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Oddly, science elsewhere doesn't stop whilst I'm holed up trying to do science. What with the Gliese 581 star system being in the news again, I thought I'd pull part of an old post out of the stacks.
Two questions come up every time the discovery of an extrasolar planet hits the headlines: "Is there life on it?" and "How long would it take to get there?" To the first, we respond, "Dunno" (or sometimes "probably not"), and to the second, "That depends upon how fast you can go." Elaborating on the latter reply will be our subject for today.
If you want to get to a star 20 light-years away in a reasonable amount of time, you have to travel at relativistic speeds, which means that (duh) Einstein's Relativity comes into play. This is such a fun problem that the "relativistic rocket" ends up being worked out time and time again. The time measured by a clock on your spaceship will differ from that measured on Earth; I'll use T to denote the former (which we call "proper time") and a lowercase t to denote the latter. The answer involves the hyperbolic trigonometric functions, and in particular the hyperbolic cosine:
This function takes in a number x and spits out the corresponding hyperbolic cosine, which is like the cosine we saw in high school but slightly stranger (it's defined in terms of hyperbolas instead of circles). The proper time — the time measured by clocks and people on the ship — can be found using the inverse cosh. Writing d for the distance traveled, c for the speed of light and a for the acceleration the people in the ship feel, the formula is
What's a good value for a? Well, a comfortable acceleration would be 1 g, what we feel sitting on Earth. Plugging in 9.8 m/s2 for a gives (somebody should double-check my arithmetic) the result that T is 3.6 years.
That's for people on the ship. What about the folks back home? Well, the formula for little t is
which with the same values for d, c and a as before gives t = 20.5 years. Remember, the destination star is 20 light years away (meaning, of course, that light gets there in just 20 years). Accelerating at 1 g for three years, ship time, you pick up so much speed the people back on Earth see you going almost the speed of light!
If you want to slow down so that you stop at Gliese 581 instead of whizzing by, the simplest solution is to turn your relativistic rocket around at the midpoint of your journey and start blasting in the opposite direction. This trip will take just about not quite twice as long, ship time, but because so much of the journey takes place at near light-speed, the people on Earth won't see much of a difference.
The details of building a vessel which can travel this fast for this long and keep everybody alive inside are left as an exercise to the interested reader.
Physical Science
Comments
When Blake makes a post saying he's on vacation for a while and won't be posting, it's a signal that he'll have a new post tomorrow.
Posted by: John Armstrong | April 21, 2009 9:25 PM
Hey, nice equations.
Posted by: Greg Laden | April 21, 2009 9:37 PM
Reruns don't count! :-) All I had to do was copy, paste and invert the colours on three GIFs.
Posted by: Blake Stacey
| April 21, 2009 9:41 PM
What I like to do when there's an interesting new planet found is to fire up Starstrider, travel to the relevant star, face Sol, and imagine how the aliens might describe the location of Sol on their night sky with reference to their own set of constellations. At least, that's what I always did when I had a CRT monitor. I now have an LCD monitor (because the old one is dead and you can't buy anything else these days) and it really, really sucks for interstellar travel. You can't get a good simulation of the sky at night with a monitor that thinks "black" is just another word for "darkish silvery sheen". End abbreviated rant.
Anyway, here are some instructions for people looking for Sol in the night sky of a planet orbiting Gliese 581.
To help you find the right constellation, draw an imaginary line from the Pleiades to the Hyades. The perpendicular bisector of that line points you to a nearby row of bright stars arranged in a gentle curve, forming the stem of a constellation that may look to you like a stem with a leaf on the end, assuming you have leaves.
Counting from the end of the stem, there's a bright star (we call it 5 Tauri), a dim star, three bright stars, a dim star where the stem joins onto the leaf, another bright star on the leaf (we call it Menkar), and then five more stars on the leaf before you're back to the stem again.
Going back to the three bright stars in the middle of the stem, the third one is Sol. It is about half way between 5 Tauri and Menkar on your night sky, and has an apparent magnitude of 3.7.
Posted by: Adrian Morgan | April 22, 2009 4:45 AM
More quickly than light? Eaaaaaaaasy! It is enough to switch off the light!
Close eyes: you are there already!
Posted by: humorix | April 22, 2009 9:07 AM
Adrian, have you checked local thrift stores? I remember when I lived in Gainesville, FL, it seemed like the thrift stores were rotten with CRTs. The Salvation Army had dozens of them, all marked either 10 cents or free. Call around, I'm sure you can find a CRT somewhere.
Posted by: Gary | April 22, 2009 7:22 PM
I plugged numbers in and I don't get twice as long if you flip the ship over - I get essentially sqrt(2) times as long (i.e. about 40% longer ship-time if you want to stop there rather than whizz by).
(The first half of the trip takes about sqrt(1/2) as long as the whole trip, and then you double that for the decelerating phase)
Did I err?
Posted by: efrique | April 23, 2009 3:17 AM
I ran the numbers again and got a ratio of about 1.66, which I suppose rounds up to 2. The post above has been modified to clarify accordingly. It gets larger the longer the trip, getting up to 1.89 if you're crossing the galaxy and 1.91 if you're visiting Andromeda.
My results check with the figures given in the Physics FAQ, so if I'm totally screwed up then at least I'm in good company.
Posted by: Blake Stacey
| April 23, 2009 11:52 AM
Never one to turn down a challenge, I decided to see what we could do to get there with (what I believe to be) realistic constraints on velocity and drive efficiency. The end result: for something about as massive as a fully fueled Saturn V, we could send a ship with the core (ie, not including fuel and fuel tanks) about the size of three Nimitz-class aircraft carriers and a crew complement about 1/10th a carrier's for a round trip of 40 years proper time, including a four year exploratory/refueling phase at the destination. See the link for a more detailed write-up.
Posted by: W. Kevin Vicklund
| April 23, 2009 6:39 PM
Don't forget time dilation's ugly stepsister; relativistic mass.
The faster you go the greater your mass, the more energy you need to accelerate more. Cool calculator for this stuff here.
Posted by: tresmal | April 26, 2009 10:29 PM