Eclipse Phase Space Warfare

Anders Sandberg

April 26, 2011

Abstract

noindent This essay analyses the physics of spacecraft and space com-bat in Eclipse Phase. Based on the technological assumptions explicitlyand implicitly made in the game together with known physics, variousconstraints on space warfare can be concluded. In general, the space bat-tlefield is extremely high-energy, high-loss and dominated by the forcesthat can estimate the locations of enemy assets accurately despite mas-sive interference. FTL quantum entanglement communication provides abig but not decisive advantage to forces able to afford it.

Contents

1 Introduction 31.1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Ship performance 4

3 Energy requirements 43.1 Reactor sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Exhaust temperature . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Detection 94.1 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Thermal emissions . . . . . . . . . . . . . . . . . . . . . . . . . . 104.3 Exhaust emissions . . . . . . . . . . . . . . . . . . . . . . . . . . 104.4 Sunlight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.5 Sensor sizes and sky scanning . . . . . . . . . . . . . . . . . . . . 114.6 Ship spectra and detectability distance . . . . . . . . . . . . . . . 124.7 Scanning limitations . . . . . . . . . . . . . . . . . . . . . . . . . 124.8 Particle emissions . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.9 Railgun projectiles . . . . . . . . . . . . . . . . . . . . . . . . . . 144.10 Active sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.11 Stealth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.11.1 Anisotropic radiation . . . . . . . . . . . . . . . . . . . . . 154.11.2 Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.12 Parallax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.13 Hiding from many eyes . . . . . . . . . . . . . . . . . . . . . . . . 164.14 Spoofing and jamming . . . . . . . . . . . . . . . . . . . . . . . . 17

1

5 Weapons 195.1 Independent weapon buses . . . . . . . . . . . . . . . . . . . . . . 195.2 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.2.1 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2.2 Beam optics . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2.3 Heating beams . . . . . . . . . . . . . . . . . . . . . . . . 215.2.4 Explosive beams . . . . . . . . . . . . . . . . . . . . . . . 21

5.3 Railguns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.4 Particle weapons . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.5 Missiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.5.1 Kinetic impactor rods . . . . . . . . . . . . . . . . . . . . 225.5.2 Nukes and antimatter . . . . . . . . . . . . . . . . . . . . 235.5.3 Rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.5.4 Relativistic missiles . . . . . . . . . . . . . . . . . . . . . 245.5.5 Leashes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.5.6 Nanoweapons . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.6 Fighters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.7 Networked warfare . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6 Armor 266.1 Kinetic impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.2 Beam impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7 General strategy 28

8 Cloud combat 298.1 Weapons vs. sensors . . . . . . . . . . . . . . . . . . . . . . . . . 29

9 Hitting a dodging enemy 30

10 Conclusions 3210.1 Deep space combat . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.2 Orbital warfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

List of Tables

1 Ship engine types . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Ship properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Energy storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2

1 Introduction

Eclipse Phase is a mostly hard SF role-playing game with a setting stretchingacross the solar system (and some ex-oplanets). Space travel is not uncom-mon and in the past there have beenmilitary conflicts in space. While per-sonal combat is described space com-bat is not; the game states that:

Spacecraft have few statsin Eclipse Phase, as theyare primarily handled assetting rather than vehi-cles. Note also that nostats are given for space-craft weaponry. It is highlyrecommended that spacecombat be handled as aplot device rather than acombat scene, given the ex-treme lethality and dangerinvolved.1

While this is sensible advice for aroleplaying game, it is a direct chal-lenge for players who like to con-sider how space combat might work,the strategies involved and how thesemight affect the characters inside thegame. For example, is space com-bat heavily stealth oriented or re-quires ships with massive armor barg-ing through enemy point defenses?What are fighter craft (and fighter pi-lots) useful for? How much of anadvantage does a space habitat haveagainst an attacker?

In the following I will analyse whatfollows from the assumptions made inthe game, as well as extrapolationsfrom known physics and technology.

Technology in Eclipse Phase hasachieved dense energy sources allow-ing fast spacecraft, fast optical pro-cessing running human-level cognition,

and antimatter weaponry. But spacestrategy is limited by lightspeed de-lays (somewhat modified thanks toquantum entanglement FTL commu-nications), the limitations of materialsbased on molecular bonds, high visi-bility of accelerating major crafts andfinite reaction mass resources. Manytechnologies in the game are close tothe limits set by physics, which simpli-fies analysis somewhat.

1.1 Acknowledgements

This essay was inspired by past dis-cussions at the Eclipse Phase Forum2,where many bright ideas were sug-gested.

The Atomic Rocket page3 andRocketpunk Manifesto4 have been animportant source of inspiration, refer-ences and opinions influencing this es-say.

The basic ship performance num-bers were kindly supplied by JSnead.I would also like to thank him for hav-ing done proper design calculations be-hind the scenes, simplifying this workimmensely.

1Eclipse Phase core book, p. 346.2http://www.eclipsephase.com/forum3http://www.projectrho.com/rocket/index.html4http://www.rocketpunk-manifesto.com/

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Table 1: Ship engine types

Engine Acceleration G Acceleration m/s2 Isp sHydrogen-Oxygen Rocket 4+ 39.3 450Metallic Hydrogen 3 29.5 1,600Plasma Rocket 0.01 0.1 20,000Fusion Rocket 0.05 0.5 100,000Anti-Matter 0.2 1.96 200,000Rocket Buggy 0.5 4.9

2 Ship performance

The key ability of spaceships is thatthey can accelerate long and stronglyenough to reach high velocities orchange their velocity (v) radically.This requires expelling reaction massat a high velocity. A key value isthe specific impulse (Isp), how muchmomentum each kilogram of reactionmass can impart on the ship. Thefaster the reaction mass is emitted, thehigher the Isp. However, this doesnot necessarily mean a higher acceler-ation. Available engines typically havea high acceleration for low Isp and viceversa: the high acceleration engines areless able to achieve high v since theywaste much fuel, while high Isp enginescannot produce high accelerations dueto energy limitations.

The achievable velocity change is

v = ve log

(m0m1

)where ve is the exhaust velocity (ve =

gIsp, g is the Earth surface gravity),m0 the initial total mass of the space-craft and m1 the remaining payloadmass after all reaction mass has beenused. Higher velocities require eitherhigher exhaust velocities or exponen-tially more fuel.

Ships with v > 80 km/s typicallydo not have to worry about launch win-dows, while slower ships need to plantheir trajectories so that the origin anddestination are in the right alignment.

Ships are limited by how muchremaining reaction mass they retainwhen making course corrections (es-pecially defensive ones) en route. Iwill assume ships (especially warships)keep a fraction of their reaction massbudget in reserve, giving them a frac-tion of the total v for defensive oroffensive course changes5.

Table 1 describes the performanceof the basic ship engine types. Table 2lists the basic Eclipse Phase spaceshipproperties.

3 Energy require-ments

Some estimates of the powers availablecan be gained from considering space-

ship performance. A ship thruster re-quires P = (1/2)mv2e W of power,where m is the mass flow in kilograms

5The Eclipse Phase core book (p. 283) suggests many ships burn a quarter to a third ofthe reaction mass during the initial burn. In practice this is rather costly, as reaction massnot used will make the trip longer and require extra reaction mass for the develeration burn.Saving this much fuel is rational only if drastic course corrections may be needed, or the fuelvery cheap compared to the cost of arriving later. Many commercial ships likely retain verylow fuel margins, and may rely on tugships that help them slow down at the destination.

4

Table 2: Ship propertiesShip1 Size (m) Cross-

section(m2)

FullyLoadedMass(tons)

EmptyMass(tons)

Enginetype

Total vkm/s

Maxaccel-eration(Gs)

Acceler.(m/s2)

Maxpower

Destroyer 150x50x50 7,500 40,000 25,000 AM 800 0.2 1.96 800 GW

FastCourier

75x13x13 975 1,000 300 AM 1,600 0.2 1.96 20 GW

BulkCarrier

150x25x252 3,750 90,000 4,500 Fusion 40 0.002 0.001964 9 GW

StandardTrans-port

150x25x253 3,750 /12,000

10,000 4,500 Fusion 400 0.02 0.01964 10 GW

Fighter 4.5x3x3 14 7 3 MH 11 3 29.46 16.8 MW

SCUMBarge

300x70x70 24,000 180,000 80,000 Plasma /fusion

80 / 400 0.003 /0.015

0.02946 /0.1473

5.4 GW /135 GW

LLOTVHO HI

25x16x16 400 450 26 HO 11 2 19.64 203 MW

LLOTVOHO LO

25x16x16 400 450 26 HO 7 2 19.64 203 MW

LLOTVMH HI

19x12.5x12.5 237.5 450 26 MH 17 2 19.64 720 MW

LLOTVMH LO

19x12.5x12.5 237.5 450 26 MH 8 2 19.64 720 MW

SLOTVHO HI

17x11x11 187 150 11 HO 11 2 19.64 67.5 MW

SLOTVHO LO

17x11x11 187 150 11 HO 7 2 19.64 67.5 MW

SLOTVMH HI

13x8.5x8.5 110.5 150 11 MH 17 2 19.64 240 MW

SLOTVMH LO

13x8.5x8.5 110.5 150 11 MH 8 2 19.64 240 MW

GeneralExplo-rationVehicle(GEV)

6x2.2x2 13.2 5.5 3 MH 3.6 0.1 0.982 0.44 MW

Missile4 1x0.1x0.1 0.1 0.02 0.002 MH 31 2005 1,964 3 GW1 Abbreviations: AM = Antimatter, HI = HIgh velocity configuration, HO = Hydrogen-Oxygen chemical rocket, LLOTV = Large

Lander and Orbit Transfer Vehicle, LO = LOw velocity configuration, MH = Metallic Hydrogen rocket, SLOTV = Small Landerand Orbit Transfer Vehicle.

2 Plus externally mounted cargo pods.3 80 m wide and high with rotating booms fully extended.4 Own design. Payload can be a small ( 1 kt) nuclear warhead, a 3 megaton antimatter warhead, kinetic impactor projectiles or

attack nanotechnology.5 Short burst launch or evasion acceleration.

Table 3: Energy storageEnergy source Specific power Power density Specific energy Energy density

Fission 2.5 kW/kg 12.5 MW/m3

Fusion 200 kW/kg 1 GW/m3

Antimatter 372 kW/kg 1.86 GW/m3 4.5 PJ/kg 22.5 EJ/m3

Chemical fuels 10 MJ/kg 20 GJ/m3

Nuclear isomers 10 GJ/kg 100 TJ/m3

5

per second and ve is the reaction massspeed. The thrust force F = mveproduces an acceleration a = mve/Mon the spacecraft (of mass M). Fora given acceleration and reaction massspeed the power is P = (1/2)Mave.Expressed in terms of Isp, F = Ispm

g(where g 9.82 m/s2 is the gravita-tional acceleration at Earths suface)a = gIspm

/M , m = aM/gIsp, andP = (1/2)gaMIsp.

This allows us to estimate thepower requirements of spacecraft ifwe know their mass. Consider afusion-powered spacecraft. It has a =0.05G 0.5 m/s2, Isp = 100, 000 s.A fully loaded bulk carrier with totalmass 90 106 kg will hence require anoutput of 9 GW during full thrust, anenergy output of 100 W per kg of shipmass. In practice bulk carriers likelysacrifice exhaust velocity and powerfor economy, so the output will be farbelow this level. Antimatter-poweredships produce comparable energy out-puts (at least in terms of propulsion,since the mechanism is relatively simi-lar). The Destroyer, weighing 40,000tons, implies a reactor power of 800GW.

As a comparision with existingtechnology, a Nimitz-class aircraft car-rier produces 190 MW and a major nu-clear power plant can reach 8 GW. Itis probably safe to assume that largeships have reactors that can producepower up to the terawatt range Mostof this energy is likely only availablefor propulsion rather than poweringweapons due to the problems of con-verting it to electricity. There are alsogoing to be efficiency losses leading tolarge amounts of waste heat. Assum-ing 90% efficiency still requires hun-dreds of gigawatts of cooling. Hencewarships are unlikely to use their re-actors at full power during battle, inorder to avoid having to unfold very

noticeable and vulnerable large radia-tor surfaces. The amount of availableweapon power is still going to be verylarge.

As shown below, there are goodtactical reasons for wanting to accel-erate quicker than allowed by fusionor antimatter drives. This can beachieved by using high g-thrusters suchas metallic hydrogen: while the mainengines aim at a very high exhaust ve-locity to keep reaction mass require-ments down while achieving a high v,these engines are intended to use lowvelocity reaction mass to make a fewbrief but strong changes in ship veloc-ity. The total v is negligble sincethey cannot be sustained for long, butthey allow rapid evasive maneouvers.Also, main engines can in some casesbe supplied with more reaction massthan normal to produce short burstsof acceleration.

3.1 Reactor sizes

What is the size of capital ship re-actors? For fission reactors the spe-cific mass is around 40 kg/kW, al-though advanced vapor core reactorsmight go down to 0.4 kg/kW. Accord-ing to R.W. Bussard6 the fusion reac-tor specific mass could be 0.05 kg/kW.For the transports 10 GW reactor theweight would be 500 tons.

The destroyer is antimatter pow-ered, and assuming the whole reactor isabout the size of the containment sys-tem we get a specific mass of 0.0025kg/kW for antimatter power.

Assuming a density of the reactorto be about 5000 kg/m3 a 10 GW fu-sion reactor would be 100 cubic meters,or a 4.6 4.6 4.6 cube. In practicethe reactor will be far more extended,since these estimates mainly deal withthe core. Containment, control, cool-ing systems etc. will probably be at

6http://www.askmar.com/Fusion_files/FusionElectricPropulsion.pdf

6

least ten times as large (but also oflower density). Similarly, older reac-tors will also be much larger.

It should be noted that small vehi-cles like fighters and missiles likely useother energy sources. An antimatter-powered 1 ton reactor could provide upto 370 MW of power, but for maxi-mal acceleration various chemical fu-els (such as burning metallic hydro-gen) would be more effective. Giventhe available technology high energy-storage densities are possible, proba-bly on the order of 10 MJ/kg (corre-sponding to computed explosives suchas dinitroacetylene, octanitrocubaneand octaazacubane and still smallerthan lithium-fluorine combustion). Foreven higher densities nuclear isomerstorage might be possible.

This table sums up the above con-siderations. The specific power ofchemical fuels and nuclear isomers de-pends on the rate of reaction and canin principle become very high when re-leased explosively.

3.2 Exhaust temperature

A sizeable fraction of the energy out-put is going to be present as heat in theexpelled reaction mass. Rocket noz-zles of chemical rockets can achieve 60-70% efficiency as heat engines convert-ing heat into velocity. The remainingenergy will largely be carried away byreaction mass. If the efficiency is andthe power is P , then the exhaust willhave temperature

Texhaust (1 )P/Cm

where C denotes its specific heat ca-pacity in J/kg K.

For the Destroyer, expelling 40kg/s hydrogen at 60% efficiencyTexhaust = 560, 000 K. This is a hardUV source and not far from a particle

beam weapon. At distance d, assumingthe exhaust radiates spherically, the in-cident energy is (1 )P/4pid2 W/m2.In this case it is 25 MW/m2 at one kilo-meter distance, enough to vaporise thesurface of steel7. This is why cooling isso essential for accelerating spaceships.

Jons law: Any propulsion systempowerful enough to be interesting, ispowerful enough to be a weapon.

3.3 Cooling

A key problem for all spacecraft withhigh power is cooling since space is aperfect thermal insulator. While someengines (metallic hydrogen, fusion, an-timatter) carry away a sizeable frac-tion of the power as heat in the ex-haust, most ship power plants will pro-duce vast amounts of waste heat thatmust be removed8. As a rough ap-proximation, assuming 50% efficiency,the same amount of power the reac-tor produces for the engines and otherforms of usable work is also producedas waste heat. Radiators radiate wasteheat into space but have an upper ac-ceptable temperature Tmax. This re-quires a total radiator area larger thanArad = P/T

4max where is the emis-

sivity (likely chosen close to 1) and = 5.67 108 W/m2K4 is Stefan-Boltzmanns constant.

Using liquid lithium as a coolantgives Tmax = 1600 K. For P =800GW Arad =2.15 million m

2, re-quiring multi-kilometer fins (or dropletradiators, where sheets of droplets ofmolten metal are allowed to drift fromemitters to collectors). The fighterrequires 45 m2 of radiators when us-ing full power, not too different fromfighter plane wings. Using thermalconduction in 3000 K tungsten thefighter can reduce the radiator area to3.6 m2.

7http://panoptesv.com/SciFi/DamageAverage.html8There will also be separate radiators for cooling low-temperature sections of the ship such

as the life support system, but they are negligble compared to the main radiators.

7

In practice radiators of militaryships are foldable, fully used onlywhen doing main accelerations andcompletely folded back when in bat-tle mode for maximum protection andminimum emissions. Since most bat-tles are very short this is usually justa minor problem, but if the situationpersists overheating begins to becomean issue (see section on cooling and in-ternal storage of heat). Damage thatprevents full unfolding after a battleseverely limits course changes.

Calculations by Nemtos9 for morerealistic tapering cooling fins find thatthe total flux that can be radiated is(4/5)LT 4 where L is the width ofthe fin. This gives a total mass ofL0/3, where 0 = (8/5)L

2T 3/is the thickness of the central con-densing channel, is the density and is the thermal conductivity of thefin material. Using liquid potassiumas coolant (T = 1000 K), pyrolyticgraphite ( = 400 W/m, = 2200kg/m3) as a heat conductor and whiteceramic ( = 0.95) as a surface mate-rial allows radiating away 43 kW/m2.Smaller fins were much more efficientin terms of energy release per weightbut require more extended pipe sys-tems since they need to be extendedfurther out.

Liquid droplet radiators were es-timated to require about 50% of thecooling fin mass. Since the dropletsloose energy faster when they are hotit is more effective to build compactradiators with a flight time around asecond. Given some of the limitationsof droplets screening each other it wasconcluded that it could radiate away20 kW/m2 when using liquid tin atT = 1000 K. Droplet radiators arehence preferable over cooling fins whenmass budgets are an issue, such as insmall ships.

Another cooling method is to heatcoolant and dump it into space. Us-ing hydrogen 14.30 103 J per kilogramand Kelvin can be removed10. Heatingmetallic hydrogen to 10,000 K wouldremove 143 MJ per kilogram. As anexample, the 800 GW Destroyer wouldneed to vaporize 2,797 kg/s for cool-ing. This is impractical for normalcooling, but acceptable during combatwhere radiators are not available (andthe detectable coolant emissions areovershadowed by the main engines).The fighter requires 0.06 kg/s (assum-ing 50% efficiency). It only got about4 tons of MH fuel, so it gets just 18hours of cooling even if it doesnt useany hydrogen for propulsion.

Cooling through expelling coolantsis particularly useful for cooling lasersand railgun weapons during battle, es-pecially if they are disposable. It isworth remembering that if you need XJoules to harm your enemys ship youwill have to dissipate X(1) Joule ofheat at home, where is the conversionefficiency of the weapon.

9http://nemtos.ouvaton.org/techfiles/Cooling_Systems.pdf10I am ignoring the complications of different thermal capacity at different temperature

here.

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4 Detection

Detection of enemy spacecraft and as-sets is of central importance. Unlikeon a planet there is no intervening ma-terial that blocks radiation emissions,but the sheer volume of space compli-cates things. There are often tradeoffsbetween how wide part of the sky canbe scanned, how exact positions can bedetermined and the time it takes.

4.1 Sensors

Passive sensor systems collect energyfrom particular directions, sum the to-tal energy and try to tell whether thereis a statistically significant differencefrom the background.

The signal to noise ratio (S/N) fora sensor collecting photons such as aCCD sensor is

S/N =Ft

(Ft+Bnpt+Btnp +Dnpt+R2np

where F is the average photon fluxfrom the source (photons per secondper square meter), t is the time inter-val of the measurement, B is the fluxper pixel per second from the sky back-ground, Bt is the flux per pixel fromthe telescope itself, D is the dark cur-rent flux (due to the CCD array itself),R is the readout noise per pixel and npis the number of pixels11. Typicallyuseful observations begin to be possi-ble at S/N > 5, although guessing thatsomething might be there is possible atS/N = 2 or 3 (but estimates of energyand other properties will have 50% er-rors). In Eclipse Phase sensors can beassumed to be nearly perfect - B,Bt,D,and R are small, and every photon iscaught and turned into a measurableelectron.

For sensors with very low noise,a dark sky background and a rela-tively bright source the S/N ratio isFt. The time needed to reach a use-

ful S/N is on the order of 1/F sec-onds. If the sensor looks for pho-tons of wavelengths in an interval (its bandwidth), has a collecting ra-dius r and the source has a flux ofFs photons/m

2/ m/s then the timeis on the order of 1/piFsr

2. As rincreases the sensor can tell whetherthere is something there faster.

As an example, for infrared light = 1 m (most sensitive to a 3000K body), a bandwidth of = 1 m(a broadband sensor) and radius r = 1m the sensor needs about 300,000 pho-tons in order to detect the target in onesecond.

If the sky background is significantcompared to detector noise, then thetime needed scales as Bnp/F

2. Thistends to scale as 1/r2 for large tar-gets and 1/r4 for targets so small theyare limited by diffraction: larger pho-ton collectors are significantly faster.Typical backround sky fluxes in the so-lar system are between 109 106Watt/m2 per steradian in the mi-crowave to UV range of interest tospacecraft detection12. For lukewarm(300 K) objects the heat radiation inthe zodiacal light is the main confusingfactor, while hotter (3,000 K) objectsare confused by the background of re-flected sublight, faint stars and galacticcirrus. In order to be visible againstthe backround the flux density fromthe target needs to be above 12,000 -3 108 photons per square meter at thedetector. [develop!]

The time it takes to achieve a given

11http://www.physics.mq.edu.au/current/undergraduate/units/ASTR278/10_ASTR278_

JL_5_Sensitivity.pdf See also The Design and Construction of Large Optical Telescopes,ed. Pierre Y. Bely, Springer 2003

12Ch. Leinert, S. Bowyer et. al. The 1997 reference of diffuse night sky brightness, Astron.Astrophys. Suppl. Ser. 127, 1-99 (1998)

9

S/N is

tdetect = (S/N)2 1

F

(1 +

B

F

)The Planck radiation law gives that aspherical object of radius r and tem-perature T at distance R will producea spectral flux of

F =2pic

4(ehc/kT 1)( rR

)2

wavelength photons per square me-ter (here I assume is small enough;for broadband detectors F needs tobe integrated over all sampled wave-lengths). If there are several partsof the object of different temperaturetheir spectral fluxes can be added to-gether. Note that for best performancethe detector needs to look at a smallangle of the sky, since the backgroundflux will grow with the angle.

In order to calculate whether detec-tion is possible we need some estimatesof the thermal emissions of spaceships.

4.2 Thermal emissions

A spacecraft or other object radiat-ing at power P uniformly in all di-rections will produce a total flux ofFtot = P/4pid

2 W/m2 at distance d.A ship of temperature T and surfacearea A will radiate AT 4 W of ther-mal radiation, or Ftot = AT

4/4pid2.Typically a ship on full power will

have extended radiators at tempera-ture Tmax (and enough area to handlethe power). For reasonable Tmax be-tween 1,000 and 3,000 K the peak fluxis between 1-3 m IR radiation.

A ship that is merely coasting willhave an energy output much belowthese levels, but still significant. Eachbiomorph onboard produces around100 W, not to mention life support. Arough guess at the energy dissipationis about 1 kW per crew member. Thiswould put the Destroyer minimal en-ergy dissipation at 9 104 W and the

fighter at 1 kW. These limits can prob-ably be pushed for short spans, espe-cially by using heating internal coolingreserves (see below). The internal en-vironment will also be maintained ata temperature around 300 K throughheating or cooling, and this will likelycontribute to a harder to shield surfacetemperature.

4.3 Exhaust emissions

An accelerating ship will be leaving along trail of energetic hydrogen, chem-ical exhaust or plasma behind it, andthis will have detectable black-body ra-diation. Even if the ship itself is per-fectly caumoflaged the thermal emis-sion (and its doppler shift, allowing acalculation of relative velocity to theobserver) will be detectable.

Exhaust temperatures go down as

Te(t) =1

3

3A(t C)/K

where t is the time, A is the area ofa one second parcel of exhaust, K isthe thermal capacity of it (J/K kg)and C = K/AT 30 is a constant setso that at time 0 the temperature isthe initial exhaust temperature T0. Forhigh temperature exhaust it is a goodapproximation to treat it as releasingnearly all its energy instantly at tem-perature T0. This tends to dominateother radiation sources, especially forshort-wavelength emissions.

4.4 Sunlight

Internal heat becomes dominantroughly around the orbit of jupiter.Inside that orbit reflected sunlight is asignificant source of radiation of totalpower P = SA/R2 W at solar distanceof R AU, sunward ship area A and so-lar constant S = 1.366 103 W. If theship or object has albedo it will re-flect Preflect = SA/R

2 W, which is

10

can be very visible. The reflected sun-light might however be made highlydirectional, for example by placing aplane mirror in front of the object.

The absorbed sunlight, Pabsorb =(1 )SA/R2 kW, will turn into heatthat is emitted largely homogeneously.The resulting temperature due to theabsorbtion (assuming equilibrium anda spherical object of radius r) will be

T =

[S(1 )4R2

]1/4where is the emissivity (typicallyclose to 1 for dark bodies and down to0.02 for polished silver). In Earth orbitfor a high reflectivity object with =0.9999 with low emissivity = 0.02 theequilibrium temperature is 74 K. For ashiny metal object with = 0.75 and = 0.9 T = 168 K.

Inner system civilian spaceshipsand equipment are often made white orreflective to keep solar heating down,while outer system ships can be anycolor they like. Military ships will havereflective mirrors and dark, absorbentcoloring in order to be stealthy duringcertain phases of battle.

4.5 Sensor sizes and skyscanning

Ideally sensors should cover the entiresky and watch continously, but theyare limited by the conflicting demandsof having large apertures that cancollect many photons (a large lightbucket), a narrow angular field ofview to avoid too much backgroundnoise and the physical practicalities ofwhere to attach sensors to spacecraft.For habitats and defense systems it ispossible to put large numbers of bigsensors in place covering most of thesky, but a spacecraft and in particulara mobile asset such as a missile will nothave much space.

As an example, the WMAP satel-lite has a 52.8 arcminute beam sizefrom its 2.24 m2 sensors: this corre-sponds to 1 part in 67,827 of the entiresky. If it were to make a quick one sec-ond scan of each part it would need 18hours to do a full sky scan.

For a large ship in Eclipse Phase af-fixing a few square meter size sensorsdoes not appear to be a major problem;assuming 1% of the ship surface is usedfor sensors would allow the Destroyerto have 75 m2 sensors and the fighter0.14 m2. If we assume 8 sensors onthe Destroyer (one scanning each oc-tant of sky) they would have a 9.4 m2

area each. These would be the highsensitivity deep scan sensors: in directbattle more numerous, disposable sen-sors would be deployed for point de-fense control against incoming missiles.

Full sky scans at long distance donot have to be instant. Assuming a fullsky scan takes an hour, standard trans-ports can move 1,400,000 km, destroy-ers 2,900,000 km, fighters 40,000 km,fast couriers 5,800,000 km and missiles110,000 km. This is more than enoughto detect them before they can getclose. Once detected sensors can trackthem more intently, estimating theirtrue speed, course and other proper-ties.

A sensor covering one octant of thesky in one hour will watch 0.000436steradians of sky per second, a field ofview of about 1.2 degrees side.

One issue is detecting that some-thing is an interesting target and notjust random debris, a space habitat ora remote star. A first step is to com-pare the position with a detailed ob-ject catalogue, which allows the sensorto ignore all known objects. The nextstep is to compare the spectrum to pos-sible ship profiles. Active spaceshipshave different emissions from debris -infrared from internal energy produc-tion, short wavelength emissions fromthe drive, hot radiators, doppler effect

11

from high velocity etc. The scannerwill distribute its scanning time be-tween investigating objects of interestat length and jumping past empty orblocked directions.

4.6 Ship spectra and de-tectability distance

Putting the above considerations to-gether, we get the following approxi-mate fluxes from the ships of EclipsePhase.

I here assume the target ship is 1AU from the sun, 1 AU from the detec-tor, has albedo 0.5, that the radiatorsare fully extended and the ship is accel-erating maximally. The emissions con-sist of thermal emissions from the radi-ators, emissions from the ship (largelydue to absorbed sunlight), emissionsfrom the exhaust and reflected sun-light. The sensor is assumed to be a0.000436 steradian sensor with 1 m2

area, bandwidth 1 and desired S/N =5. Background noise is assumed to be106 W/m2 sr.

Manned ships can be detected overinterplanetary distances using an IRsky scan that takes one hour. Typi-cally ships have a characteristic multi-temperature spectrum: one peak forthe hot engine exhaust, one for the ra-diators, one for the reflected sunlight.

Reflected sunlight and the radia-tors are the biggest contributions: aship that has folded most of the radia-tors, reflects away sunlight with a mir-ror, powered down to a minimal leveland cooled the surface to a few Kelvinis significantly harder to detect. TheDestroyer is just barely detectable at0.09 AU distance (13,000,000 km) inthis case, and the fighter at 60,000 km.This means that stealth can be prof-itable, at least in the light of cloudcombat: a silent approach allows asurprise launch of attack assets in acloud large enough to be hard to evadefor the enemy. The fighter is notquite stealthy enough to reach a targetwithout being discovered, but it has achance to close to a very short rangeand launch a barrage of missiles.

4.7 Scanning limitations

These estimates assume overload-freesurroundings. As soon as the bul-lets start to fly any sensors this sensi-tive will be blinded (quite possibly de-stroyed) if they look anywhere close todetonations.

Note that not all directions areavailable for scanning: thermal sensorspointing at the sun or nearby plan-ets will be blinded. In deep spacethis is a minor problem, but in thevicinity of planets distributed sensorsare necessary to keep watch over localspace. The number of objects is alsofar larger, turning scanning into moreof a pattern recognition problem thana detection problem.

A ship that is accelerating is mostly

blind backwards due to the exhaustcloud. The blind angle is determinedby how fast hot particles from the ex-haust spread laterally relative to howfast they move backwards. It is afew times13 = 2 arctan(

kT/mv2e)

where T is the exhaust temperature, mthe mass of the exhaust particles andve the average exhaust speed. For fora hydrogen-oxygen engine at 3000 Kwith ve = 450, 000 m/s 0.3 whilefor an antimatter rocket with a millionK hydrogen plasma moving back at 108

m/s 0.1. Hence the blind angleis a few degrees across.

Sensors must also be placed so theyare not blinded by unfolded radiatorfins. These cover a far larger partof the sky but can be avoided by for

13Since the Maxwell-Boltzmann distribution extends beyond its average value.

12

Figure 1: Quanta received by a 1 m2 sensor at 1 AU from different ship types atfull acceleration. The red line is the zodiacal light background, giving a roughestimate of the noise.

Figure 2: Time until a 1 m2 sensor at 1 AU can confidently detect different shiptypes at full acceleration. The red line is 1 second scanning time.

13

example placing sensors at their ends(gaining parallax information in addi-tion).

4.8 Particle emissions

Antimatter annihilation does not justproduce the desired gamma photons,they also produce pions and muonsthat decay while radiating neutrinos.Fusion reactors also produce neutrinosfor some fusion reactions (pure helium3 reactions avoid it, but reactions withhydrogen may release neutrinos). Thismeans that even if a ship hides itsplasma tail it will radiate a neutrinosignature. Given Eclipse Phase tech-nology such as emergency farcasters weknow that neutrinos can be detectedover interplanetary distances. How-ever, it may be hard to get a good posi-tion from neutrinos, so the enemy willjust know there is an active antimatterreactor somewhere. Note that muondetectors would be effective at detect-ing active antimatter annihilation overdistances of a few kilometers, helpingmissiles to zoom in on active antimat-ter reactors.

4.9 Railgun projectiles

Railgun projectiles can in principlebe detected by their heat emissions.When accelerating a projectile to ve-locity v, (1/2)mv2 J of work is done.A fraction f ( 1 but > 0) of thiswill turn into heat. The temperaturebecomes T = (1/2)fv2/C, where C isspecific heat capacity ( 500 J/kg Kfor metal). If f = 1% and v = 10km/s, the result is bright 1,000 K pro-jectiles. For v = 100 km/s f mustbe much less, since otherwise the pro-jectile would be a vaporized mess. Iff = 104 the faster projectile will also

be 1,000 K.

The only thing making them hardto see is their small area. Assumingthe visible area is 10x10 cm, thenthey can be detected 4,487,940 km [check, update - old equation insteadgives 13.4*0.1*500 = 670 km away.That gives you 67, 6.7 or 0.67 secondsto point defence them. At least forprojectiles slower than 100 km/s thisis pretty OK for the defender. ]

current railguns have plasmaflashes 23,000-35,000 K, blackbody ra-diation 1.6-8.1 MW/cm2

4.10 Active sensors

Active sensors are a dead give-awayof where you are (unless they man-age to mimic natural EM activity),but the sensors can be put on an ex-pendable buoy (and triangulation oftargets from dispersed sensors is sig-nificantly more accurate). Stealthingagainst radar/twave/lidar/Xdar on allwavelengths is not going to be practi-cal.

Unfortunately active sensors havea shorter range than passive sensorssince the radiation emitted decreasesas the square of the distance and thenthe reflected radiation also decreaseswith the square of the distance, giv-ing a return signal that scales like1/d4. In order to double the range thepower has to be increased 16-fold. Theradar equation describes the limitingdistance where an active sensor can de-tect a target:

dradar =

[PSG

22

64pi3Pm

]1/4where PS is the power emitted, G isthe antenna gain, is the wavelengthused, is the radar cross section of the

14A way around this is to use bistatic radar, where the signal emitter and receivers are indifferent locations: sensors close to the target will receive a stronger signal. This requires thatthe sensor cloud is at least as large as the basic radar range to work.

15http://www.fas.org/spp/military/program/track/pavepaws.htm

14

target and Pm is the minimum receivedpower that can be detected14.

A current missile and satellitetracking system like PAVE PAWS15

uses a peak power of 582 kW, a fre-quency of 420 Mhz ( = 1.4 m) andhas a range of 5500 km. Scaling itup to a 1 GW radar would increase therange about 6-fold, to 35,000 km. Notethat shorter wavelengths have shorterranges: a teraherz version would havea range of just 113 km, while one using30 kHz would have a range of 650,000km.

While having a long range may ap-pear useful, it also includes more po-tential targets and more clutter. Ina space battle active sensors are moreuseful for pinpointing nearby incomingprojectiles and direct defensive fire onthem.

There is also a trade-off betweenrange and resolution. The angular res-olution is 1.22/L, making the kHzradar useless for detecting direction.The range resolution is c/2B whereB is the signal bandwidth, 1/.Shorter, more high frequency pulseshave higher bandwidth; a radar with 1m range resolution needs a frequencyin the 150MHz band. Point defenseradar needs very accurate position andDoppler measurements and will hencehave a short range.

4.11 Stealth

Reducing the profile of a ship or assetrequires reducing emissions that can beseen with passive sensors, and prevent-ing signals from active sensors frombouncing back with revealing informa-tion.

Stealthing against active sensorsworks if you can absorb the signalwell enough or reflect it in a safe di-rection, reducing the radar cross sec-tion. Thanks to metamaterials and ad-vanced materials science this is oftenpossible - for particular frequencies.

It is generally not possible to stealthagainst all frequencies, so if the en-emy uses the wrong sensors the invis-ible object will be obvious. Some mil-itary ships can reconfigure the surfacemetamaterials to adapt to expectedopponent strategies but the process isnot instantaneous, taking minutes tohours.

4.11.1 Anisotropic radiation

Averaged over time the total power ra-diated by an object must equal the to-tal power generated. It is possible tocool a ship surface (at an energy cost)and radiate the heat into particular di-rections, and to store heat into tanksfor a while. However, these stealthmethods have serious limitations.

A ship of power P that emits itspower as a blackbody will have a sur-face temperature T = [P/A]1/4 whereA is the total surface area. Usingonly a fraction f of this area increasesthe temperature of the hot surface byf1/4 and the flux will be P/f W/m2.

[extend]

4.11.2 Cooling

Cooling the surface using a cold reser-voir at temperature TC has maximumtheoretical efficiency = T/(T TC).The amount of work needed to reducethe heat of the surface is W = Qwhere W is the work if the heatpump and Q = KT is the changeof heat in the surface (K is the ther-mal capacity). Putting this togetherthe energy cost of cooling from Thot toTcool is

W = K

ThotTcool

T/(T TC)dT

= K

[TC ln

(Thot TCTcold TC)

)+ Thot Tcold

]A typical spacecraft temperature in

Earth orbit varies between 173 K and

15

393 K depending on light and shadow.Using an average of Thot = 283 Kand cooling to Tcool = 3 K using areservoir at TC = 1 K would takeW = K[2.15+280] J of energy. Assum-ing a radius 20 meter spherical shipwith specific thermal capacity around1 kJ/kgK, density 1.2 g/cm3 and thatonly the top cm need to be cooled, Kis about 60,000 and the total energycost for cooling down to stealth tem-perature is 17 MJ.

During stealth mode the power hasto be fed directly to the cooling tank.It will last for KCM/P seconds beforeincreasing in temperature by 1 Kelvin(where KC is the specific thermal ca-pacity of the cooling substance, M thetank mass). Since TC is so low it willbe enough for a few Kelvins of increaseto get above the desired stealth tem-perature Tcold (and the efficiency ofcooling drastically decreases). Usinghydrogen, which has the best specificheat capacity KC 12 kJ/(gK) andP = 30 kW (power of a one-man he-licopter), the ship uses up 2.5 kg ofcoolant per second. A one hour stealthepisode would require 1,440 kg (21 m3)of coolant. Assuming a more energeticship of P = 140 MW (Boeing 747) therate is 12,000 kg/s, and the above ra-dius 20 ship would at most (assumingit to be entirely filled with coolant) last187 seconds.

[cooling lasers are too inefficient- compare heat capacities. Electro-magentic thermal radiation has ef-fective volumetric heat capacity of32pi5k4T 3/15(hc)3. ]

4.12 Parallax

Determining the position of somethingin space will be dependent on resolv-ing its location. The resolving powerof a telescope is = 1.22/D where is the wavelength and D the diame-ter of the telescope. Parallax distanceerrors are d /2. For the ideal

case of a target at orthogonal distanced to a spaceship of length L with twotelescopes at the sides, 2L/d andwe get d 0.305d2/DL2.

For a baseline of 100 m, looking at = 105 (300 K blackbody radiation),D = 1 m and d = 10 km the uncer-tainty in distance is about 3 mm. Atarget at 1000 km distance will how-ever have uncertainty of 300 metersand at 10,000 km the uncertainty ismore than 30 km - far too much for anyuseful targeting of even a ship-sized ob-ject.

Turning the formula around, as-suming umax to be the maximumacceptable distance uncertainty, themaximum range where targets can behit is

dparallax = LDumax/0.305

Note that increasing L increasesdmax proportionally: having sepa-rate sensors imaging the same targetfrom widely separated vantage pointsgreatly extends the range from which itcould be hit. By using multiple sensorsand data fusion this range can be im-proved further to some degree. Largertelescopes and shorter wavelengths aremuch less effective.

4.13 Hiding from manyeyes

If the enemy is known to be watch-ing from a particular direction it mightbe possible to reduce detection prob-ability by carefully aligning a mirrorto hide the emissions of the ship (andavoid reflecting other bright sources),avoid accelerations and use stealthingagainst active sensors. Similarly it issometimes possible to approach (or de-part) from an observer along the direc-tion towards the sun, planets, stars oreven hiding in the zodiacal light (as-suming very low emissions; it has apower around 0.0005 W/m2).

16

However, as the parallax sectionshows, there is a great advantage inhaving multiple sensors scattered overa long distance. If the sensors are adistance L apart and the angular di-ameter of the background object is ,then the hiding ship must be more dis-tant than

dhide = (L/2) tan(pi/2 /2)to hide from both. In the case of theSun in Earth orbit, for L = 100mdhide =10.7 km, while a 10 km base-line forces a hiding distance of morethan 1,000 km. At Jovian distancethe distances are about 5.2 times largerand at Saturn 9.2 (as a rule of thumb,just multiply by the distance to thesun in AU). Hiding in planetary lightis harder: using Venus or Jupiter tomask an approach near Earth has adhide 300 km for 100 m baseline anda 30,000 km distance for a 10 km base-line. Using red giant stars will notwork within 400,000 km even for near-sighted L = 100 m spaceships16.

Multisensor detection depends notjust on failure to keep away from back-grounds that produce contrast, butalso whether the sensor network candetect a discrepancy and flag it as in-teresting. The larger the number ofsensors the higher chance one of themwill detect something interesting, butat the same time the amount of datato be processed and the number of er-rors will increase. If there are N sen-sors and each has a probability p persecond of generating a false positive(crying wolf) the probability persecond of avoiding false alarms will bejust (1 p)N . In practice very sophis-ticated data fusion algorithms can beused to get robust estimates, handlingfaulty or even suborned sensors (c.f.

the Brooks-Iyengar algorithm, whichworks up to N/3 faulty sensors). How-ever, the best algorithms also requiresignificant network bandwidth as eachsensor needs to communicate with ev-ery other.

[fake blackbodies] [metamaterials][occultation probability] [directionalradiation] [changing direction]

4.14 Spoofing and jamming

While correctly imitating the exhaustplume from an accelerating spacecraftis hard (the luminosity, spectrum anddoppler shift need to match the orig-inal, and this requires essentially thesame engine and performance as theoriginal) in the high-noise environmentof a space battle it is likely possible toproduce distracting or apparently sim-ilar phenomena. This might misleadsensors, targeting systems or point de-fenses.

[spoofing doppler] [spoofing back-ground] [confusing sensors]

It is easy to clutter radar and IRby releasing chaff that reflects signalsstrongly or in the right wavelengths.Exactly how well chaff works dependson the signal processing abilities of theenemy.

In particular, sensors are easilyblinded by bright detonations or de-liberate scorch attacks with beamweapons. This either permanentlydamages them or leaves them blind un-til they recover. Having replacementsensors that can be opened when thecurrent one are down will be necessary,but still introduces a short delay of ob-servation. Assuming a fully functionalC3I system one side can time closingsensor ports with the arrival of en-ergy from their own detonations, giv-

16Of course, the ship or object has to have an angular diameter much less than the back-ground for this trick to have a chance.

17As a simplistic example, sensors could be watching during even seconds and attacks timedto occur during odd seconds. In practice the pattern would have to be pseudorandom andtake lightspeed delays into account.

17

ing itself an advantage. This is easierwith QE signalling, but even withoutit some synchronization is possible17.

[overwhelming point defenses withlarge clouds of objects

If point defenses can drill X me-ters of object per second, then attack-ing with more than that (per second)will allow a hit on the ship. ]

18

5 Weapons

The forms of weapons available forranged space battle are lasers, railguns,particle weapons, missiles and fighters.Nanoweapons can be included in rail-gun and missile payloads.

5.1 Independent weaponbuses

Weapons can either be placed on theship, or on free-flying weapon buses.Placing offensive assets outside theship has the advantage of allowingshots at closer range and without riskto the main ship, but also limits theamount of energy they can providesince they are too small to contain re-actors.

Chemically stored energy is on theorder of 20 MJ/kg (20 GJ/m3) whilenuclear isomers could reach 10 GJ/kg(100 TJ/m3). A volume V weaponwith energy density and efficiency (likely < 0.5) will be able to fireN = V/Eattack attacks of energyEattack. The remaining (1 )V en-ergy becomes waste heat; the resultingtemperature (T = (1 )/C whereC is the heat capacity in J/m3) of theweapon will scale proportional to theenergy density of storage. Using largeamounts of energy makes the weaponheat up significantly, making it hard tohide (in addition to energy release fromorientation thrusters). Most weaponsare hence intended to fire just a singleburst attack and then coast away onthe recoil.

Missile buses also have this disad-vantage, but to a lesser degree: mis-siles can be launched quietly from astealthed bus using springs or cold gas

and their engines activated at a dis-tance from the bus. The main useof a missile bus rather than individ-ually drifting missiles is that the buscan be equipped with maximal stealth,hopefully drifting close to a tacticallyimportant volume before releasing themissiles18 (at the price of risking los-ing all missiles to an unexpected at-tack: some bus designs have an emer-gency eject function that releases themissiles prematurely if the bus comesunder attack).

[railguns in space can be made long][snap open phased arrays - since

disposable device, little need keep lowprofile after firing]

5.2 Lasers

[Expand, update]Laser weapons work by either heat-

ing a target to an unsustainable tem-perature (which requires a long lockon the same spot providing more than108 W/m2 over a second or more), arapid energy impulse causing a localplasma detonation (requires on the or-der of 1013 1014 W/m2) or drillingthrough the outer shell (requires an en-ergy density of LEeap W/m

2 where Lis the desired drilling distance per sec-ond and Evap is the energy needed forvaporisation per cubic meter).

5.2.1 Sources

Laser beams can be generated usingsingle laser cavities or phased arrays.Laser cavities contain a gain mediumwhere atoms, molecules or free elec-trons are placed in an excited energystate and then stimulated to decay toa lower energy state, releasing electro-magnetic waves that trigger other de-

18There is some potential for game theory in whether to launch all missiles or leave oneas a surprise later launch. A revealed missile bus is easy to hit and hence has little valueto store a surprise in, but in a situation where long-range defenses are busy (since there areapproaching missiles) it would be a low priority target. This leads to a mixed strategy equilib-rium where the missile side randomly leaves surprise missiles and the defender side randomlydecides whether to shoot at the worthless target.

19

cays and shoots out as a beam. Thisis relatively simple but has the prob-lem that to function the cavity needsto be resonant: the waves must beable to bounce between the front andback to produce a resonance, and thismeans the cavity itself will need to re-sist the laser power. Worse, puttingthe medium into an excited state in-volve big energy losses that also heatsthe medium. Hence large amounts ofcooling are needed.

Phased arrays make use of manysmaller lasers or antennas, producing abeam by combining many small com-ponents accurately. For lasers phasedarrays need to be manufactured usingnanotech metamaterials.

5.2.2 Beam optics

(Gaussian) laser beams have a diver-gence angle of /piw0 where is thewavelength and w0 its smallest width.If the laser is produced by a lensof width L it will produce a spotsize at distance d w = d/2piL andwith intensity piPL2/2d2 W/m2 ifP is the total beam power. Short-wavelength lasers remain sharp overlonger distances than long-wavelengthlasers, and in space it is possible to goall the way down the the vacuum fre-quencies of UV around 108 m thatare strongly absorbed by air and othermatter. Large lenses allow tightly fo-cused beams. However, with the nan-otechnology available in Eclipse Phasephased array lasers are possible: manysmall elements producing parts of thebeam, possibly focusing it closer to thesource if needed (this also allows higherpower densities at the target than atthe source, always a nice thing for aweapon). With a size L array focusedon the distance d target the focal widthis w0 = 2d/piL and the length of thefocal region is L2/42.

[example, showing that the regionis usually long]

When firing a laser the ideal beamwidth at the target is wopt = 2(v)d/c,the uncertainty in target lateral veloc-ity v times the lightspeed delay be-tween the target and laser (if there areQE-linked sensors shortening the de-lay this is reduced further down to aminimum of (v)d/c). If the velocitymeasurement is perfect there is still av < 2ad/c due to unknown accelera-tions since last observations (with QEthe factor 2 approaches 1). So the op-timal beam width will be

wopt = 2(v)d/c+ 4ad2/c2

However, close to the laser theactual beam width will be limitedby the focal width of the beam,giving a beam width at the tar-get of min(wopt, 2d/piL. The dis-tance beyond which velocity uncer-taity dominates is d = (c2/2piaL) (c(v)/2a). Typically this dependson the beam wavelength, with IRlasers being uncertainty-limited andUV lasers diffraction-limited over com-bat distances.

Now, this suggests another reasonyou dont want to fight close to planets:stationary defence stations can easilyset up pretty big phase array lasers,and then they can blast you very well.Ships could in principle unfold big ar-rays too, but I expect it is hard to bothpower them and keep them accuratelypointed while dodging incoming lasers,projectiles and missiles.

This kind of laser arrays still havethe problem that if you are uncertainof exactly where the enemy is (and weare talking about meters here) you willmiss him. So my previous calculationsstill apply - the Titan moon lasers canvaporize nearly anything, but if youare more than a few thousand kilome-tres from a sensor that pinpoints youand flying evasively, they will not beable to hit you.

[Energy requirements, size require-ments, range]

20

[calc neergy needed for 1 m pene-tration - kills smaller ships, scale up10 to hurt big ships]

[cycling time] [flash cooling]

5.2.3 Heating beams

A beam pulse of length tp that has acircular radial spread at the target rarrives after a delay td from observa-tion. During this time the target hasaccelerated with acceleration a, hav-ing an unknown relative velocity atd,which will have become a(td + tp) atthe end of the pulse. Assuming thebeam hits, it will move across the sur-face since the firing system is not prop-erly taking the unknown componentsof target motion into account. The to-tal area heated will be pir2 + 2ra(td +tp).

The velocity smear will reduce theheating, and preclude damage if thearea is more than K times the in-tended. This gives the requirement

td + tp < (K 1)pir/2a.

An 1 m2 object that of mass Mand specific thermal capacity K that isheated by power P Watt/s for tp sec-onds will reach temperature Ptp/MKif it cannot radiate away the heat.The time needed to reach a damag-ing temperature Tdam is roughly tp =MKTdam/P shorter pulse lengths re-quire proportionally higher power, upto the limits set by the firing array.

In reality the target will reach anequilibrium temperature where the in-flux equals the thermal radiation. P =T 4 Teq = [P/]

1/4 If Teq is too lowthere will not be any real damage. Therequired power is P > Tdam

4 (persquare meter).

A rough calculation of the timeneeded to reach this temperature isTeq = Ptp/MK tp = MK[1/P

3]1/4

tp needs to be shorter than this in or-der to avoid large energy losses.

If the vaporization energy per kgis Evap, in time tp it can vaporize to adepth z = Ptp/pir

2rhoEvap (assuminga tdtp tp[P/pirhoEvapz]

Diagram of r tp plane for fixedintensity r must be larger than diffrac-tion limit r must be smaller than limitof damaging power tp limited by heat-ing of array - too fast gets too hot tplimited by equilibrium temperature

r2 tp[P/pirhoEvapz] - able toget to damage depth

constraint due to targeting

5.2.4 Explosive beams

A very high power does damage not byheating the target but by vaporisingthe surface layer, creating a pressurewave fed by the beam.

5.3 Railguns

Rick Robinsons First Law of SpaceCombat: An object impacting at 3km/sec delivers kinetic energy equal toits mass in TNT.

Railgun projectile speeds: cur-rently a few kilometers per second,comparable to normal inter-spacecraftvelocity differences (even running intoa stationary pebble will do signif-icant damage to a ship). It is plausi-ble that in Eclipse Phase ship-launchedrailgun projectiles will be significantlyfaster, between 10 and 1000 km/s.Railgun projectiles need to be tens orhundreds of km/s in order to hit flee-ing ships, but can often move moreleisurely.

The kinetic energy from a 10 km/s1 kg impact is 50 MJ. At this point thekinetic energy starts to become biggerthan any (chemically) explosive forcethat can put in the projectile. 100

21

km/s is 5 GJ (about one ton of TNT)and 1000 km/s projectiles release 500GJ (100 tons of TNT).

A railgun projectile of massmmov-ing at speed v takes (1/2)mv2/ J tolaunch at efficiency . Conversely, withan energy E the projectile gets velocityv =

2E/m.

multi-barrel railguns can be madeshorter 6 km/s, 20000 G accel, 100kg projectile 90 m, 4 barrels, tempera-ture increase 450 K sleeve thickness 10mm, sleve radius 0.09 m, outside ra-dius 0.07m

They will penetrate to a distanceabout equal to the projectile lengthtimes the ratio of projectile to armourdensity (Newtons penetration law).This is actually a problem: the at-tacker wants to deposit all the energyinside the ship, so they must tune theprojectile length to the target armour.Too heavy projectiles will go straightthrough the ship. Sometimes havingno armour at all is the optimal strat-egy (just hope they do not hit any an-timatter containers). Too light projec-tiles and all energy gets deposited out-side the armour. [Merge with armorsection discussing this?]

[Energy requirements, size require-ments, range]

5.4 Particle weapons

Particle weapons produce beams ofheavy relativistic particles. Giventhe existence of personal particlebeams and fusion engines (which areessentially propelling proton-electronplasma in a beam) larger particle beamweapons are plausible. The advan-tage of particle beams is that theydeposit their energy deeper into thetarget, producing a stronger detona-tion and depositing Bremsstrahlungand secondary particles into sensitivenanosystems.

Unlike lasers they are hard to fo-cus and will tend to disperse over longdistances. Beam divergence angles areon the order of = 4.5 108T/Zwhere T is the beam temperature andZ is the atomic number of the beamparticles19. If a beam power P andinitial radius r is directed at a tar-get at distance d the intensity will be P/4pi(r+ kT/Z)2 PZ/4pik2Td2W/m2. The halving distance (wherethe energy per square meter has de-clined to half) for a proton beam (Z =1) is 280 km, for a mercury ion beam2,500 km.

[check this, compare destroyer en-gine]

[Energy requirements, size require-ments, range]

5.5 Missiles

Missiles can presumably accelerate atleast a few 100 G.

missiles that burst into clouds ofshrapnel as they approach or if hurt[calc velocity needed to cover target]

number of fragments that can bezapped on approach

[calc saturation with how manymeters laser point defenses can burnthrough - indicates how many missilescan be shot down per second and whatthe overwhelming number is]

can act as vector denial system

[Energy requirements, size require-ments, range]

5.5.1 Kinetic impactor rods

Kinetic missiles are little more thanmines, making use of ship delta vrather than their own acceleration.Missiles typically have small delta v (afew km/s), but this is enough to do se-rious damage.

either straight - armor piercingsideways - higher imact area, good for

19http://www.projectrho.com/rocket/rocket3x1.html#particle

22

weakly armored targets usually built sothat a hit disperses rods

5.5.2 Nukes and antimatter

Both nuclear warheads and antimatterhave roughly the same effect: a mas-sive release of gamma rays (and someparticles, especially from enhanced ra-diation warheads), with a small ex-panding cloud of plasma. The kineticimpact, heat blast and EMP that oc-curs in planetary environments is ab-sent making the kill radius small, onthe order of kilometers. A high preci-sion hit is hence required.

An x kt detonation provides an en-ergy of 4.19 1012x/4pid2 W/m2 at dis-tance d in about a microsecond. In or-der to produce impulsive shock dam-age (the vaporisation of material movesfaster than the speed of sound in thematerial) on the order of 1013 to 1014

W/m2 is required, producing a minus-cule kinetic kill distance of

dkkill = 5.77x

meters. The most likely effect is tomelt part of the facing surfaces, pos-sibly vaporising a thin layer. This mayor may not be disabling.

However, the amount of particlesis likely enough to kill biomorphs andsensitive nanotechnology at a longerdistance. Using the loose estimates inNuclear Rocket20 a conventional nu-clear weapon produce an X-ray flu-ence of 2.6 1027x/d2 and neutron flu-ence of 1.8 1023x/d2, and a unshieldedbiomorph will receive a dose of 1.78 109x/d2 Grays acute x-ray dosage and7.2108x/d2 Grays of neutron dosage21.

In order to exceed 20 Grays (immedi-ate disorientation) the ship needs to becloser than

drkill = 9433x

m. This does not take radiation pro-tection from the spaceship into ac-count: if the ship armor absorbs afraction fX of x-rays and fn of neu-trons the distance changes to drkill =max(9433

fxx, 6000

fnx) m. For a

ship with 5 cm of steel and 5 cm car-bon in the hull fx 0.62 and fn 0.8522, giving drkill = 7428

x (still

dominated by the x-ray damage in thiscase).

Using nuclear shaped charges(Casaba-Howitzer) jets of plasmatravelling at 10,000 km/s can be gen-erated, transmitting up to 5% of thetotal energy of the detonation as ki-netic energy in a cone with half-angle0.1 radians23. Assuming it can be di-rected accurately at the target, thiswould have a

dkkill = 667x

meters and (unshielded) radiation killout to

drkill = 21, 000x

m. Since the beam diameter is 0.2d m,it has a fairly broad cross-section forhitting a spaceship.

Nuclear warheads have the advan-tage of guaranteed stability, but anti-matter packs significantly more punchper weight. Theoretical limits on fu-sion warhead mass are 1 kg per kilo-ton (current warheads closer to 500 kgper kiloton). A warhead requires a 45

20http://www.projectrho.com/rocket/rocket3x1.html#nuke21Antimatter weapons would instead deliver nearly all energy as gamma rays.22f 2

i ti/vi , where the sum is over all materials, each with thickness ti and half-value

thickness vi for the relevant kind of radiation. This is at best an approximation since itignores the effects of secondary radiation, Bremsstrahlung from charged particles and othercomplications.

23Winterberg, Thermonuclear Physics, p.41, 122.24http://www.projectrho.com/rocket/rocket3x1.html#nuke

23

kg/kt missile with a volume of 0.036xm3/kt (based on extrapolations fromthe US trident missile24). Antimatterwarheads have 43 megaton yield per kgof mass, requiring 10 times the amountof shielding. Using previous numbersimplies a missile mass of 0.01 kg/ktand 8 106 m3/kt. So small chemi-cal missiles are likely not possible, butsmall antimatter candles might bepossible.

Given the stability risks of antimat-ter thermonuclear devices are likelypreferred on ships not using antimatterengines and not expecting significantcombat. Antimatter warheads how-ever have the advantage that if pointdefenses destroy them at close rangethey will go off anyway, doing damagethe less volatile nuclear warheads willnot do.

http://www.5596.org/cgi-bin/

nuke.php has a nuclear effect calcula-tor.

5.5.3 Rocks

It is entirely feasible to boost small as-teroids into orbits that will impact tar-gets. The upside is that the mass canbe significant, the damage enormous,and the asteroid is hard to stop (es-pecially if equipped with some pointdefenses). The downside that the im-pact can usually be predicted weeksor months ahead. Rocks represent arelatively minor threat to space habi-tats and aerostats, which can be movedout of the way. They represent a sig-nificant threat to planetary or aster-oid habitats that are not defended bydefense arrays. However, since theywould tie up the defense array (andproduce fragments) as they get withinrange they also provide an ideal timefor a conventional attack.

5.5.4 Relativistic missiles

More of a strategic weapon than a re-alistic space battlefield weapon, rel-ativistic missiles attempt to hit tar-gets at long distance using projectileswith a sizeable relativistic mass. Theirmain benefit is the impossibility of pro-tecting oneself from them, since theywould not be detectable from the tar-get until shortly before impact andthere is no way of stopping a verylarge mass moving fast (it would passthrough armor as per Newtons impactlaw, and any effect that splinters it willnot be able to make the fragments de-viate very far before it hits).

Fortunately, given the energy re-quirements of launching relativisticmissiles launch is very visibile, and QEallows monitoring of possible launchsites giving information long before themissile reaches its target. There arealso some doubts on whether relativis-tic missiles are feasible or not in EclipsePhase.

5.5.5 Leashes

A leash is a warhead that is attachedto the hull of the enemy ship, readyto explode on command or if tam-pered with. In principle just placinga missile with a functioning warheadon or near the ship works, but in prac-tice the extra reassurance from tam-perproofed antimatter containment ispreferred. Actually leashing a ship isquite a coup, and allows the leasher todictate terms to the leashed.

5.5.6 Nanoweapons

Nanomachines encased in diamondoidshells can resist accelerations up to108 1010 G, making delivery by highvelocity impactors possible if they canbe slowed just before impact. This canfor example be done by placing themfar back in a penetrating warheadorhaving an explosive charge accelerate

24

them in the opposite direction just be-fore impact. The main problem withnanoeweapons is that the amount thatcan be transfered is relatively smallcompared to the amount of defensivenanosystems onboard (see section 7)and that hives generally cannot surviveimpacts. This makes direct attacksusing disassemblers weak. The mainnanoweapon payloads tend to be sab-otage nanites or proteans constructinga position tracking transmitter.

5.6 Fighters

essentially a roving missile bus

5.7 Networked warfare

Generally, I expect space battles toinvolve extended networks of decoys,drones, munitions, sensors and what-not. The communications issues areserious: I expect it to be worth themoney to use FTL quantum com-munication to keep everything linked,untraceable, fast and unjammable.Qubits are a very strategic resource(incidentally, I doubt they can bestored without some very good nano- bad news for the Jovians, who proba-bly have to break a few rules to get it).Primitive forces that have lightspeed-limited networks are at a serious dis-advantage, and must also ensure thatthe enemy cannot detect *where* thecloud sends its messages (OTP en-crypted neutrino broadcasts in all di-rections instead?)

Burnsides Zeroth Law of spacecombat: Science fiction fans relatemore to human beings than to siliconchips. While the polities of the so-lar system have very good reasons notto like AIs, AGIs and infomorphs con-trolling weapon systems the militarybenefits often outweigh these consider-ations. On the strategic level there isa tension between avoiding developing

systems similar to the ones implied inthe Fall and keeping a military edge.

25

6 Armor

If we assume that the cost of drive sys-tems is large compared to the cost ofweapon systems, then it is clear thatwarships will tend to be as protected aspossible. This can be achieved throughheavy armoring or making them ableto evade (through stealth, maneouver-abilityor point defenses) attacks.

Armor needs to be able to han-dle hypervelocity impacts and beamweapons.

6.1 Kinetic impact

Any impacts faster than a few km/swill produce inertial stresses muchlarger than the material strength (evenfor extremely hard materials like dia-mond): for all practical purposes thearmor behaves like a liquid. This isnot just a military problem but alsoan issue for any ship moving at orbitalvelocity, since impacts with microme-teorites and other debris is potentiallydangerous (at speeds of 800 km/s 1gram impacts correspond to explosionsof 76 kg of TNT).

One way to handle this is Whippleshields: a relatively thin outer bumpershield placed a distance from the wallof the spacecraft. The incoming par-ticle will be shocked and may disin-tegrate, spreading its energy across alarger area of the inner wall. Shieldscan be stacked, further dispersing theenergy, and the space between themfilled with shock or radiation absorb-ing material. They are light but in-crease the spacecraft size; some shipsuse unfoldable Whipple shields. Theouter shields tend to be damaged bymicrometeors and conflict, but are ex-pendable and can be cheaply repairedafterwards.

Newtons law of impact depth

states that a projectile of length Lwith density P impacting a targetwith density A will penetrate to adepth L(P /A). This is applicablefor hypervelocity impacts where mate-rial cohesion can be ignored. Whilehigh density projectiles are possible(osmium achieves 22610 kg/m3, 2.9times steel) just extending the projec-tile into a spear guarantees deep pen-etration. This will also tend to pene-trate Whipple shields, since the desin-tegrating parts will just make way formore incoming spear.

Heavy armor needs to be thickerthan spear projectiles unless it is signif-icantly denser. This is hard to achieveacross a whole ship since the mass be-comes prohibitive: even a heavily ar-mored ship will just armor vital sys-tems (reactor, cooling, antimatter con-finement, possibly crew battlestations)and rely on redundant or repairablesystems elsewhere. Another approachis to use thick low-density shieldingin the form of water or reaction masstanks that would anyway be present.A hit will destroy the compartment butdissipate the energy.

Another approach is to have verylight armor and allow projectiles topass through, relying on redundantship systems to survive the damageand nanoswarms to repair it. How-ever, high velocity impacts turn intorapidly expanding clouds of shrapnel.A velocity v impactor of mass m willrelease 0.5mv2 J of kinetic energy. Ifall of it turns into kinetic energy offragments (and assuming them to havea Maxwell-Boltzmann distribution25)the average lateral velocity of the frag-ments will be v/

2, corresponding to

a approx35 cone. Hence this strat-egy works best for very long and thinships, where little of the volume can be

25There are m/mf fragments, where mf is the fragment mass. Each gets E = 0.5mfv2

kinetic energy if it is divided evenly. The average speed in the lateral direction isE/mf ,

producing the above formula.

26

affected.

6.2 Beam impact

[ Flash damage, impulse kill, drillingThe energy needed to drill through anobject is within a factor 2 of the heat ofvaporisation of the object best vapor-ization energy for mass carbon, 29.6kJ/g (boron even better) des 5 g/cm2(increase density?), burning a 1cm2hole requires 148 kJ and 20 millisec-onds combat conditions need largerspot to remain focused - 10 cm2 spotsaccepts uncertainty velocity 5 m/s ]

6.3 Conclusions

One approach is to add maximal ar-mor, making the target hard to dam-age. [how handle laser effects?]

Self-repair using nanomachines canrestore Whipple shields fairly rapidlyafter a battle, but is too slow to mat-ter during a battle.

27

7 General strategy

The conflict between different mili-tary assets can be described using theLanchester (square26) model:

Let B represent the number of as-sets of side blue and kb is the de-struction efficiency, how many enemyassets one blue asset can neutralize perunit of time. Let R and kr representthe red side. The both sides will suf-fer attrition like

B = krRR = kbB

Blue will win by attrition (i.e. reduceR to 0) if kbB

2 > krR2. There is a size-

able advantage in having more offen-sive assets, bigger than merely lethalassets. Concentrating forces so theycan focus their fire gives a big advan-tage (but the concentration needs to bediffuse enough so that no volume effectweapons can strike at them, somethingthe model does not include). The sidewith fewer assets still have a chanceof winning if they can make strikesthat disrupt the command and controlstructure of the other side, for exampletaking out the main communicationslinks (again outside the basic attritionmodel).

In practice the Lanchester equa-tions will be just a crude approxima-tion, since there exist weapons that de-stroy groups of enemies, some weaponshave synergistic effects (detonationsthat temporarily blind enemy sensorsyet provide active sensor informationabout enemy asset locations), there aresensors and other assets that do not at-tack yet are valid targets, and there aredifferent kinds of weapons that work

differently well against different tar-gets. The key point likely remains: theside that can manage to rapidly reducethe enemy assets early on has a largeadvantage.

In traditional warfare the sidewith smaller numbers can improve itschances if it splits up into hard-to-findunits and make local raids where ithas numerical superiority27. This ishard to do in deep space combat, sincevisibility is high, but an approxima-tion is to use missile buses and fightersto rapidly deploy local superior forcewhen needed. In orbital warfare it iseasier to do guerilla tactics, mak-ing surprise ambushes and retreats intounmonitored volumes.

26The name comes from the square in the winning criterion, the equations themselves arelinear. Ironically there are also the linear Lanchester model that describes conflicts wherethe loss rate is proportional to the product of the force sizes (and hence forms a pair of nonlin-ear equations). This model describes situations where combat occurs between pairs of units,and here the advantage in number is reduced to a linear relation.

27S.J. Deitchman, A Lanchester model of guerilla warfare, Operations Research 10:6, 818-827

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8 Cloud combat

Much of deep space battles occursbetween distributed assets driftingthrough space, forming two (or more)clouds passing through each other atmultikilometer per second speed. Iffriendly assets have not cleared enoughenemy assets ahead of the ship, it willlikely be hit.

Assets can be made very low-emission by cooling before launch,mechanical low-energy launching(springs, slow railguns, even bolalaunching), internal heat storage, lowemission electronics, being folded to-gether and equipped with stealth cov-ering. This can also cause a hard-todetect change in ship velocity thatcould grow into positional uncertaintybefore the battle.

A cloud launched at time T beforecombat at velocity vs will have radiusvsT . The passage through the cloudof a ship of velocity v takes 2vsT/vseconds. vs is presumably restrictedto less than 1 km/s if silent launch isused. If v = 10km/s and T = 1 daythe passage takes 4.8 hours, while forv = 100km/s passage takes 29 minutesand for v = 400 km/s just 7 minutes.

However, after the first side haslaunched their cloud the other side canchoose to do a maneouver with v >vs, avoiding the cloud and being ableto launch a cloud of protective assets.Obviously the first side can themselveschange the velocity to make this cloudworthless, and so on. If continued thisturns it into a pure ship-to-ship bat-tle. This is the preferred strategy fora side with limited spaceborne assetsand plenty of v to burn.

If a ship moving with velocity vtowards another ship launches an as-set cloud spreading with velocity vswhen at distance d, the cloud willhave radius vsd/v when it reaches theother ship. To evade it, the shipneeds to change its velocity by at

least vs (favoring rapid, possibly de-tectable launches). However, the den-sity of assets in a wide cloud willbe lower. If assets have a ranger and N assets are launched in thecloud, on average a ship will be withinrange of 3Nr2v2/4v2sd

2 assets if pass-ing straight through the cloud. Thisfavors a late and slower launch, pro-ducing a denser cloud. If M assetswithin range are needed to achieve awin, they should be launched withindistance

3N/4M(rv/rs), but if the

enemy is expected to have a highenough v budget earlier and sub-optimal launches are needed - muchhinges on accurately estimating howmuch fuel the enemy can spend on eva-sion and the performance of their en-gines.

8.1 Weapons vs. sensors

A cloud of density has average dis-tance to the nearest asset < d >=k1/3, where k 0.5622.

If there is resources C to produceassets per unit volume, and sensorscost 1 and weapons x, C = s + xwwhere s is the sensor density and wis the weapon density. Assumine afraction f of the resources are used tomake weapons, w = fC and s =((1 f)/x)C. The average delay be-tween a sensor detecting a target anda weapon has a chance to hit it is:

t = k

[1/3s +

1/3w

c+1/3w

v

]where v is the weapon velocity. Themiddle term represents a delay as in-formation is transmitted at lightspeedto the nearest weapon. If a QE link ispresent it is zero.

t = kC1/3[x1/3(1 f)1/3 + f1/3

c+f1/3

v

]To minimize t the expression in thebracket has to be minimal, leading to

29

the optimal weapons allocation

f =1

1 + (1 + c/v)3/4x1/4

If x is very low (cheap weapons)most assets should be weapons. Forweapons as costly as sensors 50%should be weapons if v c while 62% if they are lasers: a railgun ormissile-based system will tend to havenumerous weapons, hoping to be closeto a detected target. For more expen-sive weapons a smaller fraction is opti-mal.

[check QE term, check other case]

9 Hitting a dodgingenemy

Suppose a vessel observes another ves-sel at distance d and known velocityand fires immediately with a projectilewith velocity v (v = c for a laser)28.The time it will take for light from theenemy to reach the vessel and for theprojectile to reach the vicinity of theenemy is

t = d(1/c+ 1/v)

During this time the enemy will be ac-celerating with acceleration a < amaxin a random direction to escape a hit.After time t it has moved up to

rmax = (1/2)amaxt2

and will be within a sphere of radiusrmax around the position it would havehad if it had not accelerated. The at-tacking vessel does not know where in

the sphere the enemy is, but we willassume it knows rmax (for example byobservations of past accelerations andship type).

A projectile will hit anything alongits path through the sphere within itseffective cross section (this includesthe area A of the ship and the radius rpof the projectile, A+2piArp; if Nshots are optimally fired is multipliedby N). This is equivalent to selectinga point on the cross-sectional disk seenby the launching ship: the area of thedisk is pir2max, and an area will be af-fected. The evading ship will attemptto distribute its probability across thedisk uniformly29), so the probability ofhitting will be

phit = /pir2max =

4

pia2maxd4(1/c+ 1/v)4

When phit 1 the evader has a goodchance of avoiding a hit. While a highacceleration ability is useful, increasingdistance has a far greater effect.

The critical distance devasion wherephit approaches unity is

devasion =

[4

pi

]1/41

amax(1/c+ 1/v)

Inside this distance the probability ofhitting is large. The first term variesslowly with and is of the order unity.For = 1 m2 it is 1.06, for = 100m2 it is 3.36 and for = 1000 m2 5.97.

As an approximation, the enemyis possible to hit if d < devasion v/amax for projectiles and if d