Thermal Radiation Studies for an Electron-Positron Annihilation Propulsion System

Jonathan A. Webb†

Embry Riddle Aeronautical University Prescott, AZ 86301

Recent studies have shown the potential of antimatter propulsion to produce specific impulses on the order of 107s. One such concept requires a radiation shield to absorb pure electromagnetic radiation in the form of 511 KeV photons. Different shield materials are analyzed and a thermal analysis is completed on the most probable shield material “tungsten”. The required shield size, mass, and maximum rocket thrust is determined as a function of the shield properties.

Introduction Recent studies have shown that it is possible to design a very high specific impulse rocket engine utilizing electron-positron annihilations. A spacecraft equipped with an !+

ee engine will annihilate it’s fuel to produce gamma rays that will be captured into a momentum transfer shield at the extreme end of a rocket system. Two cases have been considered to capture momentum from the resultant photons as shown in Figure 1.(1)

In the first case !+

ee beams are injected into a hemispherical dish that extends from 22 !! "# , and directed towards a collision at the center of the dish. At this point the

!+ee pairs annihilate producing two 511 KeV ! -rays back to back. Half the resultant photons

collide with the perfectly absorbing shield, while the other half are ejected into space from the annihilation point. The annihilation process is isotropic which produces a uniform distribution of photon collisions across the dish. Each collision with the shield causes a transfer of momentum to the dish boosting the spacecraft from one velocity to a higher velocity. This process yields an Isp of (~9.75x106s), assuming ideal conditions. In reality the incident photons would cause a scattering effect inside the shield. The photons will collide with atoms of the shield material causing a spray of electrons and lower energy photons to be ejected at random angles. Some of these photons and electrons will be ejected at angles opposite to the direction of motion causing some momentum to be lost. This will serve to decrease the specific impulse and thrust. Currently Monte Carlo simulations are being developed to determine the efficiency lost due to the scattering of electrons and photons.

The second configuration involves the use of a parabolic shield that extends from 5π/6 to -5π/6. In this case the shield is assumed to be a perfect reflector of the 511 KeV photons. This reflection process creates a large increase in the momentum vector of the incident photons. This concept yields a much larger Isp and thrust than with the previous case. Although this is by far the most efficient of the two cases current technology lacks the ability to reflect 511 KeV photons. The Isp from this case is (~3.06x107s).

Figure 1. Photon absorbing and reflecting shields. Dashed line represents reflecting shield.

A detailed velocity profile of these two configurations has shown incredible burnout velocities for relatively small amounts of fuel (i.e. electrons and positrons) Figure 2,3.(XX) The velocity profile graphs are derived from the equation.

!"

#$%

&

+

'=

x

xcv1

1 (1)

where

!!

"

coscos

1

1#$

%&'

(

)=

##

$

%

&&

'

(=

f

i

M

Mx (2)

and! is a running parameter iMpM used to graph the function, !cos is the mean angular

distribution of the momentum that is transferred to the dish in the forward direction by the γ-rays. In the case of the absorbing shield the average momentum capture is 2/π, and is 1.985 for the reflecting shield.

Figure 2. Velocity profile of the absorbing shield concept. α is the fraction of the spacecrafts fuel used as propellant, β is the burnout velocity as a fraction of the speed of light, <cos θ> = 2/π.

Figure 3. Velocity profile of the reflecting shield concept. α is the fraction of the spacecrafts mass used as propellant, β is the burnout velocity as a fraction of the speed of light, <cos θ> = 1.985.

Thrust and specific impulse for an electron-positron engine is written as

( )cmF &2

cos!= (3)

g

cI sp

2

cos!= (4)

Equations three and four are derived in further detail in a previous study at Embry Riddle Aeronautical University(1).

Shield Selection Due to the technical problems associated with reflecting 511 KeV photons, near term propulsive applications of this type of rocketry will use the absorbing shield concept. The design of the shield is based on the nuclear and thermal properties of different materials. The shield must be able to absorb as much of the incident energy as possible without melting and also be light weight. The ability of a shield to absorb energy is based on the radiation lengths of various materials. A radiation length is the thickness of a certain material required to absorb a certain amount of energy. The amount of radiation lengths required to absorb 100% of the energy incident upon a radiation shield is based on the equation;

0

0

xx

eEE

!

= (5) As is seen in figure 4 it requires roughly 5 radiation lengths of a material to absorb the majority of the incident energy.

Absorbed Energy Vs. Radiation Lengths

0

20

40

60

80

100

120

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Radiation Lengths (#)

Ab

so

rbed

En

erg

y (

% o

f in

cid

en

t en

erg

y)

Series1

Figure 4. Graph of Absorbed Energy Vs. # of shield radiation lengths

In order to efficiently absorb all the photon momentum incident upon the shield a shield must be made of a material with small radiation lengths and a low density. Much of the energy form the photons will also degrade into heat which will cause the shield to melt, so the shield should have a high melting point. On an initial investigation Tungsten appears to be the best candidate with a melting point of 3600 K a density of 19.3 gm/cm3 and radiation lengths of 0.35 cm giving the shield a total thickness of 1.75 cm at 5 radiation lengths. Assuming the shield is hemispherical the dimensions of the shield are then examined as a function of its area and radius. This is done by using basic geometrical equations of spherical volume and radius.

Shield Area Vs. Shield Radius

0

2

4

6

8

10

12

14

0 200 400 600 800 1000 1200

Shield Area (m^2)

Sh

ield

Ra

diu

s (

m)

Series1

Figure 5. Graph of shield area vs. radius

Knowing the area of the shield we can then determine the shield mass;

Shield Mass Vs. Inner Area (5 rad lengths)

0

50

100

150

200

250

300

350

400

0 200 400 600 800 1000 1200

Shield Area (m^2)

Sh

ield

Ma

ss

(M

t)

Series1

Figure 6. Graph of shield mass Vs. inner area

Shield Analysis It is important to note that equations 3 and 4 are assuming perfect efficiency and that the momentum will be completely absorbed in the direction of the photon impact over the shield. Unfortunately this is not the case, the photons incident on the shield will strike electron in the tungsten atoms. Following the collisions, the impacted electrons will be stripped from the atoms and ejected at random angles. The photons will exit the atom also at random angles with a lower energy and the same process will occur with electron-electron, photon-electron collisions until all of the energy is absorbed in the shield, Compton Scattering, Brehmstralling, and the Photo-electric effect will result from the process. The random diffraction angles of the photons and electrons will serve to change the cosine average and in some cases photons will be reflected in the backwards direction.

Figure 7. Diagram of a photon collision and electron ionization inside the tungsten shield

Before we can move any further in our analysis we must understand how much energy our shield is storing that needs to be dissipated. We can determine this through the equation;

mcm

QT =! (6)

The thermal energy stored as a function of shield area/radius is shown below.

Shield Inner Area Vs. Thermal Loading (Constant 3300 K)

0

20

40

60

80

100

120

140

160

0 200 400 600 800 1000 1200

Shield Area (m^2)

Th

erm

al

En

erg

y (

GJ

)

Series1

Figure 8. Graph of thermal energy stored in a tungsten shield vs. shield area

The thermal properties of the Tungsten shield place practical limitations on the propulsion system. The shield can not be heated past its melting point by the thrusting gamma rays. This places a limit on the maximum amount of propellant which can be injected into the hemisphere at

any one time. A constant shield temperature limits the propellant mass flow rate and thrust to a maximum level. The only option which exists to augment the propellant mass flow rate is to increase the surface area of the shield which the photons are incident upon. Tungsten melts at 3600 K and has an emisivity that ranges from 0.032 at 300 K to 0.354 at its melting point. The maximum steady state thrust at which the shield can cool itself by radiating photons at an equal power density to the incoming radiation can be found through the equation:(2) The power at which the shield can radiate electromagnetic energy as a function of shield area is written as; 4

TAP != "# (8)

Radiated Power Vs. Shield Inner Area

0

500

1000

1500

2000

2500

0 200 400 600 800 1000 1200

Shield Inner Area (m^2)

Rad

iate

d P

ow

er

(MW

)

Series1

Figure 9. Graph of shield area vs. radiated power

2

42

c

TAm

!"=& (7)

Combining equations 7 and 3 derives the maximum thrust that can be obtained before the shield will melt down.

c

TAF

4cos !

="#

(8)

Shield Inner Area Vs. Thrust (Radiative Cooling)

0

0.5

1

1.5

2

2.5

0 200 400 600 800 1000 1200

Shield Inner Area (m^2)

Th

rust

(N)

Series1

Figure 10. Graph of shield area vs. thrust

Unfortunately one problem arises in the radiative cooling technique, the shield geometry. Photons radiated from the shield may be radiated at any angle from the shield surface. Some fraction of the radiation emitted from the surface of the shield may be re-radiated into the opposite side of the shield, not allowing the shield to cool by the proper amount. This will serve to decrease the propellant mass flow rate and thrust in order to prevent the shield from melting.

Figure 11. Shield with electromagnetic energy re-radiated back into its structure

As a result it may be more appropriate to make the shield flat as opposed to hemispherical. Unfortunately this means that if a photon is ejected at a large angle from the annihilation point it may miss the edge of the shield and decrease the cosine average.

Figure 12. Schematic of flat shield in relation to a photon ejected from the annihilation point.

The angle of photon ejection form the shield can be determined from the equation shown below.

22

1sin

DR

R

+=

!" (9)

Where D is the distance of annihilation from the shield and R is the maximum radius of the shield. This can be plugged into the cosine average shown as;

!=max

min

cos1

cos"

"""

#" d (10)

Combining the two equations results in;

22

2cos

DR

R

+=!

" (11)

Assuming the annihilation point is 2 cm from the shield, the cosine average as a function of shield radius can be seen in figure 13. As is seen the cosine average remains very close to the 2/π average until the shield radius becomes smaller than 0.2 meters. Unfortunately the shield area is smaller for a flat shield which means that the shield can not radiate as much energy and the thrust level will drop form the hemispherical case.

Shield Radius Vs. Cosine Average (Small Shield)

0.585

0.59

0.595

0.6

0.605

0.61

0.615

0.62

0.625

0.63

0.635

0.64

0 0.2 0.4 0.6 0.8 1 1.2

Shield Radius (m)

Co

sin

e A

vera

ge

Series1

Figure 13. Cosine average vs. flat shield radius assuming annihilation point is 2 cm from the shield.

In any of the radiative techniques the thrust to weight ratio becomes so unacceptably small that it would take hundreds of years to accelerate to peak velocities, although the specific impulse is exceedingly high. A method needs to be developed to increase the thrust to a level that accelerates a spacecraft much quicker. A possible method of augmenting the thrust is through evaporative cooling using liquid Hydrogen or some other working fluid. If we assume that the shield is raised to a peak temperature of 3300 K and we use liquid Hydrogen stored at 16 K we can solve for the resulting energy transfer, thrust and specific impulse. First we find the normalized heat transferred (Power) from the shield to the LH2. Since the shield area is indeterminate, we will find the power as a function of shield area.

( )2LHshield TThAQ !=& (12)

plugging in parameters listed for liquid Hydrogen

2689640

m

WxAQ =&

The power convectively transferred as a function of internal shield area is shown below:

Figure 14. Energy Transferred to liquid Hydrogen as a function of Tungsten shield inner area. This graph also gives us an idea of how much the Tungsten shield will be cooled.

Next we find the mass flow rate of LH2 required to maintain the above energy transfer rate(3).

( )2

2

LHshieldp

LHTTC

Qm

!=

&

& (13)

combining equations 1 and 2 and solving generically we arrive at:

p

LHC

hAm =

2& (14)

solving eq.3 with the parameters given for liquid hydrogen and as a function of shield area we obtain:

!"

#$%

&=

s

kgxAmLH 021.0

2&

The required mass flow rate as a function of inner shield area is shown below:

Figure 15. Mass flow rate of liquid Hydrogen required into the shield as a function of inner shield area.

Next we find the thrust created by the expanding Hydrogen and its effect on the engines specific impulse. First we must find the resulting expansion rate of the Hydrogen gas. Knowing this in conjunction with the mass flow rate we can then find the thrust via the equation below:

222 HHHVxmF &= (15)

We can find an approximation for the rate of expansion from the Kinetic energy theorem shown in equation 5.

2

2

2

1

HVmE = (16)

Assuming that the thermal energy absorbed by the Hydrogen becomes kinetic energy we can rewrite equation 5 to find the initial rate of expansion for the Hydrogen gas.

2

2

2

LH

H

m

EV

&= (17)

Where mm

LH=

2&

Assuming that equation 5 is normalized to a one second time frame EQ =& we find

( )22

2 LHshieldpH TTCV != (18) We find that regardless of the shield area, with the properties given for this case, the Hydrogen will expand at a velocity of:

s

mVH

32.81042=

Combining equations 1, 3,and 7 we arrive at

( )22

2 Hshieldp

p

H TTCC

AhF != (19)

Figure 16. Thrust produced from the expanding Hydrogen gas over the Tungsten shield as a function of shield area/mass flow rate.

Equation 8 describes the thrust imparted on the engine from the expanding Hydrogen. The combined thrust from the Hydrogen expansion and antimatter annihilation is written as:

( )2

22

cos

Hshieldp

peeT TTC

C

AhcmF !+= !+&

" (20)

The thrust produced from the absorption of photons over a Tungsten shield is only a partial function of shield area, making a graph of its effect on thrust irrelevant. Next use the following equation to find the specific impulse imparted by the Hydrogen via:

gm

FI

LH

spH2

2&

= (21)

Combining equations 8 and 10 we arrive at:

( )

g

TTCI

Hshieldp

spH

22

2

!= (22)

We find that regardless of the shield area the Hydrogen will expand with a specific impulse of: sI

Hsp826

2

= Next we need to find the Hydrogen expansions effect on the net specific impulse of the engine.

( )gmm

FFI

eeLH

eeH

sp

+!

+!

+

+=

&&2

2 (23)

Combining equations 3, 9 and 12 we obtain:

( )

( )AhmCg

TTCAhcmCI

eep

Hshieldpp

sp+

!+=

+!&

&

2

22cos2

" (24)

For large thrusts we find that the thrust to propellant mass ratio is so insignificant compared to the Hydrogen that the specific impulse of the rocket is always 826 seconds. This is a significant reduction from the millions achieved with the radiative technique, but is still over twice as high as current engines. The engine in effect become a nuclear thermal engine very similar to the NERVA engines of the 1960’s but instead of using Uranium fission for thermal energy this concept uses positrons. An antimatter thermal rocket will have the same performance as a nuclear thermal rocket, but will not require the mass of the reactor as in the nuclear thermal case, this may serve to slightly increase the thrust to weight ratio. A possible method to increase the thrust to weight ratio if this concept is to slice a shield into ten separated shields with a combined thickness of 5 radiation lengths. This will effectively multiply the shield area by 20. In the case of a shield 10 meters in radius the shield will have a surface area of 10,000 m2 a shield mass of 170,000 kg, a convective energy transfer of 6,800 MW and a thrust of 1.70 MN.

Figure 17. Simplified schematic of a sub-shield concept for a positron thermal rocket

Figures 18-19 show the performance of a positron heated thermal rocket for ΔV’s of 0-20 km/s. The velocities studies are in the range of total ΔV’s required for one way Hohman transfers to all eight planets within the solar system. It should be noted that the mass of positrons required in figure 19 may be available in the very near term future.

Initial Mass in Low Earth Orbit/Hydrogen Propellant Mass Vs. dV (400 Mt Payload)

0

500

1000

1500

2000

2500

3000

0 5 10 15 20 25

Change in Velocity (km/s)

IML

EO

/Hyd

rog

en

Mass (

Mt)

IMLEO

Liquid Hydrogen Mass

Figure 18. Total mass in low earth orbit and Hydrogen propellant required vs. change in velocity for a positron thermal rocket.

Positron Mass Vs. Burnout Velocity

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25

Change in Velocity (km/s)

Po

sit

ron

Ma

ss

(m

icro

-gra

ms

)

Series1

Figure 19. Mass of positrons required vs. change in velocity

Conclusions

From studies it seems very feasible to cool a tungsten radiation shield radiatively, unfortunately this technique yields very high specific impulses this will limit the thrust of a spacecraft to impossibly small values. Convectively cooling a shield with a working fluid such as Liquid Hydrogen seems to be very possible. This technique will hugely augment the thrust of a spacecraft, but will lower the specific impulse of the engine from millions of seconds to 826 seconds. On the positive side, this lowers the positron requirements to a level that can be achieved with present technology. The best solution to augment thrust levels without degrading specific impulse seems to lie in the reflecting shield concept. This will allow for tens of millions of seconds of specific impulse and very high thrust levels. Unfortunately at present the technology required to reflect 0.511 MeV photons does not exist. Further research should be conducted into this area to open up the solar system to high mass aggressive manned spacecraft exploration.

Bibliography 1Smith, D., Webb, J., (2001) The Antimatter Photon Drive A Relativistic Propulsion System, AIAA Paper 2001-3231. 2Holman, J., (2002) Heat Transfer Ninth Edition, Boston, MA. McGraw Hill Inc. 3Humble, R., Henry, G., Larson, W., (1995) Space Propulsion Analysis and Design, New, York, NY McGraw Hill Inc.