with the moon at apogee leading to major changes in the orbit over time.

Or the payload might be ejected from the Earth-Moon system. This is exactly what happened with NASA's STEREO satellites.

The two satellites were launched together on October 26, 2006 into highly elliptical orbits with apogees beyond the Moon's orbit. The initial orbits of the two spacecraft were nearly identical. The two spacecraft swung rather close to the Moon on December 15, 2006. STEREO-A passed closer to the Moon than did STEREO-B and thus received a greater gravity assist than did STEREO-B. The fly-by gave STEREO-A enough energy to eject it from the Earth-Moon system. STEREO-A now orbits the Sun with a period of about 344.5 days.

STEREO-B did not receive quite that much energy from its December 15 fly-by. It instead remained in orbit about the Earth+Moon system, re-encountering the Moon on January 21, 2007. This second fly-by gave it the extra boost needed to escape the Earth-Moon system. STEREO-B now orbits the Sun with a period of about 389 days.

 

[QuantumPion]

NASA now has five satellites on an escape trajectory from the solar system: Pioneer 10 & 11, Voyager 1 & 2, and New Horizons. None of these spacecraft were launched from Earth with anything close to solar system escape velocity. These vehicles instead attained escape velocity by using gravity assists.

Actually I believe New Horizons was indeed launched directly into a solar escape trajectory, although it also received a gravity assist from Jupiter to increase its speed.

 

[D H]

Thanks. I stand corrected; New Horizons did indeed launch directly into a solar system escape trajectory.

 

[mheslep]

I'm curious if a single ballistic shot from inside the troposphere is even possible without liquifying/vaporizing known materials. If escape velocity is 11.2 km/s, what must be the much greater velocity at the end of the gun barrel so that velocity above the troposphere still exceeds 11.2 km/s? 20 something km/s?

 

[D H]

I'm curious if a single ballistic shot from inside the troposphere is even possible without liquifying/vaporizing known materials.

Of course not. That's why I said in post #2 that using a cannon is not a good idea.

 

[mheslep]

Of course not. That's why I said in post #2 that using a cannon is not a good idea.

It's that obvious? Some tungsten low cross section rod couldn't survive? Not that it would have any useful purpose.

 

[guss]

I think it could survive, obviously not the initial acceleration if done too suddenly, but a water-droplet shaped piece of the proper steel or an even tougher osmium could survive it fairly easily, i would think.