Relativistic speed of a rocket with constant thrust

[l0st]

Taking Mfb's suggestion, to show the inherent issue with effective interstellar travel, you only need show the implausibility of even the implausibly best approach.

Thus, given the desire to travel L light years in Y years per traveler, with instant acceleration and instant deceleration (and direct conversion of mass to unidirectional photons), the mass ratio needed (to very good approximation as long as Y² << L²) is:

starting mass / ending mass = 4L² / Y²

For 1000 light years in 1 year per traveler, this is already 4 million.

You really got me worried about perspectives of deep space interstellar travel with this one. I guess that bothered my conscience, so this morning I got up with idea, that potentially we could recapture those photons at the starting point, if the whole trip is planned well.

 

[PeterDonis]

All, the economics of space colonization and interstellar travel are off topic for this thread. Please limit discussion to the physics question asked in the OP.

 

[PeterDonis]

I'm writing an article about the curious fact, that we do not find any traces of alien civilisations or have proof of them coming to us.

While this in itself is useful background for the physics discussion in this thread, the article topic itself is out of scope for this thread, and this area of PF.

 

[PeterDonis]

Moderator's note: a number of off topic posts (having to do with the Fermi Paradox, economics of space exploration, etc.) have been deleted.

 

[pervect]

I don't know that there is still interest in the mathematics of a case of a rocket with a variable proper acceleration, but there's an approach I find convenient.

One needs shared definitions of three things: proper time, denoted by ##\tau##, proper acceleration, denoted by a, and rapidity, denoted by w.

Wiki has articles on proper time, proper acceleration, and rapidity. One needs to understand all three terms to follow the argument.

The key features of rapidities is that, unlike velocities, in special relativity rapiditiy add. Thus one can write

$$w(\tau) = \int dw = \int \frac{dw}{d\tau} d\tau$$

This works with rapidites (and does not work with velocities), because in 1-space, 1-time, special relativity, rapidities add linearly, while velocities add according to the "relativistic velocity addition law".

The remaining step is to note that ##\frac{dw}{d\tau} = a/c##, the rate of change of rapidity with respect to proper time is proportional to the proper acceleration. This can be conveniently done by considering the instantaneous frame of the rocket, and noting that in that frame dv=dw/c and ##dt=d\tau##. But dv/dt in the instantaneous rest frame of the rocket is just the proper acceleration of the rocket.

So then we can solve the problem of the non-uniformly accelerating rocket by writing

$$w(\tau) = \int \frac{a(\tau)}{c} d\tau$$

This follows from the chain rule, and the additive nature of rapidity.

As a check, we can consider the case where a is constant, where we get ##w = a\tau##, We use the basic conversions of rapidity to velocity and vica-versa (see the wiki article for a discussion or look for a clearer one). As the equations imply, there is a 1:1 correspondence between velocity and rapidity, if we know the velocity, we can compute the rapidity, and vica-versa. These basic conversion equations are:

$$w = \tanh \frac{v}{c} \quad v = c \, \tanh^{-1} w$$

substituting ##a(\tau) = a## into the integral, we find

$$w = \tanh v/c = a \tau / c$$

which matches such references as http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html for the constant proper acceleration case.

 

[Flisp]

I don't know that there is still interest in the mathematics of a case of a rocket with a variable proper acceleration, but there's an approach I find convenient.

...

which matches such references as http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html for the constant proper acceleration case.

Thank you! It is always helpful if one can double-check the results from differnt approaches.

 

[Alan McIntire]

Convert units to where c =1 (light year/year)
Acceleration in units of ( 1 light year per year per year which works out to about 0.97 Earth g)
distance in light year units.

Then with acceleration g light years per year per year
distance in light years
Speed of rocket as a fraction of c=1

hyperbolic cosine = cosh =( e^x + e^-x)/2
hyperbolic sine = sinh =(e^x -e^-x)/2
hyperbolic tangent= tanh = sinh x/cosh x

If you have a scientific calculator on your computer, there'll be a function for HYP SIN, HYP COS , HYP TAN
Suppose g = 1(.97 Earth g) for rocket time 2 years
then EARTH time will be sinh( gt) = sinh(1*2) =3.626... Earth years will have gone by in 1 'rocket' year.
Velocity of the rocket will be sinh(gt)/cosh(gt) = sinh(1*2)/cosh(1*2)= tanh2 =.964 times the speed of light after accelerating at 1 light year per year per year for 2 years.
DISTANCE traveled in light years will be [ cosh(gt) -1] = cosh2 - 1 = 2.762 light years.

For a second example , suppose you accelerate at a rate of 1.5 light years per year per year for 1.5 rocket years.
Earth time passed will be sinh(1.5*1.5)= sinh 2.25 = 4.69 Earth years
speed of rocket relative to Earth will be tanh 2.25 =.978 c
distance traveled from Earth will be [cosh(2.25) -1] =3.796 light years,

 

[thorpie]

So if we
1 - made our spaceship a large parabolic dish, say 1 km across,
2 - with the dish pointed away from our current star towards our destination star,
3 - with a small atmosphere in the dish,
4 - with a solar receiver on the rear that receives concentrated solar energy sufficient to accelerate at 1 g at the start
5 - somehow made a solar concentrator to send said solar energy to space ship
Then after 1 year the spaceship approaches light speed and relativistic effects start reducing the acceleration - i.e. the apparent energy of the solar light being collected is reduced because the apparent wavelength of the solar light is greater.
If the ship rotates and then the same process is applied as the ship approaches its destination star the ship is slowed and becomes stationary at much the same solar orbit as it started. During acceleration and deceleration could get an atmosphere of say 0.1 atmospheres and could go out and enjoy the open air. Wouldn't life be grand!

 

[Ibix]

@Alan McIntire - you've made the same mistake others, including myself, made on this thread. The OP asks about constant thrust, not constant acceleration.

@thorpie - you are basically describing a solar sail. The problem is that they yield very low accelerations and their performance drops off very rapidly as you move away from the sun. Even if you try to use lasers in solar orbits to drive your sails, there are significant problems with delivering much power to a target as small as a solar sail - both beam-spread and simple accuracy are problematic. Half way to the nearest star is two light years, approximately 10¹³km. Hitting a 1km target at that distance (with a 4 year lag on your targetting data by the time the beam gets there) would be very very difficult, even one that wants to be hit.

 

[Flisp]

I finally finished my article, that started started my questions into this matter. http://florianb.se/the-possibilities-and-impossibilities-of-interstellar-flight/, thank's again Physics Forum for your support.