View Full Version : How to calculate constant thrust trajectories

m74z00219

2009-Feb-01, 06:51 AM

Hi all. I was pondering how to calculate the path length and trajectory time for a craft moving from one circular orbit to another with constant acceleration.

I know Hohmann transfers don't apply because it's constant acceleration and I assume you have to reverse your acceleration to match your objectives velocity (eg, a planet).

Is there a way to calculate path length and trajectory for a fixed acceleration, known initial velocity (velocity of parent planet), and final velocity (velocity to match, that of the destination).

I have know idea how to do this. Someone mentioned something about calculus of variations to me, but they weren't too sure.

Any thoughts involving by hand calculation or computational methods would be greatly appreciated.

Thanks,

m74

frankuitaalst

2009-Feb-01, 12:53 PM

Unfortunately there's no way to do this analytically ...

Even the most elementary 2 body problems can't be solved exactly .

You might first consider describing the constant thrust ... Thrust in the direction of travel , or thrust in the direction of the goal ... or ...?

I think the best way to tackle the problem is to write down the necessary differential equations and try to solve them numerically .

A general descripton of the problem can be found on the Nasa/JPL portal : Basics of space flight.

http://www2.jpl.nasa.gov/basics/

Murphy

2009-Feb-02, 03:24 AM

Well, if you haven't already, you should look into the Equations of motion (http://en.wikipedia.org/wiki/Equations_of_motion).

The Tsiolkovsky rocket equation (http://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation)

And perhaps visit websites like this one http://www.projectrho.com/rocket/, which aims to design realistic Sci-Fi (look up the entries on Spacecraft missions, Delta V, Rocket engines, etc).

Hope that can help.

mugaliens

2009-Feb-02, 08:25 PM

I was pondering how to calculate the path length and trajectory time for a craft moving from one circular orbit to another with constant acceleration.

...

Is there a way to calculate path length and trajectory for a fixed acceleration, known initial velocity (velocity of parent planet), and final velocity (velocity to match, that of the destination).

What may work is a low-thrust transfer (http://en.wikipedia.org/wiki/Hohmann_transfer_orbit#Low-thrust_transfer). This assumes the use of thrusters such an ion engines, which constantly impart energy, and thus, altitute. No deceleration is required. The engine is always firing in line with the satellite's velocity, and since it gradually spirals to it's target altitude, once it reaches that altitude, it simply shuts off. It begins with a circular orbit, and if the engine is shut off at any point, it enters a circular orbit at the altitude at which the engine was stopped. The greater the thrust to weight ratio, the steeper the spiral.

Calculating this is actually fairly simple. While the fuel usage does exist, it's usually minimal to the mass of the satellite. The thrust is constant and so imparts a fairly constant change in momentum, from which velocity, and thus altitude, can be figured. Since the rate of change of momentum is constant, the time it takes is simpe function of that rate and the difference in potential energy between the initial and final altitudes.

Calculating the point of insertion into the higher orbit, on the other hand, is more complicated.

For movement between planets, there is an Interplanetary Transfer Network (http://en.wikipedia.org/wiki/Interplanetary_Transport_Network), which is dynamic, in that it's constantly changing. Interesting reading...

Princeton Press has a pretty good book on it, called Capture Dynamics and Chaotic Motions in Celestial Mechanics: With Applications to the Construction of Low Energy Transfers (http://press.princeton.edu/titles/7687.html). It's a bit heady reading, but, uh...

Wiki's got a good list of orbits (http://en.wikipedia.org/wiki/List_of_orbits), which may help you find what you're looking for.

Good luck!

m74z00219

2009-Feb-03, 06:45 AM

What may work is a low-thrust transfer (http://en.wikipedia.org/wiki/Hohmann_transfer_orbit#Low-thrust_transfer). This assumes the use of thrusters such an ion engines, which constantly impart energy, and thus, altitute. No deceleration is required. The engine is always firing in line with the satellite's velocity, and since it gradually spirals to it's target altitude, once it reaches that altitude, it simply shuts off. It begins with a circular orbit, and if the engine is shut off at any point, it enters a circular orbit at the altitude at which the engine was stopped. The greater the thrust to weight ratio, the steeper the spiral.

That first part is very intriguing (the continuous increase of velocity without need for acceleration in the opposite direction). So using this very simple method you would eventually have the appropriate velocity to rendezvous with any higher orbit (or lower orbit if your accelerating opposite your velocity). This is nice because it's simple and it seems that assuming you have a fixed acceleration, that this would also be the shortest time wise as well. Although it is a bit complicated by the fact acceleration would actually increase if a constant thrust is maintained. I will look at the equations. Thanks.

Calculating this is actually fairly simple. While the fuel usage does exist, it's usually minimal to the mass of the satellite. The thrust is constant and so imparts a fairly constant change in momentum, from which velocity, and thus altitude, can be figured. Since the rate of change of momentum is constant, the time it takes is simpe function of that rate and the difference in potential energy between the initial and final altitudes.

Calculating the point of insertion into the higher orbit, on the other hand, is more complicated.

For movement between planets, there is an Interplanetary Transfer Network (http://en.wikipedia.org/wiki/Interplanetary_Transport_Network), which is dynamic, in that it's constantly changing. Interesting reading...

Princeton Press has a pretty good book on it, called Capture Dynamics and Chaotic Motions in Celestial Mechanics: With Applications to the Construction of Low Energy Transfers (http://press.princeton.edu/titles/7687.html). It's a bit heady reading, but, uh...

Wiki's got a good list of orbits (http://en.wikipedia.org/wiki/List_of_orbits), which may help you find what you're looking for.

Good luck!

Haha, I should have read the second paragraph before writing much of what I said. You know, I've read about the Interplanetary Transfer Network. It's seems that it would be the cheapest in terms of time and fuel use, but is not very useful for time sensitive projects (perhaps science can wait, but what about movement of goods? yes, I'm a bit of a futurist). I really like the idea of cyclers. Ships that follow an orbit between earth and resonant orbits.

http://clowder.net/hop/railroad/railroad.html

Or even Buzz Aldrin's concept of cyclers. Also very interesting.

Thanks,

M74

PS: I'm glad we're passed the argument and back into the physics.

m74z00219

2009-Feb-03, 06:54 AM

Just had a thought. Why would you accelerate parallel to your initial velocity vector rather than perpendicular? Wouldn't it be faster to accelerate perpendicular to your original velocity vector? Also wouldn't it use less fuel?

IsaacKuo

2009-Feb-03, 12:57 PM

Acceleration parallel to your current velocity is always most efficient at increasing or decreasing your orbital energy.

The mathematics are not as simple as mugaliens suggests. The important thing you need to keep in mind is the difference between speed and energy. A particular change in speed corresponds to different changes in energy, depending on your current speed.

You do not need to assume a slow spiral orbit, except in the case of using a low acceleration drive to go from LEO to GEO or vice versa. If you're interested in interplanetary trips, then a slow spiral orbit is never a good idea--unless you're using an EXTREMELY feeble drive like a solar sail. (Today's feeble solar-electric drives are already capable of better trajectories.)

The basic math formula you must keep in mind is that for small delta-v:

energy difference = delta-v * current velocity

This assumes you are thrusting in the direction of your current velocity.

So choose a particular small delta-v step to start your calculations. Using the above formula, you can calculate your new orbital energy. The tough part will be estimating your new altitude at the end of the step...I'm not sure exactly how to do this part. It's easy if you assume an slow spiral--but that's not a good assumption for interplanetary trips. Well, after you figure out your new altitude you can calculate the potential energy of that altitude. Subtract this from your orbital energy, and you get your new velocity.

mugaliens

2009-Feb-03, 08:09 PM

That first part is very intriguing (the continuous increase of velocity without need for acceleration in the opposite direction). So using this very simple method you would eventually have the appropriate velocity to rendezvous with any higher orbit (or lower orbit if your accelerating opposite your velocity). This is nice because it's simple and it seems that assuming you have a fixed acceleration, that this would also be the shortest time wise as well. Although it is a bit complicated by the fact acceleration would actually increase if a constant thrust is maintained. I will look at the equations. Thanks.

It's counterintuitive, but even though the satellite is contantly accelerating in the direction of it's travel, it's velocity actually decreases! What happens is that the thrust results in an ever-increasing altitude. Higher altitude, lower velocity required to maintain circular orbit. It's total energy increases as it climbs - that's the end, net result.

I've read about the Interplanetary Transfer Network. It's seems that it would be the cheapest in terms of time and fuel use, but is not very useful for time sensitive projects (perhaps science can wait, but what about movement of goods? yes, I'm a bit of a futurist).

The interplanetary network includes all (infinite) orbits, both super-efficient as well as the super-fast, although the later have been largely ignored as there's been little requirement to zip around. The primarily constraint is reaction mass, which is very expensive to lift into space.

PS: I'm glad we're passed the argument and back into the physics.

Me, too!

Just had a thought. Why would you accelerate parallel to your initial velocity vector rather than perpendicular? Wouldn't it be faster to accelerate perpendicular to your original velocity vector? Also wouldn't it use less fuel?

No. When employing slow, constant acceleration, accelerating parallel to one's velocity when you're in a circular orbit is the most efficient. Accelerating perpendicular will result in a higher orbit, but only so long as the acceleration continues! The higher orbit will never continue to increase as you're adding no increasing energy into the satellite.

mugaliens

2009-Feb-03, 08:12 PM

The mathematics are not as simple as mugaliens suggests.

...

You do not need to assume a slow spiral orbit, except in the case of using a low acceleration drive to go from LEO to GEO or vice versa. If you're interested in interplanetary trips, then a slow spiral orbit is never a good idea--unless you're using an EXTREMELY feeble drive like a solar sail.

A constant-acceleration ion drive was assumed. A slow spiral is the only option given the "feeble," but extremely efficient ion drive.

m74z00219

2009-Feb-03, 09:04 PM

mugaliens, you must be familiar with the VASIMR drive, right? I'm trying to understand the basics of it to extrapolate what the tech would be like in 100 years for a story i'm writing.

http://en.wikipedia.org/wiki/Variable_specific_impulse_magnetoplasma_rocket

ok, easy math question. The rocket equation can be used to determine the delta v. Thrust or force i= dp/dt. Is is correct to rewrite it the following way?

dp/dt = [dm/dt]*[dv/dt] Would this be a nice simple way to calculate a ship's thrust? Or am I over simplifying?

I guess I'm just trying to determine the theoretical maximum thrust and delta v possible and perhaps the most optimal combination of thrust and delta v.

Thoughts??

Oh, this slow spiral should apply to any ion tech, right? Even the vasimr?

m74z00219

2009-Feb-03, 09:07 PM

Oh wait, dp/dt should be dp/dt = mdv/dt + vdm/dt ....

IsaacKuo

2009-Feb-03, 11:46 PM

A constant-acceleration ion drive was assumed. A slow spiral is the only option given the "feeble," but extremely efficient ion drive.

Not with human tech ion drives. Our current technology ion drives are already capable of much better than slow spiral maneuvering, on an interplanetary scale.

BTW, an ion drive is not constant-acceleration. It's closer to being constant thrust (although, if we assume the use of solar power it's not even that).

Constant thrust and constant acceleration are different things. The latter is more or less a spherical cow with no practical application. The former is a good approximation of many rocket drives. Acceleration varies inversely with mass, while thrust remains constant.

mugaliens

2009-Feb-04, 08:36 PM

mugaliens, you must be familiar with the VASIMR drive, right?

Yes (http://en.wikipedia.org/wiki/Variable_Specific_Impulse_Magnetoplasma_Rocket).

I'm trying to understand the basics of it to extrapolate what the tech would be like in 100 years for a story i'm writing.

I do enjoy a good story!

ok, easy math question. The rocket equation can be used to determine the delta v. Thrust or force i= dp/dt. Is is correct to rewrite it the following way?

dp/dt = [dm/dt]*[dv/dt] Would this be a nice simple way to calculate a ship's thrust? Or am I over simplifying?

Tsiolkovsky's rocket equation (http://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation)is best written as vdelta=ve*ln(m0/m1), where m represents the initial total mass (0) and the final total mass (1), v is the effective exhaust velocity such that ve=Isp*g0.

Between these two equations you have your mass and your acceleration, from which you can derive the ship's thrust (F) using F=ma.

Let me know if you need any additional help. The Tsiolkovsky link contains additional equations for energy and power.

...this slow spiral should apply to any ion tech, right? Even the vasimr?

It'll apply to any rocket capable of long-term, sustained thrust. Naturally, for something with a high thrust, the rocket will only sustain a fraction of the spiral before it will have exceeded escape velocity, at which point the spiral trangentially transfers to a hyperbolic departure.

m74z00219

2009-Feb-04, 10:02 PM

Thanks for the clarification Isaac. I did figure that acceleration would increase with decreasing mass, if the assumption is that the thrust is constant.

Thanks mugaliens. I do actually have one question regarding that second equation.

v_e = I_sp * g_o

I don't see how that's your acceleration if g_o is a constant. Besides, how does g_o describe the acceleration if it's gravitational acceleration at sea level?

PS

Also, I'd be happy to share with you the first 4 chapters if you're interested. Those first chapters involve androids, noomen (humans uploaded into an artificial but humanoid infrastructure), social commentary, etc. Perhaps you have an email I could forward it to? PMing will probably destroy formatting.

IsaacKuo

2009-Feb-05, 03:53 AM

v_e = I_sp * g_o does not describe your acceleration. It's merely a formula to convert between specific impulse (Isp) and average exhaust velocity (v_e). The two measures are identical, they just use different units.

In order to calculate your acceleration, you use one of Newton's equations:

f = ma

Solving for acceleration, you get:

a = f/m

Typically, your thrust force is constant, so acceleration varies inversely with current mass.

And just to reiterate--neither VASIMR nor any other ion drive will use slow spirals for interplanetary travel. None of them. They will only use slow spirals for maneuvers near a planet (if at all).

If you're wondering about technology for 100 years from now, the big question is how much space infrastructure there is. If we have any sort of refueling capability in Earth orbit, VASIMR is actually pretty slow in comparison to refueled chemical or thermal rockets. However, if you assume that there's very little development of space technology in the next century, then VASIMR can be slightly faster than thermal rockets (but at much greater expense).

wolfekeeper

2009-Apr-29, 04:33 PM

Spirals are inefficent.

The main problem is Oberth effect: for any delta-v you gain a particular speed, but the increase in kinetic energy is non linear because it's a square law on speed- you get more energy from the same delta-v at high speed than low.

The upshot is that you should do a 'burn' (for want of a better term with ion drives) near to the planet, at high speed.

That means that spiralling out in ever increasing orbits is a bad idea- the orbital speed is decreasing the whole time.

The optimal way to use continuous thrusters like ion drives isn't known AFAIK, but elliptical orbits are somewhat more efficient.

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