Reaching light-speed while simulating gravity

[Fyraltari]

So, a spaceship is accelerating at a constant rate of 1 g (9,80665 m·s^-2) whatever its current speed in order to simulate an Earth-like gravity for the benefit of its crew.

How long would it take for it to reach light-speed (minus, say 1 m·s^-1)?

The naive answer would be to divide that speed by the acceleration to get sligthly less than a year (0.969 year, I'm not converting that in months). But that does not account for relativity, does anyone know what the right equation would be like?

[HandofShadows]

Simple answer is that the ship will never reach light speed. How fast the ship can go will be determined by how much thrust it puts out. When getting close to the speed of light it will start to take more energy to accelerate. At some point well before you hit .99% of light the ship will not be producing enough thrust to continue to accelerate.

[Fyraltari]

That's why I said "whatever its current speed". I know that's not possible but what if we overlook that?

[Tvtyrant]

So, a spaceship is accelerating at a constant rate of 1 g (9,80665 m·s^-2) whatever its current speed in order to simulate an Earth-like gravity for the benefit of its crew.

How long would it take for it to reach light-speed (minus, say 1 m·s^-1)?

The naive answer would be to divide that speed by the acceleration to get sligthly less than a year (0.969 year, I'm not converting that in months). But that does not account for relativity, does anyone know what the right equation would be like?

Pretty sure the answer is "infinite time for an observer." I have no idea for the crew itself.

[Anymage]

A bit of googling gives me this. Particularly from the table they include, after one year an observer on earth will see you going at ~77% c while after two years they'll see you going at ~97% c.

So the simple answer of "they'll never really hit c" is accurate, but they'll still get really close pretty early on.

[Fyraltari]

A bit of googling gives me this. Particularly from the table they include, after one year an observer on earth will see you going at ~77% c while after two years they'll see you going at ~97% c.

So the simple answer of "they'll never really hit c" is accurate, but they'll still get really close pretty early on.

Thank you! I got curious about that because in the orignal Planet of the Apes (the 1963 book) the spaceship uses this method except that they accelerate faster than that (it's described as being inconfortable) and it takes them five years to reach light-speed minus epsilon (no actual figures are given). So, I wondered how accurate that was.

[Radar]

Thank you! I got curious about that because in the orignal Planet of the Apes (the 1963 book) the spaceship uses this method except that they accelerate faster than that (it's described as being inconfortable) and it takes them five years to reach light-speed minus epsilon (no actual figures are given). So, I wondered how accurate that was.

It all depends on the epsilon in question. Going for each subsequent 9 after the decimal point is more and more difficult. But as was said, with just 1G acceleration, you can get pretty close in a few years.

The best epsilon we obtained was for electrons and it was 10^(-10), so the speed was 0.9999999999c. The highest ever observed was for the Oh-My-God particle. To be honest we do not know the exact speed, since only the energy of the particle could be detected (it was 51 J by the way - crazy high for a single particle), but if it was a proton (most common thing to find in space radiation) it would have an epsilon of 5*10^(-24).

The crew will also see a very interesting phenomenon in that the distance to their destination will become shorter and shorter when they accelerate. This will still be noticable even when they will not gain any speed for an outside observer, since for anyone outside the time in the spaceship will continue to slow down the more it closes in on the speed of light.

[factotum]

Might be worth reading the novel "Tau Zero" by Poul Anderson, Fyraltari--it deals with this sort of thing quite deeply. It's supposed to be pretty hard science, too, apart from the bit where a spacecraft is "orbiting" the massively dense mass that exists immediately before the Big Bang!

[Fyraltari]

I’ll keep an eye out for it but I’ve got a lot of books to read for my studies this year.

[Lord Torath]

The hypothetical Hydrogen Scoop Fusion Ram Jet was supposed to be able to maintain a 1G acceleration indefinitely using a huge (10,000 km) electromagnetic scoop to collect hydrogen from in front and funnel it into the ramjet. Of course, I read about it in one of those Time-Life books which were cutting edge about 2 years before they were actually published in the 1980s, if I recall correctly, so take that with a grain of salt. Half-way to your destination you were supposed to turn the ship around and re-engage in reverse. I'm not certain how well that would work if you're thrusting into your magnetic scoop, though.

[factotum]

The hypothetical Hydrogen Scoop Fusion Ram Jet was supposed to be able to maintain a 1G acceleration indefinitely using a huge (10,000 km) electromagnetic scoop to collect hydrogen from in front and funnel it into the ramjet. Of course, I read about it in one of those Time-Life books which were cutting edge about 2 years before they were actually published in the 1980s, if I recall correctly, so take that with a grain of salt. Half-way to your destination you were supposed to turn the ship around and re-engage in reverse. I'm not certain how well that would work if you're thrusting into your magnetic scoop, though.

You would have to reverse your engines without reversing the scoop, so just turning round wouldn't work. Anyway, what you're talking about is the so-called Bussard Ramjet (since it was first proposed by Robert W. Bussard in 1960), and the problem with it is that the interstellar hydrogen density has been found to be far lower than it was believed to be in his day, making his design as proposed completely impractical--you wouldn't get enough thrust to overcome the drag of the ramscoop.

[Yora]

I did the calculation and the result is apparently about 5 years.

This means when you accelerate for 5 years at 1G and then decelerate for 5 years at 1G, you will get so close to light speed that in the middle of the journey, time dilation will get absolutely massive. This means no matter which star you want to visit in the observable universe, for the people on board it will always be about 10 years of flight time.

[Craft (Cheese)]

So, a spaceship is accelerating at a constant rate of 1 g (9,80665 m·s^-2) whatever its current speed in order to simulate an Earth-like gravity for the benefit of its crew.

How long would it take for it to reach light-speed (minus, say 1 m·s^-1)?

The naive answer would be to divide that speed by the acceleration to get sligthly less than a year (0.969 year, I'm not converting that in months). But that does not account for relativity, does anyone know what the right equation would be like?

The relativistic formula for this is:

v = c * tanh(a*t/c)

Where v is the velocity of the rocket (measured by an outside observer), c is the speed of light, a is the acceleration, and t is the rocket's proper time. (That is, time as experienced by passengers on the rocket.)

Solving for t gives us:

t = c*atanh(v/c)/a

And substituting in 1 g for a and the speed of light minus 1 meter/second for v, gives us about 9.79 years.

[Telok]

You would have to reverse your engines without reversing the scoop, so just turning round wouldn't work. Anyway, what you're talking about is the so-called Bussard Ramjet (since it was first proposed by Robert W. Bussard in 1960), and the problem with it is that the interstellar hydrogen density has been found to be far lower than it was believed to be in his day, making his design as proposed completely impractical--you wouldn't get enough thrust to overcome the drag of the ramscoop.

Apparently we're in a sort of local bubble of low interstellar hydrogen caused by a supernova a while ago. So his figures may have been good, but locally (for your multi-light years values of 'local') we have something like 1/10th the normal density.

Edit: My memory was pretty good this time. Local bubble is maybe-ish 300 ly across and 1/10th average. en.m.wikipedia.org/wiki/Local_Bubble

[Chronos]

Both the thrust and the drag of a ramscoop would be proportional to the interstellar hydrogen density, so you can make it work at any density. Lower densities will just result in lower accelerations (and you're not going to be able to get anywhere near 1 g at any sort of realistic numbers). The problem is that the drag also scales with the speed, while thrust does not, which ends up giving you a maximum speed (assuming you're using the hydrogen for fusion) of 12% of lightspeed.

But assume, as the OP does, that we have some ludicrously over-fueled craft capable of sustaining 1 g for a very long time. Relativity makes most of the calculations difficult, but one calculation that's not difficult is the "proper speed", or the distance as measured relative to the stars, divided by the time as measured by your crew (in the same units). Proper speed has no upper bound, and the math for it works out to be the same as for the Newtonian case. In other words, if you have a star that's such-and-such distance away, and you want to know how much your crew will age on the way there, you can ignore relativity and still get the right answer.

[factotum]

The real problem I have with the Bussard ramjet concept is that it assumes that you can somehow get regular atomic hydrogen to fuse into helium, and that's actually really, really hard. Even at the temperatures and pressures at the Sun's core it takes millions of years for any particular hydrogen fusion event to happen--which is really good for a star, because it means it lasts for a really long time, but not so great if you're trying to generate energy and/or thrust from it.