Relativistic Rocket: The rest of the story...

[Janus]

Many of you have probably heard that if we had a rocket that accelerated at 1g, we could reach far off places of the galaxy in reasonably short times as measured by ship time. However, as true as this is, it is only a part of the story.

 

What I’m going to talk about here is what it would actually take in terms of fuel for a ship to be able to maintain such a constant acceleration in the first place.

 

Now, in order for you to see those time benefits mentioned in the first paragraph we need to be able to travel at relativistic speed, or speeds that are a good fraction of the speed of light. To calculate how much fuel we need we can use one of the relativistic rocket equations, namely:

 

[math] v = c \tanh \left ( \frac{v_e}{c}\ln (MR) \right)[/math]

 

Where

v is our final velocity

Ve is the exhaust velocity

MR is the mass ratio of fully fueled ship to un-fueled ship

 

Using this equation we can plot our final velocity against the fuel needed to reach that velocity as long as we know the exhaust velocity of our rocket.

 

To give you an idea, here’s an example:

 

Most the most common rocket in use today is the chemical rocket. Modern chemical rockets typically have exhaust velocities in the order of 4500 m/s. The following chart shows the relationship between velocity and mass ratio for such a rocket:

 

 

As can be seen, the mass ratio becomes impossibly high before we reach even a small fraction of the speed of light, too small of a fraction for relativistic effect to show. Obviously chemical rockets are out for exploring the cosmos.

 

So what else do we have? One of the first attempts to go beyond the chemical rocket was call NERVA (Nuclear Engine for Rocket Vehicle Application), but it was somewhat of a disappointment as it generated exhaust velocities only about twice that of a chemical rocket. Way short of that needed for interstellar trips.

 

So let’s look at three other systems and see how they stack up; Ion, VASIMR (VAriable specific Impulse Magnetohydrodynamic Rocket) and Nuclear pulse ( project Orion). Each of these has their strengths and weaknesses, but we won’t go into that here, we are just going to deal with their theoretical MR vs velocity plots. Typical ion engines can create exhaust velocities of 200,000 m/s, VASIMR, 500,000 m/s, and nuclear pulse, 5,000,000 m/s. All of which put both chemical and NERVA rockets to shame, but are they enough?

Below is the MR/velocity chart comparing these engines:

 

 

Note that the ion rocket would require pretty much all the mass of the visible universe as fuel just to reach 15% of c. The VASIMR does somewhat better with only about the mass of a million suns and nuclear pulse does the best with a MR of about 360,000 to 1.

 

But even to get a 2 to 1 relativistic factor (not even close the hundreds to one we would need for galactic exploration) we need to get up to .866c we would need about 10,000 suns worth of mass, even with nuclear pulse.

 

So far, we are coming up dry.

 

But wait a minute. The main draw back to the above engines is that a lot of the fuel is used to accelerate the fuel needed later. What if we could get around that and collect our fuel as we go along? Enter the Bussard Ramjet, which supposed to do just that. The idea is that space is not completely empty but contains a certain amount of interstellar hydrogen. If we collect this hydrogen as we go along, pinch it together and cause it to undergo fusion, we could use this fusion output to propel our ship along. We carry no fuel, thus have no MR issues to deal with, and should be able to get as close to c as we like.

 

The problem is that the BR has a draw back of its own. As we collect the interstellar hydrogen, we incur drag. The faster we go, the greater this drag becomes. At a certain point, this drag equals the thrust output of our engine and we stop accelerating. This limit is estimated to happen sometime before we reach 10% of c. Again, this is much short of what we need.

 

Even controlled fusion drives, (which, if possible, are a ways down the road.) would only generate exhaust velocities in the range of 10% of c. At this exhaust velocity we it would take a MR of about 300 billion to get up to 99% of c, where we would have a gamma factor of just 7.

 

Unless we learn how to do much better, the Galactic Empire is going to be slow in growing.

[TheFaithfulStone]

What about an anti-matter rocket?

 

TFS

[i just want to watch you do the math.]

[Boerseun]

Good thread, Janus!

 

How about learning how to shape a nuclear explosion, say, via kick-*** magnetic fields to contain the plasma? And then at the very tip of the magnetic field, you *somehow* leave a tiny hole which you can open and close at will, magnetically, of course.

 

So, you'll basically explode a nuclear bomb in a magnetically enclosed space, say, of a hundred cubic meters. And then release the pressure in the direction needed?

 

Essentially the same as nuclear pulse, but more controlled, and you eventually end up using close to 100% of the energy from the explosion.

 

In nuclear pulse, you basically have a big plate protecting your ship from the explosion, which also acts as the thrust plate. But when the bomb goes off, it spews energy spherically, of which only a tiny fraction is used to accelerate your ship. If it would be possible to contain the explosion, you'll have a hell of a lot more bang for your buck, I guess...

[Janus]

What about an anti-matter rocket?

 

TFS

[i just want to watch you do the math.]

 

If (and this is a big if) you could operate a antimatter rocket at 100% efficiency, then you could get as close to c as you like with a fairly small mass-ratio. And by 100% efficiency, I mean total energy conversion and essentially beaming photons out your tail for propulsion.

 

In practice, I don't see us ever attaining that type of efficiency, 50% to 75% percent might be more reasonable. the following chart gives the mass ratios for Antimatter engines operating at these ratings.

 

 

At 75% efficiency, a mass ratio of 21 will get you to 98% of c. This will get you to Alpha Centauri in about 10.5 months ship time. This would make travel to the nearby stars practical.

 

However, there are a couple of downsides to antimatter rockets. The most obvious is that antimatter seems to be a bit in short supply. Any antimatter we use we have to make. And unless we get a lot better at it, we jjust won't have the resources to create it in the amounts we need.

 

The other downside is the elaborate storage system needed to contain antimatter. You need magnetic "bottles" and the power systems needed to maintain them. You know that mass ratio of 21 that will get us to 98% of c that seems so reasonable? Well it might not be so reasonable. It simply may not be possible to develop containment systems that mass 1 kg for every 21 kg of antimatter contained. Heck, we might be limited to containment systems that outmass the antimatter they contain.

 

If this is the case, then this drops our attainable velocity way down. For example, if our containment system plus the rest of the ship masses twice the antimatter it contains, we would be limited to .30 c at 75% efficiency. Even if we could bump our efficiency up to 99%, this only gets us up to .38 c.

 

Antimatter rockets might not be the panacea they seem to be at first.

[Jay-qu]

damm.. So we either come up with another way of containing it or we some how collect it on the way.

 

By the way, how long is it in earth time for the anti-matter ship (the one that gets there in 10.5 months ship time) to get to alpha-centauri? Im guessing in the order of 5 years..

[Janus]

Good thread, Janus!

 

How about learning how to shape a nuclear explosion, say, via kick-*** magnetic fields to contain the plasma? And then at the very tip of the magnetic field, you *somehow* leave a tiny hole which you can open and close at will, magnetically, of course.

 

So, you'll basically explode a nuclear bomb in a magnetically enclosed space, say, of a hundred cubic meters. And then release the pressure in the direction needed?

 

Essentially the same as nuclear pulse, but more controlled, and you eventually end up using close to 100% of the energy from the explosion.

 

In nuclear pulse, you basically have a big plate protecting your ship from the explosion, which also acts as the thrust plate. But when the bomb goes off, it spews energy spherically, of which only a tiny fraction is used to accelerate your ship. If it would be possible to contain the explosion, you'll have a hell of a lot more bang for your buck, I guess...

 

Such a system might be able to reach exhaust velocities close to the controlled fusion drive I mentioned. But there are draw backs. to maintain that magnetically enclosed space will require powerfull magenets and power sources, which add to the mass of the ship greatly. this means that a greater percentage of your ships mass becomes engine, at the sacrifice of payload.

[Janus]

 

By the way, how long is it in earth time for the anti-matter ship (the one that gets there in 10.5 months ship time) to get to alpha-centauri? Im guessing in the order of 5 years..

 

4.3 ly at .98c = ~ 4.4 years.

[Jay-qu]

but thats not including accel and decceleration.. what kind of maths do you need to account for relativistic time dilation when there is a changing velocity? Also does the acceleration add any non-symetric effects to the calculation?

[CraigD]

The main draw back to the above engines is that a lot of the fuel is used to accelerate the fuel needed later. What if we could get around that and collect our fuel as we go along?

Taken a step further, what if we don’t don’t carry either fuel or reaction mass on the ship, but rather supply the ship with a continuous supply of both from one or more very large (relative to the ship) facilities remaining near the Sun?

 

Possibly the best known design for such a spacecraft is the Prometheus, from Robert Forward’s 1990 novel “Rocheworld”. This system used a 1.5*10^15 W laser to accelerate a several-thousand-ton crew module to about .2 c. using light pressure. Due to the majority of its mass being in a large outer light sail that is released to act as a reflector to decelerate the ship for the last phase of its trip, it decelerates faster than it accelerates - 0.1 g vs. 0.01 g – resulting in a 6 LY trip (to Barnard’s Star) taking about 40 years, about half of them under acceleration. These details, though well thought out, are, of course, fictional.

 

A very similar, though much miniaturized ship, appears in Charles Stross’s 2005 novel Accelerando.

 

“Beamrider” ships need not be limited to lightsails like Forward’s Prometheus. A design that streams directs a stream of ionized or atomic matter at a ship instead of or in addition to light energy, could potentially provide similar or better performance using the projected matter as reaction mass, for less energy. Specific impulse, and, equivalently, exhaust velocity, is of little significance if the ship being accelerated doesn’t carry its fuel or reaction mass.

 

In addition to the obvious challenge of building large lasers/masers and/or particle stream generators in space, “beamrider” spacecraft systems pose the additional challenge of aiming the beam accurately enough to hit a ship light-years distant, a challenge compounded by the inability of the beaming station’s inability to know if its hitting or missing the ship. Just projecting a beam in a sufficiently constant direction that the ship could maintain its position with the beam is a daunting challenge.

[CraigD]

but thats not including accel and decceleration.. what kind of maths do you need to account for relativistic time dilation when there is a changing velocity?

A consequence of Special Relativity’s Equivalence Principle is that, so long as you use the ships clock, trip duration can be calculated using Newton’s original, simple equations for motion under constant acceleration. So long as the maximum velocity doesn’t reach a large fraction of the speed of light – which, given the engineering difficulty of protecting a ship moving at such a velocity from collision with the roughly 1 hydrogen atom per cm^3 of interstellar space, is likely to be a necessary condition for interstellar spaceflight – the difference between ship’s clock and Earth’s time is small enough to be ignored for estimating purposes.

 

So, for example, a 10^17 m (a bit more than 10 light year) trip, including acceleration and deceleration at 1 m/s/s, and limited to 6*10^7 m/s (about .2 c) would take

time_total = time_accelerating + time_coasting + time_accelerating

time_accelerating = velocity_maximum / acceleration = (6*10^7 m/s)/(1 m/s/s) = 6*10^7 s (slightly less than 2 years)

distance_accelerating = 0.5 * acceleration * time_accelerating^2 = 0.5 * (1 m/s/s) * (6*10^7 s)^2 = 1.8*10^15 m

time_coasting = (distance_total – 2 * distance_accelerating) / velocity_maximum = (10^17 m – 3.6*10^15 m) / (6*10^7 m/s) = about 1.6*10^9 (about 51 years)

for time_total = about 54.5 years

 

If the ship had accelerated instantly to 6*10^7 m/s, the time dilation factor (lambda) would be (1-.2^2)^.5 = about 0.98, so the trip, as observed from Earth, takes less than 55.84 years.

[TheFaithfulStone]

So yeah, I think somebody better get on that wormhole thing.

 

:)

 

TFS

[IDMclean]

It's only practical, with one major facet. We have never observed a worm hole of any size that would be usuable by us, for the purposes we need.

 

I have often wondered about Quantum Teleportation.

 

Though wormholes are the choice of Interstellar travel for my science fiction. I have recently been contemplating the absurdity of slip drive. Imagine that you could form a bubble of EM field about your ship, such that your interstellar friction (gravitational drag) was zero. So you would be able to switch it on and instantly jump to c, relative to everything else. Sounds major Sci-fi, but hey.

[Janus]

but thats not including accel and decceleration.. what kind of maths do you need to account for relativistic time dilation when there is a changing velocity? Also does the acceleration add any non-symetric effects to the calculation?

 

True. In reality if you wanted to get to Alpha C in 10 months accelerating halfway there and decelerating the rest of the way, you'd have to pull about 8.5 gs the whole time.

In this case, the Earth time would work out to about 4.52 years.

 

You would reach a top speed of 0.989368692c (this by the way blows our MR of 21 out of the water.)

 

The equations used to figure this out are:

 

[math]d=\frac{c^2}{a}( \cosh \left ( \frac{at}{c} \right )-1)[/math]

 

[math]v=c \tanh \left ( \frac{at}{c} \right )[/math]

 

And

 

[math]t_1 = \frac{c}{a} \sinh \left ( \frac{at_0}{c} \right )[/math]

(where T1 is Earth time and t0 is ship time.

 

Now it might seem odd that it still only takes a little over 4 and a half years by Earth time. This is because that measured from Earth, the acceleration flattens out as the ship nears c and the ship spends most of its time at near C speeds.

The following chart shows the ship velocity for the first half of the trip as measured from Earth.

 

[Jay-qu]

but the acceleration wouldnt be constant.. as more mass is ejected, there is less to push.

 

Thanks craig, but I was talking about at relativistic speeds, where newtonian mechanics breaks down.

[TheFaithfulStone]

So Janus, is there anything we can do, or is the idea of interstellar travel at all as stupid as the idea of warp drive?

 

TFS

[TheBigDog]

It will require patients. You will need a ship that can hole generations of people. You may begin the journey, but your children or grandchildren or great-grandchildren will complete it. It is inevitable that this will take place. And very likely that the first ships lauched to interstallar destinations will arrive AFTER ships lauched later that may have superior technology. But that is the nature of choosing a time to do anything that will take a long time to complete. If it is a 100 year journey to the nearest habitable solar system then it would be five generations onboard a starship to arrive at the destination.

 

Bill

[Janus]

but the acceleration wouldnt be constant.. as more mass is ejected, there is less to push.

 

Generally, you would want to throttle down your engine to maintain a constant acceleration. It might get mighty uncomfortable for the passengers if the acceleration continued to increase as the voyage progressed.