Spring 2002

Week 12 Synopsis

Updated 29 April 2002

CONTENTS

Rockets

Rockets work on the principle of conservation of momentum. A good reference site on rocket propulsion is located here. Contrary to some popular opinion, rockets do not "push off" against anything. That is why they can operate in the vacuum of space.

Special Relativity

The speed of light, denoted as c, is 300,000 km/s or 186,000 mph. The Theory of Special Relativity is based on the premise (tested by experiment) that the speed of light is an absolute constant, irrespective of the velocity of the observer. All other motions are relative; no two unaccelerated observers can agree on who is at rest and who is in motion. Another good site is located here. This site has some interesting graphics illustrating the effects of Special Relativity.

At low velocities

• F=ma (Newton's law. F=Force, m=mass; a=acceleration)
• v=at (v=velocity; a=acceleration)
• E=1/2 mv2 (kinetic energy), or V=sqrt(2E/m)

At high velocities, the factor gamma comes into play. gamma = sqrt(1/(1-(v/c)2))

• When is much less than c, then gamma is very close to 1
• When V approaches c, gamma gets very large
• At high velocities, time slows down. t = t0/gamma. t is the perceived time; t0 is the time perceived by an observer at rest.
• At high velocities, mass increases. m = gamma m0. m0 is the rest mass.
• v = sqrt(2E/gamma m). Increasing E without limit does not lead to arbitrarily large velocities. V approaches but cannot exceed c.

The result of this is that if you accelerate to high velocities, time will pass slowly for you, as shown in the Table below. However, time will pass at the normal rate for an outside observer. Home won't be the same when you get back.

Round Trip Voyages at an acceleration of 1 Gravity
Elapsed time Elapsed Time Maximum Distance
on spaceship on Earth (years) Velocity Travelled (ly)
1 1.01 0.25c 0.06
2 2.08 0.46c 0.26
3 3.29 0.64c 0.59
4 4.70 0.76c 1.08
5 6.41 0.85c 1.77
10 24.2 0.99c 10.26
20 296.8 0.9999c 146
30 3613 c 1805
40 44,100 c 22,050
50 535,900 c 276,900
100 1.44x1011 c 7.2x1010

There are apparent paradoxes in Special Relativity. Some of these, such as the Twin Paradox are well known. Fear not, all the paradoxes are resolvable. Most arise from the limitation that Spcial Relativity applies only to linear motion, and not accelerated motion.

Rocket Examples

(Single stage rockets only)

V is the velocity of the rocket.
S is the velocity of the ejecta (the rocket exhaust).
Mfuel is the mass of the fuel.
ln is the natural logarithm.

How fast can you go?

• if Mfuel/Mpayload = 1, then V/S=0.7
• if Mfuel/Mpayload = 10, then V/S=2.3
• if Mfuel/Mpayload = 100, then V/S=4.6
To get a high V, you want a large S

How big is S?

• Chemical rocket (H2+O2) gives S=3 km/s
• Nuclear Fission gives S=8000 km/s
• Nuclear Fusion gives S=25000 km/s
• matter-antimatter gives S=c
A matter-antimatter rocket will in principle permit the fastest speeds for a given fuel-to-payload ratio.

How much fuel will it take to accelerate to a specific velocity at 1 gravity (10 cm/s2)?

How Much Fuel Do You Need?
one wayround trip
25000 mphchemical361521
0.1 cfission2.35.4
0.5 cfusion4001600
0.5 cmatter/antimatter0.61.7

Using the starship Enterprise (NCC1701D) as an example, every time it accelerates to 0.5c (half impulse) and then stops, it burns a mass of matter-antimatter fuel equal to 170% of its payload!

Fuel requirements are somewhat smaller for a multi-stage rocket. But most conceivable interstellar probes will not be multi-stage vehicles.

Accelerating slowly takes more time, but does not save fuel.

In any event, it takes a lot of fuel to reach any reasonable velocity. One way around this is by using a Bussard Ramjet, which scoops up fuel (hydrogen) from the interstellar medium. While the idea of gathering fuel along the trip is attractive, there are many practical problems, among them that the low density of the interstellar medium will require a scoop about the sie of a small planet, and the problem of stopping protons coming at you at close to the speed of light. See for a more detailed description of a Bussard Ramjet.

It has been suggested, by R. Edgars of the University of Wisconsin, that the reason there are no aliens visiting us is because we live in a low density part of the interstellar medium, the local bubble, and that keeps the aliens with their Bussard Ramjets away.

A site that goes into space travel in quite a bit of detail, including some unconventional solutions, is the Warp Drive, When site.

Where are They?

Once you begin to colonize the Galaxy, no matter how slowly, you can visit everywhere in a small time (relative to the age of the Galaxy). If you can travel at 1/10 the speed of light, it takes about 1 million years to cross the 100,000 light year-wide Milky Way Galaxy. If you stop at each star on the way, and construct probes to send to other stars (to the side), and each of these probes then constructs other probes, the length of time needed to visit every star in the Galaxy is the time it takes to cross the Galaxy, or about a million years (10 million if you only go at 1/100 the speed of light).

One need not send humans (or aliens) on these probes: they could be robots which can build replicas of themselves (von Neumann probes), which send back their data to the home world.

So, a space-faring civilization will be able to visit every star in a time much shorter than the age of the galaxy, and perhaps much less than the survival time of that civilization. If that is true, then why aren't they here?

Some possibilities are:

• We are the only (or first) spacefaring species. The principle of mediocrity suggests that we should not be the first.
• Space travel is not possible for biological beings. Recent data from the Mars Odyssey shows a higher than expected radiation level in interplanetary space. See this site for more specifics. Outside the heliosphere, cosmic ray levels will be even higher. However, this won't account for the lack of von Neumann probes.
• Space travel is possible, but advanced civilizations stay home.
• We are in quarantine. This could be to protect us, and let us advance at our own rate (e.g., Star Trek's Prime Directive), or to protect us, an immature species, from the psychological shock of knowing that we are not alone.
• We are an exhibit in a cosmic zoo.
• They are here: we just don't know it.
• They are here: UFOs are alien spacecraft.

A good article about the Fermi paradox and some possible resolutions is in the July 2000 issue of Scientific American.