Equation of motion of an interstellar Bussard ramjet with radiation and mass losses
aa r X i v : . [ phy s i c s . c l a ss - ph ] O c t Equation of motion of an interstellar Bussard ramjet with radiation and mass losses
Claude Semay ∗ Groupe de Physique Nucl´eaire Th´eorique, Universit´e de Mons-Hainaut,Acad´emie universitaire Wallonie-Bruxelles, Place du Parc 20, BE-7000 Mons, Belgium
Bernard Silvestre-Brac † Laboratoire de Physique Subatomique et de Cosmologie,Avenue des Martyrs 53, FR-38026 Grenoble-Cedex, France (Dated: October 29, 2018)An interstellar Bussard ramjet is a spaceship using the protons of the interstellar medium in afusion engine to produce thrust. In recent papers, it was shown that the relativistic equation ofmotion of an ideal ramjet and of a ramjet with radiation loss are analytical. When a mass lossappears, the limit speed of the ramjet is more strongly reduced. But, the parametric equations, interms of the ramjet’s speed, for the position of the ramjet in the inertial frame of the interstellarmedium, the time in this frame, and the proper time indicated by the clocks on board the spaceship,can still be obtained in an analytical form. The non-relativistic motion and the motion near thelimit speed are studied.
I. INTRODUCTION
During its motion through the space, an interstellar fusion ramjet collects interstellar ions (mostly protons) with amagnetic scoop (or ramscoop) to supply a fusion reactor able to fuse protons to obtain helium [1, 2, 3, 4]. Acceleratedreaction products are exhausted out of the spacecraft’s rear to produce thrust. An ideal ramjet could reach a velocityvery close to the speed of light c . If some energy extracted from the interstellar medium is lost in form of thermalradiation, the ramjet speed is limited to a value below c [5]. In this two cases, we have shown that analytical formulascan be obtained for the position of the ramjet in the inertial frame of the interstellar medium, the time in this frame,and the proper time indicated by the clocks on board the spaceship [6, 7]. These parametric equations are given interms of the ramjet’s speed.Moreover, it is natural to assume that a fraction of the collected interstellar gas can be lost during the work of theengine. The limit speed is then more strongly reduced [5] and the equation of motion further more complicated. Inthis paper, we show that analytical formulas can also be obtained for the position, the time and the proper time ofa ramjet with radiation and mass losses. The non-relativistic motion and the motion near the limit speed are alsostudied.The situation presented in this paper is more realistic than the case of a Bussard ramjet with a perfect fusion engine[6]. Nevertheless, it is not very probable that the kind of ramjet considered here could be ever built, even in the farfuture. Consequently, this work can be considered as an advanced exercise (from the point of view of calculus) inspecial relativity. The framework of interstellar space travel could be very attractive for undergraduate students. Thebasic equations are simple outcomes of momentum and energy conservation. Even if the solutions demand very heavycalculations, they have interesting properties to explore. However, let us note that the ramjet concept is potentiallytoo valuable to be simply discarded despite its tremendous technical difficulties. Some researchers have suggestedalternatives to the initially proposed proton-proton fusion ramjet [3, 4]: the catalytic ramjet (the use of catalyzedfusion reaction with a high rate), the RAIR (the use of nuclear fuel carried by the ship), etc. II. GENERAL EQUATIONS
In the following, all calculations are performed in the frame of the interstellar medium, considered as an inertialframe. A Bussard ramjet of constant mass M moves at speed v = βc through this medium, which contains protons atrest with a mass density ρ . The effective intake area of the ramscoop is denoted A . A fraction ǫ of the absorbed massis converted into useful kinetic energy in the hydrogen fusion reactor, a fraction λ is dissipated in form of thermal ∗ FNRS Research Associate; E-mail: [email protected] † E-mail: [email protected]
Typeset by REVTEXradiation and a fraction κ is lost in the interstellar medium (parameters ǫ , λ and κ have the same meaning than inRef. [5]). This means that, if a mass dm of protons at rest is scooped up from the interstellar medium, only an energy ǫ dm c is converted into ordered motion of the exhausted material.In the frame of the spaceship, we can assume that the thermal loss is isotropic and that the radiation carries nomomentum. For the ramjet, the energy absorbed due to the mass dm is dE = dm γ c , where γ = 1 / p − β , and theengine dissipates an energy equal to λ dE . Lorentz transformations [8, 9] imply that an energy γ λ dE = λ dm γ c and a momentum γ β λ dE/c = λ dm γ β c are lost in the interstellar medium frame.If a fraction κ of the collected interstellar gas is lost during the work of the engine, a useless mass κ dm is droppedout with no velocity in the frame of the ramjet. So, this mass has an energy κ dm γc and a momentum κ dm γβc inthe interstellar medium.During the time interval dt , a mass dm of protons, at rest, is scooped-up. The ramjet speed is then increased bythe quantity cdβ , thanks to the ejection of a mass (1 − α ) dm of helium with a speed wc , where α = ǫ + λ + κ. (1)Another interesting quantity is the fraction of matter, ω = ǫ + λ, (2)which is converted into pure energy by the engine.The conservation of momentum implies that [8, 9] M γ ( β ) βc = M γ ( β + dβ ) ( β + dβ ) c + λ dm γ ( β ) β c + κ dm γ ( β ) β c + (1 − α ) dm γ ( w ) wc, (3)where γ ( x ) = 1 / √ − x and dβ >
0. The conservation of energy leads to
M γ ( β ) c + dm c = M γ ( β + dβ ) c + λ dm γ ( β ) c + κ dm γ ( β ) c + (1 − α ) dm γ ( w ) c . (4)The collected mass is a function of the ramjet speed and is given by dm = A ρ βc dt. (5)
III. ACCELERATION
Taking into account equations (3)-(5), it is possible to compute the acceleration ϕ = dv/dt of the ramjet, measuredin the rest frame of the interstellar medium. With this aim, it is natural to define characteristic acceleration ϕ ∗ , time t ∗ and length x ∗ by the following relations ϕ ∗ = A ρ c M , t ∗ = cϕ ∗ , x ∗ = c ϕ ∗ = ct ∗ . (6)It is also useful to introduce the following notations u ′ = u (2 − u ) , ¯ u = 1 − u, (7)with 1 − u ′ = ¯ u .It is worth noting that ω is only 0.0071 for the most energetic known fusion reaction [1]. So, for all fusion reactions,we have ω ′ ≈ ω and ¯ ω ≈
1. If the particles collected by the ramjet are an ideal mixing of matter and antimatter,the mass reaction can be totally converted in pure energy ( ω = 1 and κ = 0) [6].Using the relation γ ( β + dβ ) = γ ( β ) + γ ( β ) β dβ, (8)the elimination of the reduced speed w of the exhausted reaction mass by the relation γ ( w ) (1 − w ) = 1 gives asecond degree equation in ϕ , whose physical solution is (in the following equations, the simplified notation γ meansmore precisely γ ( β )) ϕϕ ∗ = p γ − γ (cid:16)p γ − F ( γ ) − p γ − (cid:17) with (9) F ( γ ) = − λ ′ γ − κ ¯ λ γ + κ + α ′ . (10) Ε Λ Κ Ε Λ Ε Λ Κ FIG. 1: Limit reduced speed β l of the ramjet as a function of the parameters ǫ , λ and κ . Surfaces colored in dark grey to lightgrey correspond respectively to β l = 0 .
2, 0.4, 0.6, 0.8. Left: Arbitrary values of ǫ , λ and κ . Right: Values of ω ≤ . ω = 0 . This equation gives the acceleration of the ramjet as a function of its speed in the inertial frame of the interstellarmedium. If α = 0 ( ǫ = λ = κ = 0), the acceleration vanishes as there is no input of energy into the engine. Since ϕ = 0 if β = 0 ( γ = 1), an initial boosting is necessary for the ramjet. This is due to the fact that the reactionmass reaches the reactor thanks to the speed of the ramjet. In theory, a very small speed is sufficient to start theramjet. In practice, a fusion reactor could probably not operate correctly without a sufficient intake. Nevertheless,the ramjet could accelerate with an initial speed as low as 10 km/s [1]. Such a speed could be reached with usualchemical rockets or by future nuclear rockets. As expected, the acceleration also vanishes for β = 1 ( γ = ∞ ) since noobject can move at the speed of light.But this limit speed is never reached for a non-ideal ramjet. It is clear from Eq. (9) that ϕ = 0 when F ( γ ) = 0. F ( γ ) is a quadratic function in γ with two roots − Γ l and γ l such that F ( γ ) = λ ′ (Γ l + γ )( γ l − γ ) with (11) γ l = √ κ + λ ′ α ′ − ¯ λκλ ′ and Γ l = √ κ + λ ′ α ′ + ¯ λκλ ′ . (12)With a little algebra, it can be shown that 1 < γ l ≤ Γ l , (13)taking into account the condition α < γ ≥ F ( γ l ) = 0, F ( γ ) ≥ γ ∈ [1 , γ l ] and F ( γ ) ≤ γ ≥ γ l . So, γ l is a limit value: if γ = γ l ,the ramjet acceleration vanishes ( ϕ = 0); if γ < γ l , the ramjet speed increases ( ϕ > γ > γ l , the ramjet speeddecreases ( ϕ < β l = q − γ l . In the following, wewill only consider the realistic case β < β l or γ < γ l . Contrary to the case of an ideal ramjet, the velocity of a ramjetwith radiation and/or mass losses cannot be arbitrarily close to the speed of light [6] (see Figs. 1 and 2).Even with small values of the parameters λ and κ , the maximum speed can differ significantly from 1. This meansthat the slowing down of time on board the spaceship can become not large enough to allow interstellar travels in aperiod of time bearable for human beings. For values of λ + κ ≫ ǫ , the motion of the ramjet becomes non-relativistic.In our previous papers [6, 7], all equations were given as function of the reduced speed β . We find here moreconvenient to present the parametric equations of motion of the ramjet as a function of its reduced speed through thekinematical factor γ = γ ( β ). We will assume that, at a time t = 0 in the inertial frame, the position of the ramjet is x = 0 and its reduced speed is β = 0. Moreover, the clocks on board the ramjet indicate a proper time τ = 0. IV. TIME
Since dγdt = γ β dβdt = γ p γ − ϕc , (14)the solution of Eq. (9) is given by the following integral1 t ∗ Z t dt = Z γγ γ dγ ( γ − (cid:16)p γ − F ( γ ) − p γ − (cid:17) . (15)We give here the main steps of the procedure to solve this integral: • To multiply the numerator and the denominator of the fraction by the quantity ( p γ − F ( γ ) + p γ − γ − F ( γ ) at the denominator; • Using the relation (11), to write the fraction obtained as a sum of simpler fractions; • To integrate each of these new fractions and to simplify the result.The calculation is very heavy, but an analytical form can be found. To get a concise writing, it is useful to definesome intermediate quantities:Λ ( z ) = ¯ λ − z, (16)Λ ( z ) = ¯ λz − κ, (17) U ( z, γ ) = p ( z − γ −
1) + zγ − , (18) V ( z, γ ) = 2 q (Λ ( z ) − ¯ α )(Λ ( γ ) − ¯ α ) + 2¯ λ Λ ( z ) γ − κ Λ ( z ) + ¯ α ) , (19) W ± ( z, γ ) = 2 q (Λ ( z ) − ¯ α )(Λ ( γ ) − ¯ α ) ± λ Λ ( z ) γ − z Λ ( z ) + ¯ α ) , (20) S ± ( z, γ, γ ) = q Λ ( z ) − ¯ α ln ( γ ∓ W ± ( z, γ )( γ ∓ W ± ( z, γ ) , (21) R ( z, γ, γ ) = 1 √ z − z − γ ) U ( z, γ )( z − γ ) U ( z, γ ) + p Λ ( z ) − ¯ α z − z − γ ) V ( z, γ )( z − γ ) V ( z, γ ) . (22)We can then write tt ∗ = γ l R ( γ l , γ, γ ) + Γ l R ( − Γ l , γ, γ ) λ ′ ( γ l + Γ l ) − S + ( κ, γ, γ )2 λ ′ ( γ l − l + 1) − S − ( − κ, γ, γ )2 λ ′ ( γ l + 1)(Γ l − . (23)We can see on Fig. 2, how the speed of the ramjet tends toward the limit speed as time increases.It is interesting to look at the limit κ →
0. In this case, we have: γ l κ =0 −→ r α ′ λ ′ , (24) β l κ =0 −→ r α ′ − λ ′ α ′ , (25) q Λ ( κ ) − ¯ α κ =0 −→ √ α ′ − λ ′ , (26) q Λ ( γ l ) − ¯ α κ =0 −→ r α ′ − λ ′ λ ′ , (27)where α = ǫ + λ in the r.h.s. of these relations. Moreover, Γ l and γ l tend toward the same limit. In formula (23),one can see that, in the limit κ →
0, the coefficient of the functions R tends toward γ l α ′ β l and the coefficient of thefunctions S ± tends toward √ α ′ β l , where parameters γ l and β l are taken for κ = 0. So, one can see that expressionsfor the function t/t ∗ obtained in this paper when the parameter κ vanishes and given by formula (19) in Ref. [7] areidentical (the supplementary factor 1 / t (cid:144) t * Β Κ = Κ =
Λ =
ΩΛ =
ΩΛ =
ΩΩ =
FIG. 2: Reduced speed β of the ramjet as a function of the reduced time t/t ∗ spent in the inertial frame of the interstellarmedium, for the parameter ω = 0 . β = 0 .
1. The solid (dashed) curve corresponds to κ = 0 ( κ = 0 . κ , curves from bottom to top are respectively drawn for values 0.8, 0.5, 0.2 of the ratio λ/ω . V. PROPER TIME
Using the well known relation between the time and the proper time dτ = dt/γ [8, 9], Eq. (15) simplifies and theproper time is given by the following integral1 t ∗ Z τ dτ = Z γγ dγ ( γ − (cid:16)p γ − F ( γ ) − p γ − (cid:17) . (28)An analytical solution of this integral can be found with a procedure similar to the one used for the calculation of t/t ∗ . Using again the notations (16)-(22), a tedious calculation gives τt ∗ = R ( γ l , γ, γ ) − R ( − Γ l , γ, γ ) λ ′ ( γ l + Γ l ) − S + ( κ, γ, γ )2 λ ′ ( γ l − l + 1) + S − ( − κ, γ, γ )2 λ ′ ( γ l + 1)(Γ l − . (29)The link between the proper time on board the spaceship and the time spent in the inertial frame of the interstellarmedium can then be computed (see Fig. 3).Using relations (24)-(27) in formula (29), one can see that, in the limit κ →
0, the coefficient of the functions R tends toward α ′ β l and the coefficient of the functions S ± tends toward √ α ′ β l , where parameters γ l and β l are takenfor κ = 0. Again, one can see that expressions for the function τ /t ∗ obtained in this paper when the parameter κ vanishes and given by formula (24) in Ref. [7] are identical (the supplementary factor 1 / VI. DISTANCE
Using the relations cdt = dxβ = γ dx p γ − , (30)Eq. (15) can be rewritten into the form1 x ∗ Z x dx = Z γγ dγ p γ − (cid:16)p γ − F ( γ ) − p γ − (cid:17) . (31) Τ (cid:144) t * t (cid:144) t * Κ = Κ =
Λ =
ΩΛ =
ΩΛ =
ΩΩ =
FIG. 3: Reduced time t/t ∗ spent in the inertial frame of the interstellar medium as a function of the reduced proper time τ /t ∗ on board the ramjet, for the parameter ω = 0 . β = 0 .
1. The solid (dashed) curvecorresponds to κ = 0 ( κ = 0 . κ , curves from bottom to top are respectively drawn for values 0.8, 0.5, 0.2of the ratio λ/ω . Again, an analytical solution of this integral can be found with a procedure similar to the one used for the calculationof t/t ∗ and τ /t ∗ . New intermediate quantities must be defined: L = ¯ λ − ǫ , (32) A = ¯ α + ǫ , (33) Q = ¯ λ ¯ αL A (34) I ± ( z ) = ¯ λz ± (¯ α − κ ) z ∓ , (35) J ± ( z ) = ¯ λz ± (¯ α + κ ) z ± , (36) ν ( z ) = arcsin s LJ − ( z ) . (37)We can then write xx ∗ = X ( γ ) − X ( γ ) with (38) X ( γ ) = 1 λ ′ ( γ l + Γ l ) (cid:20) ln Γ l + γγ l − γ + ǫ I + ( γ l ) √ L A Π (cid:18) J − ( γ l ) L , ν ( γ ) , Q (cid:19) − ǫ I − (Γ l ) √ L A Π (cid:18) J + (Γ l ) L , ν ( γ ) , Q (cid:19)(cid:21) , (39)where Π is the incomplete elliptic integral [10]. The distance travelled by the ramjet in the interstellar medium as afunction of the proper time indicated by the on board clocks can then be computed (see Fig. 4).In the limit κ →
0, the first term of formula (39) reduces to γ l α ′ arg tanh γγ l , where parameters γ l and β l are takenfor κ = 0. Again, one can see that expressions for the function x/x ∗ obtained in this paper when the parameter κ vanishes and given by formula (32) in Ref. [7] are identical. Τ (cid:144) t * x (cid:144) x * Κ = Κ =
Λ =
ΩΛ =
ΩΛ =
ΩΩ =
FIG. 4: Reduced distance x/x ∗ travelled by the ramjet in the inertial frame of the interstellar medium as a function of thereduced proper time τ /t ∗ on board the ramjet, for the parameter ω = 0 . β = 0 .
1. Thesolid (dashed) curve corresponds to κ = 0 ( κ = 0 . κ , curves from bottom to top are respectively drawn forvalues 0.8, 0.5, 0.2 of the ratio λ/ω . VII. NON-RELATIVISTIC LIMIT
In the non-relativistic limit ( β ≪ ϕϕ ∗ ≈ p F (1) β, (40)where F (1) = ǫ (2 ¯ α + ǫ ) . (41)This number is clearly positive since ǫ > α <
1. Eq. (40) can be integrated to give xx ∗ = β − β p ǫ (2 ¯ α + ǫ ) with (42) β = β exp (cid:18)p ǫ (2 ¯ α + ǫ ) tt ∗ (cid:19) . (43)These expressions, characteristic of an exponential motion [6, 7], are valid provided t ≪ − t ∗ ln β / p ǫ (2 ¯ α + ǫ ). Let usremember that τ does not differ significantly from t at non-relativistic speed. VIII. ASYMPTOTIC MOTION
Near the limit speed, the term F ( γ ) is very small with respect to the term γ −
1, so we can write p γ − F ( γ ) − p γ − ≈ F ( γ )2 p γ − . (44)Equation (9) reduces then to ϕϕ ∗ ≈ F ( γ )2 γ . (45)It is worth noting that, in the case of an ideal ramjet ( λ = κ = 0), F ( γ ) = ǫ ′ and Eq. (45) is then the equation ofmotion of an uniformly accelerated spaceship with a constant proper acceleration ϕ ∗ ǫ ′ / M = 1000 t and an effective intake area A = 10 km , the characteristicacceleration is ϕ ∗ ≈ for an optimistic value of the particle density, let us say 10 /cm [1]. Within suchconditions, the asymptotic proper acceleration of an ideal ramjet with ǫ = 0 . β suchthat β s ≤ β < β l , with β l − β s ≪ β l . Using Eq. (8), the factor γ s = γ ( β s ) can then be approximately written γ s = γ ( β l − ( β l − β s )) ≈ γ l − γ l β l ( β l − β s ) . (46)So, we have β l − β s ≈ β l γ l ( γ l − γ s ) . (47)The condition β l − β s ≪ β l is then equivalent to γ l − γ s ≪ γ l β l < γ l . This inequality is relevant since γ l has a finitevalue when λ = 0 or κ = 0. We assume that the ramjet is at position x s , at time t s , and at proper time τ s when ithas a speed β s . Using the approximation (44), the integral for the position is given by1 x ∗ Z xx s dx ≈ Z γγ s dγλ ′ (Γ l + γ )( γ l − γ ) ≈ λ ′ ( γ l + Γ l ) Z γγ s dγγ l − γ , (48)since γ ≈ γ l under the integral. The corresponding solution is γ = γ l − ( γ l − γ s ) exp (cid:18) − λ ′ ( γ l + Γ l )2 x ∗ ( x − x s ) (cid:19) . (49)One can treat the time exactly in the same way. Remarking that γ/ p γ − ≈ /β l under the integral, the solutionis given by γ = γ l − ( γ l − γ s ) exp (cid:18) − λ ′ β l ( γ l + Γ l )2 t ∗ ( t − t s ) (cid:19) . (50)This equation can be directly deduced from equation (49), since x − x s ≈ β l c ( t − t s ) in this regime. Similar calculationsfor the proper time lead to γ = γ l − ( γ l − γ s ) exp (cid:18) − λ ′ β l γ l ( γ l + Γ l )2 t ∗ ( τ − τ s ) (cid:19) . (51)This equation can be directly deduced from equation (50), since t − t s ≈ γ l ( τ − τ s ) in this regime.Since β s ≤ β < β l , Eq. (47) holds also for β s replaced by β . Consequently, in Eqs. (49)-(51), quantities γ , γ l , γ s outside the exponentials can be replaced respectively by β , β l , β s . In the limit κ →
0, it can be checked that Eqs. (49),(50), (51) tend respectively towards formulas (31), (17), (23) in Ref. [7] (note a misprint in Eq. (31): the factor β l must be suppressed in the exponential). IX. FUNDAMENTAL INEQUALITIES
Since γ >
1, we have τ = Z τ dτ = Z t dtγ < Z t dt = t. (52)Moreover, since β <
1, we have x = Z x dx = Z t βc dt < Z t c dt = c t. (53)So, from these inequalities, we can conclude that functions (23), (29) and (38) are characterized by τt ∗ ( γ, γ ) < tt ∗ ( γ, γ ) , (54) xx ∗ ( γ, γ ) < tt ∗ ( γ, γ ) . (55)It is not evident to demonstrate these properties from the explicit form of these functions, except in the case of aperfect antimatter ramjet ( ǫ = 1, λ = κ = 0) [6]. Nevertheless, we have checked that they are fulfilled numerically inany case. X. SUMMARY
Formulas (23), (29) and (38) form the complete set of parametric equations of motion for a Bussard ramjet withradiation and mass losses, as a function of its speed. It is then easy to compute, for instance, the distance traveled bythe ramjet in the interstellar medium as a function of the proper time indicated by the on board clocks (see Fig. 4),or the link between this proper time and the time spent in the inertial frame of the interstellar medium (see Fig. 3).With radiation and mass losses, the ramjet speed cannot be arbitrarily close to the speed of light. A limit speed,only reached asymptotically (see Figs. 1 and 2), lowers the performance of a Bussard ramjet as an interstellar spaceshipor as a time machine for the exploration of the future [11].
XI. ACKNOWLEDGMENTS
C. Semay would like to thank the FNRS for financial support. [1] R. W. Bussard, Astronaut. Acta , 179 (1960).[2] J. F. Fishback, Astronaut. Acta , 25 (1969).[3] E. Mallove and G. Matloff, The Starflight Handbook , Wiley (New York, 1989).[4] G. Matloff,
Deep-Space Probes , Springer (London, 2000).[5] G. Marx, Astronaut. Acta,
131 (1963).[6] C. Semay and B. Silvestre-Brac, Eur. J. Phys. , 75 (2005).[7] C. Semay and B. Silvestre-Brac, Acta Astronaut. , 817 (2007).[8] F. W. Sears and R. W. Brehme, Introduction to the theory of relativity , Addison-Wesley (London, 1968).[9] C. Semay and B. Silvestre-Brac,
Relativit´e restreinte. Bases et applications , Dunod (Paris, 2005).[10] I. S. Gradshteyn and I. M. Ryzhik,
Tables of Integrals, Series, and Products , Academic Press (New York, 1980).[11] P. J. Nahin,