Relativistic speed of a rocket with constant thrust

[Flisp]

I try to calculate the speed curve of a relativistic rocket driven by a 100% efficient engine with constant thrust, when traveling to a distant star. All equations I can find consider constant acceleration, which of course is not working, since the ships mass decreases when the fuel is used, resulting in a low acceleration at the beginning and high acceleration/deceleration towards the end.
I tried using the relativistic momentum equation E² = p² + m² (using c = 1), and p = w * m,
(where E is energy, p momentum, m mass, and w proper speed) and assuming that the entire mass of the used fuel is transferred into momentum so that the energy of the rocket is constant. However, the momentum curve I get is steep at start and flat at the end (without deceleration). The resulting speed curve is steep at start, flattens and gets steeper again. I don't understand why. Burning constant amounts of fuel, I would expect a linear increase of momentum and an exponential increase of speed since less and less mass is accelerated by the constant force from the engine.
Where am I thinking wrong, here?

 

[Dale]

My first guess would be that you are mixing proper time and coordinate time at some point. Can you post your math, being very careful to identify everything?

 

[Ibix]

Google "relativistic rocket". Wikipedia has the maths, and there's more detail at mathpages.

 

[jbriggs444]

100% efficient engine with constant thrust [...] and assuming that the entire mass of the used fuel is transferred into momentum so that the energy of the rocket is constant

So although we have constant thrust, we do not have constant acceleration. The ship's mass varies over time so that its relativistic mass (##\gamma \ m##) remains constant.

This definition of 100% efficiency means that it is not a rocket at all. Instead, it is a race car using the internal energy of its fuel reserves to power the thrust it applies on a stationary highway.

Edit to add: There can be no exhaust stream. Any ejected mass or energy would reduce the remaining relativistic mass.

 

[Flisp]

My first guess would be that you are mixing proper time and coordinate time at some point. Can you post your math, being very careful to identify everything?

At start the speed is so low that there is virtually no difference between the two speeds. I would like to show my math, is there some equation editor?

 

[jbriggs444]

is there some equation editor?

https://www.physicsforums.com/help/latexhelp/

You can find it with Info => Help/How to => LaTeX Primer

 

[Flisp]

Google "relativistic rocket". Wikipedia has the maths, and there's more detail at mathpages.

It's all based on constant acceleration a.

 

[SiennaTheGr8]

Constant proper acceleration!

 

[DrStupid]

assuming that the entire mass of the used fuel is transferred into momentum so that the energy of the rocket is constant

That's impossible.

 

[jbriggs444]

That's impossible.

Impossible for a rocket. Not for a race car. (See post #4)

 

[DrStupid]

Impossible for a rocket. Not for a race car.

I was referring to "assuming that the entire mass of the used fuel is transferred into momentum so that the energy of the rocket is constant"

 

[PeroK]

It's all based on constant acceleration a.

You could also search for "hyperbolic motion relativity", to get an analysis of motion with constant proper acceleratio. E.g.

https://en.wikipedia.org/wiki/Hyperbolic_motion_(relativity)

You could also search for "Rindler Coordinates".

 

[Flisp]

So although we have constant thrust, we do not have constant acceleration. The ship's mass varies over time so that its relativistic mass (##\gamma \ m##) remains constant.

This definition of 100% efficiency means that it is not a rocket at all. Instead, it is a race car using the internal energy of its fuel reserves to power the thrust it applies on a stationary highway.

I'm not sure I understand what you mean. With 100% efficiency I meant that the specific impulse I_sp = 1. For instance some alien anti matter drive. When I get this right I will calculate more realistic I_sp :-)

When I calculate momentum, mass should be constant regardless the frame of referrence accoring to Energy–momentum relation on Wikipedia. The mass of the ship, however, decreases when fuel is burnt.

To calculate the new momentum I use ##E^2 = p^2+ m^2##, where E is energy, p is momentum, and m rest mass.
Some fuel is used and all its matter transformed into momentum while the ships total energy remains constant: From one moment to the next burning a certain amount of fuel I have:
##p_{1}^2 + m_{1}^2 =p_{2}^2 + m_{2}^2##
solved for p₂ I get
##p_{2} = \sqrt(p_{1}^2 + m_{1}^2 - m_{1}^2)##

Using ##w_{2} = \frac {p_{2}} {m_{2}}## I calculate the new proper speed.

 

[Flisp]

You could also search for "hyperbolic motion relativity", to get an analysis of motion with constant proper acceleratio. E.g.

https://en.wikipedia.org/wiki/Hyperbolic_motion_(relativity)

You could also search for "Rindler Coordinates".

I don't want constant acceleration, I want constant thrust.

 

[jbriggs444]

With 100% efficiency I meant that the specific impulse Isp = 1

Specific impulse of 1 means that the fuel can support its own mass against the acceleration of Earth's gravity for 1 second. Surely that is not what you mean.

 

[PeroK]

I don't want constant acceleration, I want constant thrust.

Have you tried using a photon rocket, as an efficient form of propellant?

 

[Flisp]

Have you tried using a photon rocket, as an efficient form of propellant?

It really does not matter what engine real or imagined I use. All I want to know at this point is how to calculate the speed curve using constant thrust.

 

[Flisp]

Specific impulse of 1 means that the fuel can support its own mass against the acceleration of Earth's gravity for 1 second. Surely that is not what you mean.

I meant specific impuls = 1 if using speed of light c = 1 light years / year.

 

[PeroK]

It really does not matter what engine real or imagined I use. All I want to know at this point is how to calculate the speed curve using constant thrust.

Constant in which reference frame?

 

[Flisp]

Constant in which reference frame?

The rockets frame. For an astronaut on the rocket the engine works the same regardless the current speed.

 

[DrStupid]

Some fuel is used and all its matter transformed into momentum while the ships total energy remains constant

No, it doesn't. The rocket loses energy with the exhaust (which consists of radiation in case of a specific momentum of c).

 

[PeroK]

The rockets frame. For an astronaut on the rocket the engine works the same regardless the current speed.

Ejecting photons at a constant rate should give you that. Although, as @Dale mentioned above, perhaps the problem will be relating the ship's proper time to the coordinate time in the initial rest frame.

The notes I have on this just give the speed when the ship reduces to a certain mass. Do you already know how to calculate this?

 

[jbriggs444]

I meant specific impuls = 1 if using speed of light c = 1 light years / year.

The speed of light is 1 light year per year. But what does that have to do with the question at hand?

Are you trying to say that the exhaust velocity is c? If so, that goes counter to the original stipulation that the relativistic mass of the craft never decreases.

 

[Flisp]

No, it doesn't. The rocket loses energy with the exhaust (which consists of radiation in case of a specific momentum of c).

Yes, I know, but that would not change my basicly faulty speed curve that has a typical square root shape when reasoning tells me it should have a tsiolkovsky-rocket-equation shape (flat at start, geting steeper all the time) since the engine pumps the same amount of energy per dt into the rocket, while the rocket gets lighter and lighter. Taking losses into account only changes the final max speed, not the basic shape of the curve.

 

[Orodruin]

I suggest you write down the energy-momentum relations in the instantaneous rest frame before and after ejecting a small amount of matter with a fixed speed u relative to the ship. This should give you the change in velocity in the instantaneous rest frame. You can relate this to the change in velocity via relativistic velocity addition and it should give you a differential equation for the velocity as a function of proper time.

 

[sweet springs]

I try to calculate the speed curve of a relativistic rocket driven by a 100% efficient engine with constant thrust, when traveling to a distant star.

If the rocket has a chemical engine, its mass is decreasing due to exhaust gas of burned fuel.

 

[Flisp]

The speed of light is 1 light year per year. But what does that have to do with the question at hand?

Are you trying to say that the exhaust velocity is c? If so, that goes counter to the original stipulation that the relativistic mass of the craft never decreases.

Did I say mass was constan? Well, it is not. Fuel is burnt. And while all the fuels matter is transformed into pure energy, momentum goes upp, mass goes down. Energy is constant.
Yes exhaust velocity is c.

 

[Orodruin]

If so, that goes counter to the original stipulation that the relativistic mass of the craft never decreases.

I think this was stipulated by you from the OP’s ”100% efficient”. I suspect the OP is not aware of this implication but is really imagining a rocket with an exhaust.

 

[jbriggs444]

Did I say mass was constan?

Yes, you said that [relativistic] mass is constant:

so that the energy of the rocket is constant

Relativistic mass is another name for total energy.

 

[jbriggs444]

I think this was stipulated by you from the OP’s ”100% efficient”. I suspect the OP is not aware of this implication but is really imagining a rocket with an exhaust.

Not so.

 

[Orodruin]

Not so.

I guess I missed the fine print, but I think this is rather a sign of the OP’s misunderstanding. With an exhaust you cannot have constant rocket energy since energy is lost to the exhaust.

 

[Orodruin]

And while all the fuels matter is transformed into pure energy

There is no such thing as ”pure energy”.

Edit: Let me again suggest that you try to write down what I suggested in my first post.

 

[Flisp]

Yes, you said that [relativistic] mass is constant:

Relativistic mass is another name for total energy.

If you read my original question you see that I talk about mass decreasing all the time.

 

[Flisp]

I suggest you write down the energy-momentum relations in the instantaneous rest frame before and after ejecting a small amount of matter with a fixed speed u relative to the ship. This should give you the change in velocity in the instantaneous rest frame. You can relate this to the change in velocity via relativistic velocity addition and it should give you a differential equation for the velocity as a function of proper time.

I don't know how to do that. I did take the energy-momentum relations as you can see from the equations I posted earlier, but I did not get changes in velocity but, proper velocity. What is wrong with the equations I used??

 

[Flisp]

I guess I missed the fine print, but I think this is rather a sign of the OP’s misunderstanding. With an exhaust you cannot have constant rocket energy since energy is lost to the exhaust.

But am I not right, that the loss of energy only affects the maximum final speed, not the basic shape of the curve?