extended essay online

PHYSICS || EXTENDED ESSAY:

Investigation of Relativistic Spaceships and

the Implications of Special Relativity on

Interstellar Space Travel

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Candidate # xxxxxxxxxxx

December 2009

Extended Essay – Physics

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3973 Words

P. Dowling – Physics Extended Essay

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ABSTRACT

I explore the basic concepts of special relativity and then apply them to interstellar space

travel, to analyze their implications. Some background knowledge of classical (Newtonian)

mechanics is assumed, and background knowledge on relativity is helpful but not absolutely

required.

Several types of fuel are discussed: nuclear fission(briefly), nuclear fusion and antimatter

annihilation. The conclusion is reached that fission fuel is fairly inefficient and that fusion

fuel is the better alternative within nuclear fuels. Matter-antimatter fuel annihilation is

found to be much more powerful, however it has the strong disadvantage of being much

harder to gather in sufficient amounts.

More specific aspects of interstellar travel are then investigated; I develop a method to find

required velocities that allow travelling certain distances in a given time. An equation to

find a mass-to-fuel ratio (for 100% efficient antimatter fuel) is also found :

𝑚𝑓𝑢𝑒𝑙

𝑚𝑠𝑕𝑖𝑝= γ 1 +

𝑣

𝑐 − 1 . I find that the amount of fuel required for a long distance mission

(25000 light years) are too high and unlikely to be possible to gather, however short range

targets such as Alpha Centaury (4.3ly) or even Gliese 871 (15ly) might be reached, perhaps

even with nuclear fuel.

Light Sail spaceships are also investigated. While requiring significantly less energy (and thus

fuel) than self-propelling rockets, they are not suitable as such for interstellar travel because

they are only capable of very low acceleration.

226 Words


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CONTENTS

Abstract .............................................................................................................................................................................................. 1

Contents ............................................................................................................................................................................................. 2

Introduction...................................................................................................................................................................................... 3

1. Time dilation........................................................................................................................................................................... 3

2. Relativistic phenomena and the Lorentz factor ....................................................................................................... 6

3. Planning a trip to a distant star ...................................................................................................................................... 9

4. Fuel .......................................................................................................................................................................................... 10

4.1. Nuclear Fuel ............................................................................................................................................................... 10

4.2. Antimatter fuel .......................................................................................................................................................... 12

4.3. Calculating the amount of fuel needed ............................................................................................................ 13

5. Alternative targets ............................................................................................................................................................ 15

6. An alternate solution ........................................................................................................................................................ 16

6.1. Light Sail Spaceships .............................................................................................................................................. 17

Conclusion ...................................................................................................................................................................................... 20

Appendix ......................................................................................................................................................................................... 21

Appendix 1: Simple derivation of the Time Dilation relation(The Lorentz factor) .................................... 21

Appendix 2: Calculation of the Velocity needed to travel a certain distance in a given Perceived time

(section 3) .................................................................................................................................................................................. 22

Appendix 3: How to calculate the amount of fuel needed to accelerate a rocket (Derivation, Section

4.3) ................................................................................................................................................................................................ 24

Works Cited ................................................................................................................................................................................... 26


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INTRODUCTION

Interstellar space travel has long been a topic of interest for many people. We have long

dreamed of going to distant stars or galaxies, but only fairly recently, within the last century,

the idea has begun to appear more realistic and perhaps even achievable. Humans have

been to the moon, and are planning to go to Mars next. In the very long run, we will try to go

further and further for different reasons. These include simple curiosity, searching for

extraterrestrial life, or other inhabitable planets. In this essay I will investigate how we

could, sometime in the future, achieve interstellar travel while focusing on special relativity’s

relevance to the subject.

I will explore the general concepts of special relativity as proposed by Einstein, and then

evaluate their implications on space travel. I will do this by investigating two different

models: Relativistic rockets that use fuel for propulsion, and light/solar sails, which use

electromagnetic radiation pressure for propulsion.

(Note: the term “payload” refers to any mass on a spaceship ship that is not fuel in this essay)

1. TIME DILATION

Einstein proposed that the perception of time, relative speed, mass and dimensions will dilate

or contract according to the velocity of an object or its observer, meaning that these


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measurements can diverge for different observers. This notion follows from two

assumptions. The first is that the laws of physics are observed to be the same from all

inertial reference frames. The second assumption is that the speed of light in a vacuum is

constant, regardless of the motion of the source of the light or the observer. Every observer

will measure the speed of massless particles (c) to be the same, no matter at what speed the

observer himself is moving. These assumptions are the postulates of relativity.1

We can visualize time dilation with a light clock, an apparatus that reflects a beam of light

between two mirrors to measure an interval of time. If we were to move the clock at

velocity v, and consider the system from an external viewpoint (that is at rest), we see that the

beam of light must travel a greater distance, since an additional component has been added to

the motion of the beam of light. Diagram 1 provides an illustration of this.

Diagram 1: Stationary and moving light clock

If the speed of light (c) is indeed constant, the beam would take longer to travel between the

two mirrors, because it travels a greater distance (cT) than when at rest (ct). However this

1 Conceptual Physics – Paul G. Hewitt; Chapters 15.4 & 15.5


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is impossible due to the two postulates of relativity, and time must therefore dilate to

compensate for this.

The exact factor of dilation can be derived from this diagram by rearranging the following

formula:

(cT) ² = (vT)² + (ct)² (Pythagoras theorem for the path of the beam of light)

For T (the time in the accelerated system in terms of t, the time in the stationary system)

Which gives us the following relation

𝑻 = 𝒕

√(𝟏 −𝒗²𝒄𝟐

)

𝑇 = γ × t ,𝑤𝑕𝑒𝑟𝑒 𝑡 =1

(1 −𝑣2

𝑐2

(see appendix for the full derivation)

where:

v is the velocity

c is the speed of light

t is the time in the stationary reference frame

T is the time as measured in the accelerated reference frame.

The variable γ (gamma) is known as the Lorentz factor. This factor is used very often when

doing calculations about relativistic effects, and I will talk more about it later on.


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2. RELATIVISTIC PHENOMENA AND THE LORENTZ FACTOR

I have shown that time slows down for light-clocks, and in order for the second postulate

(constancy of the speed of light) to be correct, it must also slow down for all other clocks

(biological, mechanical, etc.

This suggests that it is impossible for mass to travel at the speed of light, as time would freeze

completely once the speed of light is reached. When the clock reaches the speed of light, the

beam of light that is reflected must stop moving altogether for an observer on travelling

alongside it. This is because the light would have to move faster than c to reach the other

mirror.

Einstein’s theory is also supported by different kinds of empirical evidence. One good

example of this are Muon decay experiments. What these show is that cosmic ray muons,

which are “charged particles [which are] by-products of cosmic rays colliding with molecules in the

upper atmosphere”2, are found to decay at slower rates when they are moving at relativistic

speeds than when they are at rest, providing evidence for time dilation. They also show that

particles travelling near the speed of light have higher kinetic energy than one would predict

with classical equations. This has been demonstrated multiple times and is a reproducible

experiment.3 Particle accelerators in general are a good example of relativity, as they can

give particles very high kinetic energies, but never accelerate them to the point that they are

2 “Muons for Peace” – Mark Wolverton (http://www.scientificamerican.com/article.cfm?id=muons-for-peace)

3 There are many examples of these experiments, for instance:

“Cosmic Ray and Neutrino Tests of Special Relativity” - Sidney Coleman, Sheldon L. Glashow

(http://arxiv.org/abs/hep-ph/9703240)

“The Mean Life and Speed of Cosmic-Ray Muons” - Javier M. G. Duarte and Sara L. Campbell

(http://web.mit.edu/woodson/Public/8.13finalpapers/Duarte_muons.pdf)


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travelling at or above the speed of light(even if the should be travelling above c according to

𝐾𝐸 =1

2𝑚𝑣2).

I will now proceed with some more specific examples and some calculations as to how exactly

time dilation affects moving objects, since, in our everyday lives, we don't seem to notice it at

all, and it seems very counter-intuitive. To explain this, we need to look at the formula for

the calculation of the dilation, and hence see how strong time dilation should be on low-

velocity objects.

Einstein found that the effect could be calculated using the Lorentz transformations, or rather

the Lorentz factor, a simplified derivation of which I have shown earlier.4 As I previously

established, time dilation can be calculated using the following formula:

𝑻 = 𝛄 × 𝐭, where γ =1

√(1−v 2

c 2 ) (the Lorentz factor)

(This relation also shows why travel at the speed of light is impossible, as when v is equal to c,

the equation becomes T = t / 0, which makes no sense.)

4 Another approach of a derivation (with the same result) of it can be found in Einstein’s book “Relativity: The Special and

General Theory“.


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If we graph the Lorentz factor γ as a function,γ =1

√(1−v 2

c2 ), we get the following graph: 5

This clarifies why relativistic phenomena (such as time dilation) are essentially unnoticeable

6in our everyday lives, as the effects of it 'settle down' to Newton’s laws at low speeds, as γ is

extremely close to 1 for even the highest velocities humans move at. The effects are simply

too small to notice, since we never do or have been moving at relativistic speeds relative to

each other.

We also see that time gets “compressed” more rapidly between two velocities as the speed of

light is approached (around >0.85c), much more extreme than at speeds such as just 0.5c and

0.6c. This is very significant for this investigation, as it suggests that we would have to

5 The graph uses natural units, c has been replaced by 1, therefore velocity is to be read as a fraction of c, not 𝑚𝑠−1

6 As well as negligible for virtually all experiments involving low velovities (below 0.5c, or even 0.1c)


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travel at least at about 0.9c to even achieve sufficient time dilation to make the trips seem

significantly shorter.

3. PLANNING A TRIP TO A DISTANT STAR

In this first example, calculations of the necessary time dilation for a trip to the center of our

galaxy, (roughly 25000 light years away from earth) will be made. It can be assumed that

the trip will either involve two generations, or humans either have a greater life-expectancy

then, and 50 years is therefore an acceptable time to reach the planet for the astronauts.

Since the ship will travel at a speed extremely close to the speed of light, we can, to simplify

calculations, say that it will need approximately 25000 years to travel the distance from the

earths’ frame of reference, and 50 in its own. 25000

50= 50, Time for the ship therefore needs

to be slowed down by a factor of T=500.

Solving T = 500 for v:

T =1

√(1 −v2

c2)= 500

√(1 −v2

c2) =1

500 Rearranging

𝑣2

𝑐2 = 1 − 1

500

2

Square both sides and rearrange

𝑣 = 1 − 1

500

2

× 𝑐2 Solved for v


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v = 0.999998c

= 299791858.4 ms-¹

Even though I assumed that the ship was traveling at the speed of light to find the time taken

from a stationary reference frame (to simplify calculations), the value obtained for v is

roughly correct. At this speed, it takes 50.0003 years of perceived time for the ship to travel

the 25000 light years ( see appendix for full calculations of this).

4. FUEL

Next, I will examine two types of fuel that could be used to propel a spaceship which is supposed

to be capable of travelling at velocities of such proportions: Nuclear fuel and matter –

antimatter annihilation.

4.1. NUCLEAR FUEL

There are two types of nuclear reactions that would be suitable: fusion and fission. However,

nuclear fusion releases significantly more energy than fission, and fission therefore would be

an unlikely fuel to be used. But how much energy is released in nuclear fusion?

Commonly, nuclear fusion involves fusing deuterium and tritium into helium:

𝐻12 + 𝐻 = 𝐻𝑒2

413 + 𝑁 + 𝐸𝑛𝑒𝑟𝑔𝑦

𝐻12 has a mass of 2.014𝑢 and 𝐻1

3 has a mass of 3.01605𝑢, their combined mass is:

2.014𝑢 + 3.01605𝑢 = 5.03005𝑢


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𝐻𝑒24 has a mass of4.0026𝑢, and a neutron is 1.008665u,

7 their combined mass is:

4.0026𝑢 + 1.008664u = 5.011266𝑢

The mass defect of this fusion reaction:

5.03005𝑢 − 5.011265𝑢 = 0.018785𝑢 = 3.119 × 10−29 kg

Thus, the energy output is:

𝐸 = 𝑚𝑐2

𝐸 = 3.119 × 10−29× 𝑐2

𝐸 = 2.08 × 10−12𝐽 = 17.47 𝑀𝑒𝑉

This corresponds to 2.49 × 1014𝐽𝑘𝑔−1.

The idea of a nuclear powered spaceship to travel large distances has been brought up various

times before, the most well known of them was called project Orion. The idea was, in simple

terms, to detonate nuclear weapons below the ship to propel it, and it would theoretically be

able to reach a maximum velocitie of roughly 0.1c this way.8 This already suggests that

nuclear fuel might not be sufficiently efficient for interstellar travel such large scales as 25000

light years, however might be an option for closer targets.

We can calculate how fast we have to travel to reach a place in a certain time, but it is also

necessary to know how much energy is needed for such travel, if we want to actually plan a

trip under these conditions. Again, the Lorentz factor is used. The classical formula for

kinetic energy held by an object is 𝐾𝐸 =1

2𝑚𝑣2. Taking into account relativistic effects, this

becomes 𝐾𝐸 = γ − 1 mc2. 9 10At relativistic speed, the amount of kinetic energy held by

7 W.-M. Yao et al. (Particle Data Group), “N-Baryons” (http://pdg.lbl.gov/2006/tables/bxxx.pdf)

8 Carl Sagan’s “Cosmos – Journeys in Space and Time”

(http://www.cosmolearning.com/documentaries/cosmos/14/)

9 Pearson Baccalaureate: Higher Level Physics, Chris Hamper. Chapter 13: Relativity

10 This, again, cancels down to classical mechanics at low speeds, and is therefore not noticed by us.


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a moving object again increases dramatically as the speed of light is approached, and is

infinite at c. 11 This is another example of why mass can never reach the speed of light, it

would take an infinite amount of energy.

For a space mission, this is also a limiting factor, as the amount of energy needed to reach

‘relativistic’ velocities is dramatically higher than predicted using classical equations,

Newton’s laws. Calculating the energy needed to accelerate a certain mass to such a velocity

is fairly simple. 𝐾𝐸 = γ − 1 mc2 gives the energy held by the object at this speed, and

thus the energy one would need to give the object to reach the velocity. So to ‘catapult’ an

object with a mass of 1kg to a velocity of 299791858ms-1, approximately 2.25 × 1022𝐽 of

energy are required. In comparison, the largest nuclear fusion reaction ever produced by

humans, the Tsar Bomba hydrogen bomb12, had an energy equivalent of 50 Megaton’s of TNT,

which translates to 2.09 × 1020J of energy. 13 This is merely the amount of energy needed

to accelerate the mass immediately, something which we cannot do with an actual spaceship,

since the sudden acceleration would be too great for any astronauts on board. For an

actual spaceship, only some fuel is used every instant, and the rest of the fuel becomes

additional payload, which significantly increases the total amount of fuel needed, as I will

evaluate more clearly in part 4.3.

Before reaching a conclusion as to whether nuclear fuel is a good option, I will now discuss

the annihilation as a fuel for spacecraft propulsion.

4.2. ANTIMATTER FUEL

11 Pearson Baccalaureate: Higher Level Physics, Chris Hamper. Chapter 13: Relativity

12 Nuclear Weapons Archive: Tsar Bomba (http://www.nuclearweaponarchive.org/Russia/TsarBomba.html)

13 NIST Guide to the SI: B.8 Factors for Units listed Alphabetically (http://physics.nist.gov/Pubs/SP811/appenB8.html)


13

Matter-antimatter annihilation is a much more powerful reaction than both nuclear fission

and fusion. In the process, a particle and its antiparticle collide and annihilate, releasing a

lot of energy, often forming new particles. The energy released is given by the formula

𝐸 = 𝑚𝑐2. This is significantly more than in nuclear fusion, as not only a small mass defect is

converted into energy, but the entire mass involved in the reaction.14 For example: for just

one electron-positron pair, 511 keV, or 8.18 × 10−14J of energy is released, which

corresponds approximately 9 × 1016𝐽𝑘𝑔−1 in the ideal case (using 𝐸 = 𝑚𝑐2). This is

significantly more than the energy obtained in fusion reactions, which is approximately

2.49 × 1014𝐽𝑘𝑔−1.

This appears to be the better option, but has the disadvantage of being harder to aggregate.

Currently it is still extremely difficult to gather large amounts of antimatter, an obstacle which

would somehow have to be overcome if we wish to use particle annihilation as a fuel.

4.3. CALCULATING THE AMOUNT OF FUEL NEEDED

Calculating the amount of fuel needed for a rocket to reach a velocity is not as simple as it first

appears. Most of the fuel is carried around on the ship for a large part of the journey, and

this has to be taken into account, especially since the amounts of fuel easily surpass the mass

of the ship itself. The simplest way to calculate the amount of fuel that is required is to find

a ‘mass of fuel to mass of ship’ ratio by equating the formulas for conservation of energy and

momentum held by the system. You can see a derivation of this (which is partially adapted

14 Although in many cases this energy partly goes to the creation of new particles


14

from “The Relativistic Rocket” by Don Koks15) in the appendix, but the resulting formula for

the mass of fuel/ mass of ship ratio for a ship at a given velocity is

𝒎𝒇

𝒎𝒔= 𝜸 𝟏 +

𝒗

𝒄 – 𝟏 16

This equation enables the calculation of the fuel/payload mass ratio, taking into account

relativistic effects.

Accelerating a rocket to 299791858.4ms-¹ (from the previous example, reaching a system

25000 light years away) thus requires

𝛾 =1

1 −299791858.42 ms−1

c2

= 499.994

𝑚𝑓

𝑚𝑠= 𝛾 1 +

𝑣

𝑐 − 1

𝑚𝑓

𝑚𝑠= 499.994 1 +

299791858. 4ms−1

𝑐 − 1

𝑚𝑓

𝑚𝑠= 998.987

Nearly a thousand kilograms of matter-antimatter fuel for each kilogram of payload. For a

ship with a mass of 100,000kg, this means 99898700 kg of fuel are needed (which again

would increase the payload mass significantly, since more storage space is needed). By

current standards, it is virtually impossible to gather this much antimatter. Unless there

15 Don Koks “The Relativistic Rocket” (http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html)

16 Ibid.


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will be a major breakthrough in creating or collecting suitable antimatter, it is not going to be

feasible to use as a fuel.

Even if relatively large masses of antimatter can be obtained, an engine which is 100%

efficient is impossible. Some professional concepts estimate the efficiency of antimatter

fueled engines to reach up to about 64%, or 5.8 × 619𝐽𝑘𝑔−1. In a realistic case, protons and

antiprotons would probably be used, since they produce charged particles, 𝜋+ 𝑎𝑛𝑑 𝜋−

(along with 𝜋0 and gamma radiation, but these aren’t useful for propulsion) which can be

directed into a strong magnetic field to produce thrust. 17

It follows that my subconclusion at this point is that it is unlikely that humans will ever

achieve interstellar space travel on such large scale, even using such kinds of highly efficient

engines and matter-antimatter annihilation as a fuel. However, I will now examine whether

closer targets could perhaps be reached.

5. ALTERNATIVE TARGETS

A closer star or planet would require less time dilation, which means lower speed, and thus

less fuel. Below, you can find a table showing distance, required time dilation to reach the

destination in 50 years or less, (I used one year for values under 10ly, 5 for distances below

150ly, 50 for the remaining) the fuel-to-mass relationship and the resulting required fuel, for

a spaceship of mass 100000kg. 18 19 20

17 Robert H. Frisbee: HOW TO BUILD AN ANTIMATTER ROCKET FOR INTERSTELLAR MISSIONS

(http://www.aiaa.org/Participate/Uploads/2003-4676.pdf)

18 Alexander J. Willman: Known Planetary Systems (http://www.princeton.edu/~willman/planetary_systems/)

19 Chris Dolan: The closest star to Earth

(http://www.astro.wisc.edu/~dolan/constellations/extra/nearest.html)

20 Research Consortium of Nearby Stars: The One Hundred Nearest Star Systems

(http://www.chara.gsu.edu/RECONS/TOP100.posted.htm)


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These targets, especially the closer ones (less than 20 light years away) require much less fuel

and also much less time to reach. In fact, it appears that with sufficient advances in methods

of gathering antimatter (and spacecraft engineering), these are fairly realistic targets which

we have a good chance to someday be able to reach. At such distances, it might even be

possible to use nuclear fuel, which is much easier to gather (and can actually be extracted

from hydrogen, which could be found on some planets in our own solar system, which means

that a spaceship could travel in steps, rather than carrying all its fuel right from the beginning.

This is a largely discussed idea, but further exploration of this goes beyond the scope of this

essay.)

6. AN ALTERNATE SOLUTION

The main theme of this essay so far has been to establish whether interstellar space travel is

possible with so called “relativistic rockets”, spaceships that use fuel to propel themselves to

their destination, which they carry with them from the start. However, there are other

kinds of spaceships which might also be capable of interstellar travel, but do not have to solve

the problem of carrying fuel. I will briefly talk about one of these in this section: light sail

spaceships.

Name of star/planet

Distance in ly

Perceived time (years)

Required time dilation (γ)

Required speed (fraction of c)

Mf/ms Required amount of fuel (kg)

Alpha Centaury 4.3 1 4.3 0.973 7.45839 745839

Barnard's Star 6,0 1 6.0 0.986 10.916 1091600

Sirius 8,6 1 8.6 0.993 16.1398 1613980

Gliese 876 15,0 5 3 0.943 4.829 482900

Ursae Majoris 46 5 9.2 0.994 17.3448 1734480

51 Pegasi 50 5 10 0.995 18.95 1895000

HD80606 190,4 15.1 12.6 0.997 24.1622 2416220

OGLE-05-169L 9100,0 50 182 0.999985 362.997 36299700

Table 1: Alternate Targets and Velocities

Name of star/planet

Distance in ly

Perceived time (years)

Required time dilation (γ)

Required speed (fraction of c)

Mf/ms Required amount of fuel (kg)

Alpha Centaury 4.3 1 4.3 0.973 7.45839 745839

Barnard's Star 6,0 1 6.0 0.986 10.916 1091600

Sirius 8,6 1 8.6 0.993 16.1398 1613980

Gliese 876 15,0 5 3 0.943 4.829 482900

Ursae Majoris 46 5 9.2 0.994 17.3448 1734480

51 Pegasi 50 5 10 0.995 18.95 1895000

HD80606 190,4 15.1 12.6 0.997 24.1622 2416220

OGLE-05-169L 9100,0 50 182 0.999985 362.997 36299700

Table 1: Alternate Targets and Velocities


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6.1. LIGHT SAIL SPACESHIPS

These ships could theoretically use the radiation pressure of beams from a powerful laser (or

some type of radiation source, even stars) stationed on earth or some other planet as energy

for propulsion. According to Einstein, photons have no mass, but still have momentum.

This momentum is given by the equation 𝑝 =𝐸

𝑐. 21 The law of conservation of momentum

states that, in a closed system momentum in an elastic collision must be conserved. This

means that a Light Sail spaceship could, over time, achieve great velocities, using significantly

less energy than a fuel-powered spaceship, since only the spaceship, and no remaining fuel

would have to be accelerated.

The momentum held by a ship at a certain velocity, taking into account relativistic

phenomena, is calculated using the formula 𝑝 = 𝛾𝑚0𝑣. 22 Light has both energy and

momentum. Energy is given by 𝐸 = 𝑝𝑐, and momentum must be conserved upon impact on

the light sail. Therefore, if absorbed, the all the light’s energy goes toward accelerating the

ship, and if reflected, this increase in momentum is even doubled. We can calculate the

momentum held by a certain ship, and then calculate how much energy in form of photons

this corresponds.

A ship of mass 100,000𝑘𝑔 travelling at 99% of the speed of light (296 794 533 𝑚

𝑠) has the

following momentum:

𝑝 = 𝛾𝑚0𝑣

𝑝 = 50.25 × 100000𝑘𝑔 × 296794533𝑚𝑠−1

21 HyperPhysics: Relativistic Momentum (http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/relmom.html#c2)

22 Ibid.

𝛾 =1

1 −296 794 5332

𝑐2

= 50.25

𝛾 =1

1 −296 794 5332

𝑐2

= 50.25


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𝑝 = 1.4914 × 1015𝑘𝑔𝑚𝑠−1

We can now calculate the amount of light energy needed to supply this much momentum:

𝑝 =2𝐸

𝑐 (from 𝐸 = 𝑝𝑐, however if the light is reflected, the momentum gained by the sail is

twice as high as when absorbed)

So 𝐸 =1

2𝑝𝑐 23

𝐸 =1

2× 1.4914 × 1015𝑘𝑔𝑚𝑠−1 × 𝑐 = 2.2356 × 1023𝐽

This is a lot of energy, but significantly less than what an antimatter fueled rocket would

consume. Using the same method as in previous examples, that amount would turn out to

be the following:

𝑚𝑓

𝑚𝑠= 𝛾 1 +

𝑣

𝑐 − 1

𝑚𝑓

𝑚𝑠= 50.25 1 +

296794533𝑚𝑠−1

299792458𝑚𝑠−1 − 1

𝑚𝑓

𝑚𝑠= 98.997

For 100,000kg:

100000𝑘𝑔 × 50.25 1 +296794533𝑚𝑠−1

299792458𝑚𝑠−1 − 1 = 9899749.99kg

23 Alexander Bolonkin: High Speed AB-Solar sail

(http://arxiv.org/ftp/physics/papers/0701/0701073.pdf)


19

Converting to energy:

𝐸 = 𝑚𝑐2

𝐸 = 9899749.99kg × 𝑐2

𝐸 = 8.897 × 1023J

As we can see, the light-sail method requires about a quarter of this energy, even without

taking into account the fact that an antimatter rocket would not be 100% efficient.

Additionally, using light-sails as a propelling mechanism is technologically easier to achieve,

as antimatter is still extremely rare, while light sails are comparatively easy to make.

However, light sails have the disadvantage of accelerating very slowly.24 The acceleration of the sail

is given by:25

𝑎 = 1 + ř 𝑃

𝑚𝑐

Where a is the acceleration, ř is the sail’s reflectivity (fraction of the light that is reflected), P

is the light sources power output and m is the sail’s mass. Assuming perfect reflectivity, this

turns into 𝑎 =2𝑃

𝑚𝑐, so for our ship:

𝑎 =2𝑃

100000𝑘𝑔 × 𝑐=

𝑃

50000𝑘𝑔 × 𝑐

This is a very low value, since it is extremely hard to build lasers with a power output just

remotely close to close to 50000 ∗ 𝑐 joules. Also note that this is a classical formula, not

taking into account relativistic effects, which would significantly decrease acceleration as the

24 Ibid.

25 Solar Sailing: Technology, Dynamics and Mission Applications, Colin Robert McInnes, page 272 (section 7.2)


20

speed of light is approached.26 To achieve sufficient acceleration, an extremely powerful

laser and a light sail the size of a small (or perhaps large) world is needed, as this increases

the acceleration induced on the ship. Light sails would be useful for small spacecraft with

very light payload, such as satellites or probes, but they would hardly be suitable for manned

interstellar space travel.

CONCLUSION

From all this, I conclude that interstellar space travel is theoretically possible, time dilation

enables travelling large distances in little perceived time. Nuclear fuel is not an option for

large distances, but fusion might perhaps be used to reach closer systems such as Alpha

Centaury. Antimatter- matter annihilation also provides sufficient energy to reach closer

star systems, but an unlikely-to-gather amount is required for going far distances, such as to

another galaxy or even to close to the center of our own galaxy, therefore better methods of

gathering antimatter would need to be developed if such a mission is to be executed.

Solar sails are also only a very limited alternative due to their low acceleration; they might be

used for unmanned spaceflight, or would have to be built on an enormous scale. In the long

run, I think it is more likely that annihilation fuel is going to be used for manned interstellar

spaceflight. There are many advances that still need to be made, most significantly the

collection of sufficient antimatter, but there are many others, just to mention a few: Radiation

shielding is a big problem, both induced radiation form particles in space, as well as radiation

26 I was unable to find or derive a formula for this that takes into account relativistic effects, however the

classical formula already suggests the the acceleration is extremely low for most lasers, and relativistic effects

would only further decrease the acceleration as c is approached (more KE required)


21

created by the actual annihilation ‘below’ the ship; the spaceships would need to be of

enormous size, a project no nation could achieve on its own, and is even a major difficulty if

many nations collaborate. The storage of fuel is also difficult, as it needs to be kept in

magnetic fields, which are for one hard to build, and then also require a lot of energy to

maintain.

3973 words

APPENDIX

APPENDIX 1: SIMPLE DERIVATION OF THE TIME DILATION RELATION(THE LORENTZ FACTOR) We can derive the factor to calculate the degree of time dilation using the graph, the following

way:

(cT) ² = (vT)² + (ct)² (Pythagoras theorem for the path of the beam of light)

𝑐²𝑇² = 𝑣²𝑇² + 𝑐²𝑡² (expanding)

𝑇² = 𝑣²

𝑐2 × 𝑇² + 𝑡² (divide both sides by c²)

𝑇² − 𝑡² = 𝑣²

𝑐2 × 𝑇²

𝑡² = 𝑇² – 𝑣²

𝑐2 × 𝑇² (rearranging)

𝑡2

𝑇2 = 1 –

𝑣²

𝑐2

𝑡

𝑇 = √(1 –

𝒗²

𝒄𝟐) (Taking the square root of both sides)


22

𝑻 =𝒕

𝟏 −𝒗𝟐

𝒄𝟐

where:

v is the velocity

c is the speed of light

t is the time in the stationary reference frame

T is the time as measured in the accelerated reference frame.

APPENDIX 2: CALCULATION OF THE VELOCITY NEEDED TO TRAVEL A CERTAIN DISTANCE IN A

GIVEN PERCEIVED TIME (SECTION 3)

Time as perceived on the ship in terms of time from a reference frame which is at rest is given

by

𝑇 =𝑡

1 −𝑣2

𝑐2

We are trying to find the velocity, and we have a given perceived time. However, we don’t

know the time as perceived in the stationary rest system, as it actually depends on the

velocity. This significantly complicates exact calculations, and I have not been able to find a

way around this.

What I did do to solve this problem was to approximate T to be equal to the distance, but in

years, since the ship would be travelling at extremely close to the speed of light, and being

exact would not greatly change the result. The actual time taken measured from the rest

reference frame would be just slightly more than 25000 years in the example I was using.

So now we have the formula (Times in years have been converted into seconds)

T =1.5778 × 109𝑠𝑒𝑐

√(1 −v2

c2)= 7.889 × 1011𝑠𝑒𝑐


23

√(1 −v2

c2) =

1.5778×109sec

7.889×1011𝑠𝑒𝑐 Rearranging

𝑣2

𝑐2 = 1 − 1.5778 ×109𝑠𝑒𝑐

7.889×1011𝑠𝑒𝑐

2

Square both sides and rearrange

𝑣 = 1 − 1.5778×109𝑠𝑒𝑐

7.889×1011𝑠𝑒𝑐

2

× 𝑐2 Solved for v

v = 0.999998c

= 299791858.4 ms-¹

We can now check the validity of this value by substituting it into the original formula,

𝑇 = 𝛾𝑡.

First, T is now 𝑇 =𝑑

𝑣, or 𝑇 =

25000 ×𝑐

299791858 .5= 7.88924 × 1011𝑠𝑒𝑐𝑜𝑛𝑑𝑠 = 25000.049𝑦𝑟𝑠.

Already we can see that this is close to the first estimated value, so we can expect a roughly

accurate result.

Now 𝑇 = 𝛾𝑡, so

𝑡 = 𝑇 × 1 −𝑣2

𝑐2

= 7.88924 × 1011𝑠𝑒𝑐𝑜𝑛𝑑𝑠 × 1 − 299791858.42 𝑚𝑠−1

𝑐2

= 1.57786536 × 109𝑠𝑒𝑐𝑜𝑛𝑑𝑠

= 50.00060393 𝑦𝑒𝑎𝑟𝑠

This is very close to 50 years, and precise enough for our purposes. The calculations in table

1 of section 5 are based on this method, and just like in this case, the margin of error was very

small. The errors were found to be negligible, as an error of less than one year (which it was

in every case, no perceived time for a certain velocity turned out to be more than 0.05 years

more than the expected value) is not large enough to affect whether the target could be

reached or not.


24

APPENDIX 3: HOW TO CALCULATE THE AMOUNT OF FUEL NEEDED TO ACCELERATE A ROCKET

(DERIVATION, SECTION 4.3)

If the engine of the ship that is to be used is fueled by antimatter – matter reactions which are

100% efficient, meaning that all the mass is converted into energy (gamma ray photons), the

amount of which can then be calculated E = mc2. We also assume that all the energy is

converted into kinetic energy 100% efficiently and is used to accelerate the ship (which in

reality is impossible, but I will assume this for simplification). We can calculate a “mass of

fuel / mass of ship” ratio by equating the formulas for conservation of energy and momentum

held by the system.

The ship’s mass is assigned the variable ms, whereas the mass of the fuel, antimatter + matter,

is given the variable mf. We know the mass of the ship, and we are trying to find the mass of

the fuel. Applying the law of conservation of energy, the total energy held by the ship and fuel

before takeoff (from a stationary reference frame) is E0 = (ms + mf) c2, and is equal to the

total energy after the fuel has all been converted to energy (which we assign the variable

Efinal), which is Efinal = γmsc2 + Ef. 13

Thus, via conservation of energy:

(ms +mf)c2 = γmsc2 + Ef. This will be our first equation.

As I mentioned before, the momentum of the ship must also be conserved.

The momentum of the ship at rest is P0 = 0. After all the fuel has been used up, and the

energy was used to create force towards the opposite direction of the ships travel, the

momentum of this energy (assuming it is in the form of light) is plight = Ef/c, where c is the


25

speed of light. This was calculated using the formula E = pc for the energy-to-momentum

relationship of a photon. 27

Since at the end of our trip, momentum of the ship is zero, we can write the formula for the

final momentum of the ship the following way: 0 = γmsv * Ef/c.

Our two equations are therefore

(𝒎𝒔 + 𝒎𝒇) 𝒄𝟐 = 𝜸𝒎𝒔𝒄𝟐 + 𝑬𝒇 and 𝜸𝒎𝒔𝒗 −

𝑬𝒇

𝒄. = 𝟎.

We can rearrange both for 𝐸𝑓 in order to set both formulas equal, and then solve them for

𝒎𝒇

𝒎𝒔, which is the ratio of the mass of fuel needed to accelerate one kg of payload to the final

velocity. 28

𝐸𝑓 = 𝑚𝑠 + 𝑚𝑓 𝑐2 − 𝛾𝑚𝑠𝑐

2 (1)

𝐸𝑓 = 𝛾𝑚𝑠𝑣𝑐 (2)

(1)=(2) :

𝑚𝑠 + 𝑚𝑓 𝑐2 − 𝛾𝑚𝑠𝑐

2 = 𝛾𝑚𝑠𝑣𝑐

𝑚𝑠𝑐2 + 𝑚𝑓𝑐

2 = 𝛾𝑚𝑠𝑣𝑐 + 𝛾𝑚𝑠𝑐2 (expanding bracket and rearranging)

𝑚𝑠 + 𝑚𝑓 =𝛾𝑚𝑠 𝑣

𝑐+ 𝛾𝑚𝑠 (divide both sides by 𝑐2)

𝑚𝑠 + 𝑚𝑓 = 𝛾𝑚𝑠( 𝑣

𝑐+ 1) (rearranging)

27 Higher Level Physics – Chris Hamper

28 Don Koks: The Relativistic Rocket


26

𝑚𝑓 = 𝛾𝑚𝑠 1 +𝑣

𝑐 − 𝑚𝑠

𝒎𝒇

𝒎𝒔= 𝜸 𝟏 +

𝒗

𝒄 – 𝟏 (Divide both sides by ms)

Which gives the ratio of fuel mass to ship mass.

WORKS CITED

BOOKS/PRINTED SOURCES (SOME FULLY OR PARTIALLY AVAILABLE ONLINE)

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Hamper, Chris. “Higher Level Physics.” N.p.: Pearson Baccalaureate, 2009. Print.

Hewitt, Paul G.” Conceptual Physics.” N.p.: Prentice-Hall, 2002. Print.


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McInnes, Colin Robert. “Solar Sailing: Technology, Dynamics and Mission Applications.” N.p.: n.p., 2004.

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28

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