A closed set of fluid equations with radiation reaction force (RRF) are constructed from the moments of the appropriate single particle kinetic equation describing a relativistic degenerate (high density) electron plasma. The closure, in analogy with the Maxwellian closure for non-degenerate plasmas, is affected via a parametrized Fermi-Dirac distribution. It is shown that the degeneracy increases RRF just as will be predicted from the so-called “Compton Rocket” effect.

A relativistically hot plasma, exposed to intense radiation, experiences a higher radiation pressure force compared to a cold plasma. The phenomenon known as the “Compton Rocket” was originally discovered by O'Dell.1 In O'Dell's work, the force acting on the plasma was derived using a phenomenological, test-particle approach. Energy-momentum conservation is employed to address particle–photon interaction, and the resulting force is integrated over the distribution function. It was shown that incident high-frequency photon flux, entering a hot plasma, undergoes Thomson scattering that redirects it back toward the primary source. The plasma exhibits a tendency to propel itself away from the radiation source, drawing momentum primarily from the anisotropic loss of its internal energy. Applying similar but a relativistic covariant formalism, it was demonstrated in Refs. 2–4 that this process is accompanied by plasma cooling. Note that the force acting on the plasma fluid may not only accelerate the plasma but also induce deceleration at velocities surpassing certain (model-dependent) bulk flow values. This deceleration occurs due to the photon flux aberration, commonly known as the radiative drag. Methods and results obtained in Refs. 1–4 have been employed to study the dynamics of producing relativistic outflows (jets) from highly luminous radiation sources, such as active galactic nuclei (AGNs) or compact galactic objects.5 

In this paper, we explore how the radiation reaction force (RRF) would be modified, likely, intensified by the “Compton Rocket” effect. Not considered in the original calculation, the RRF, by increasing the degeneracy and effectively raising the electron degeneracy “temperature” (even in a thermally cold plasma), might be a stimulant for a stronger effect.

The basic model we investigate is contained in the manifestly covariant relativistic hydrodynamic equations that were derived taking into account the radiation reaction force (RRF) for an electron plasma.6,7 The model equations are simply the moments of the relativistic kinetic equation with RRF in the Landau–Lifshitz (LL) form. In the limit where the plasma is immersed in the field of incoherent photon fluxes, the derived equations can be readily reduced to those presented in Refs. 1–4.

It is interesting that in Ref. 7, we did demonstrate that RRF, when radiation consists of coherent ultra-strong electromagnetic (EM) pulses, can lead to an effective acceleration of electron (electron–positron) plasma. Since the relativistic hot plasmas have excess internal energy, the radiative thrust of RRF increases just as in “Compton Rocket” effect.

Before getting into detailed calculations, let us review the phenomenology of the compact astrophysical objects—neutron stars, white dwarfs, and AGN—in which the electron number density is believed to be in the range from 1026 to 10 34 cm 3. The average inter particle distance can be considerably smaller than the electron thermal de Broglie wavelength. The electron gas in this case is degenerate, must obey Fermi-Dirac statistics, and plasma particles are weakly coupled.8,9

The electrons in a high density degenerate plasma can have relativistic energies, even when they are thermally cold. The Fermi energy of degenerate electrons is ϵ F = m e c 2 ( γ F 1 ) with γ F = 1 + p F 2 / m e 2 c 2, where pF is the Fermi momentum determined by the rest-frame electron density n via p F = m e c ( n / n c ) 1 / 3,10 where n c = m e 3 c 3 / 3 π 2 3 = 5.9 × 10 29 cm 3 is the normalizing critical number-density. If n n c, the electrons acquire non-thermal energy that could become relativistic when n gets larger. When electrons have relativistic speeds, the power radiated due to acceleration increases dramatically. Relativistic electrons emit radiation anisotropically effectively enhancing the radiation reaction force acting on the plasma.

It has been widely accepted that x-ray emission might appear from accreting white dwarfs and pulsars. Details of this process are discussed in the review article of Mukai11 and references therein. The accretion matter, falling onto the star's surface, generates high-frequency radiation via Bremsstrahlung. This radiation penetrates the star's interior, which is mostly made up of very dense degenerate plasma. Consequently, the investigation of relativistic degenerate plasma (present in the interiors of compact objects) immersed in strong radiation, is a subject of considerable astrophysical interest—it could, in fact, be an excellent laboratory for the manifestation of the Compton Rocket effect.

The closed set of fluid equations is derived by taking moments of kinetic equation and affecting the closure using a local Fermi-Dirac distribution. The inclusion of RRF in the relativistic kinetic equation is done through modifying the equation of motion for a single electron:
m c d u d s α = e c F α β u β + g α ,
(1)
where u α is the αth component of the contravariant reduced four-momentum u α = [ γ , γ u / c ] , γ = ( 1 u 2 / c 2 ) 1 / 2, where u is the particle velocity, d s = cdt / γ , F α β is the electromagnetic tensor [ u α u α = 1 since the metric is g α β ( 1 , 1 , 1 , 1 )], and g α is the contravariant component of the radiation reaction force. The following expression for g α was first derived, perturbatively, by Landau–Lifshitz;8 it was, however, later shown by Spohn12 and Rohrlich13 to be exact for a single point particle. Following Ref. 6, LL force is written conveniently as
g α = 2 e 3 3 m c 3 F α β x γ u β u γ + σ c ( T ¯ α β u β T ¯ β γ u α u β u γ ) ,
(2)
where σ = 8 π e 4 / 3 m 2 c 4 is the Thompson cross section and T ¯ α β = 1 4 π [ F α γ F γ β + 1 4 g α β F γ δ F γ δ ] is energy-momentum tensor of the EM field.

The LL equations [Eq. (1)] offer several advantages over the Lorentz–Abraham–Dirac formulation (LAD).14 Specifically, in contrast to LAD, these equations are of the second rather than third order, and they do not permit a self-force in the absence of an external field. Additionally, as demonstrated in Ref. 15 (see also Refs. 7 and 16), Eq. (1) possesses exact solutions for plane electromagnetic fields. We would like to emphasize that from expression (2), one can easily recover the recoil force acting on electrons during Thomson scattering of photons, provided that the photon energy in the particle rest frame ( ω) is much less than the electron's rest mass energy ( m c 2).

Having delineated the benefits of the LL equation, our focus now shifts to the kinetic depiction of a radiating plasma. This kinetic model remains valid under the condition that the average interaction energy of particles is markedly lower than their average kinetic energy. This condition applies not only to classical hot and dilute plasmas but also to highly dense degenerate plasmas.17 Assuming the validity of the weak coupling condition among plasma particles, the kinetic collisionless equation for a single-particle electron (or positron) distribution function, influenced by RRF as follows: evolves as
p α f x α + p α ( e c F α β p β + m c g α ) f = 0 ,
(3)
where p α = m c u α is the particle four-momentum.
The standard moments of the distribution function: the flux
Γ α = c d 3 p p 0 p α f
(4)
and the energy-momentum tensor
T α β = c d 3 p p 0 p α p β f
(5)
follow the standard flux conservation law, and the equation of motion
Γ α x α = 0 ,
(6)
T β α x β e c F α β Γ β = m c 2 g α ,
(7)
where
g α = d 3 p p 0 g α f .
(8)
Note that the averaged radiation reaction force
F rad α = m c 2 g α = σ c T ¯ α β Γ β σ m 2 c 3 T ¯ β γ M β γ α
(9)
is determined in terms of the higher moment (a typical feature of moment equations)
M α β γ = c d 3 p p 0 p α p β p γ f .
(10)
To truncate the chain of momentum equations, the Fermi-Dirac thermodynamic closure is applied. This closure assumes that the distribution function can be approximated by the Fermi-Dirac distribution with varying parameters, such as density, temperature, and fluid element velocity, and can be written as
f e q ( p ) = g ( 2 π ) 3 [ exp ( μ T + p α U α T ) + 1 ] 1
(11)
with the following local parameters: g = 2 s + 1 [and g = 2 for electrons (positrons)], the chemical potential μ, the temperature T (measured in energy units), and the hydrodynamic four velocity U α: U α = ( c γ v , γ v V ) , γ v = ( 1 V 2 / c 2 ) 1 / 2 ( U α U α = c 2 ). This distribution yields the flux four vector: Γ α = n U α, while for energy momentum tensor of fluid, we have
T α β = w U α U β c 2 g α β p ,
(12)
where w = n ε + p is the enthalpy per unit volume, n ε is the proper internal energy density of the fluid, and p is the pressure. As shown by Cercignani and Kremer,18 for these parameters, the distribution (11) yields the following general relations:
n = 4 π ( m c ) 3 g h 3 J 21 ,
(13)
n ε = 4 π m 4 c 5 g h 3 J 22 ,
(14)
p = 4 π 3 m 4 c 5 g h 3 ( J 22 J 20 ) ,
(15)
where
J n m ( T , μ ) = 0 d ϑ sinh n ϑ cosh m ϑ exp ( μ T ) exp ( m c 2 T cosh ϑ ) + 1 .
(16)
One may also calculate the third moment:
M α β γ = A 1 U α U β U γ + A 2 g { α β , U γ } ,
(17)
where A1 and A 2 are found to be
A 1 = 4 π m 5 c 3 g h 3 ( 2 J 23 J 21 ) ,
(18)
A 2 = 4 π 3 m 5 c 5 g h 3 ( J 21 J 23 ) .
(19)
Now constructing and using T ¯ α β M α β γ = A 1 T ¯ α β U α U β U γ + 2 A 2 T ¯ α γ U α, the equation of motion (7) becomes
T α β x β e c F α β n U β = F rad α
(20)
with radiation reaction four force F rad α given by
F rad α = 2 e 3 3 m 2 c 4 F α β x γ T β γ + σ c T ¯ α β U β ( n 1 m 2 c 2 2 A 2 ) σ m 2 c 3 A 1 T ¯ β γ U β U γ U α .
(21)
The preceding system of covariant equations is valid for arbitrary, physically allowed values of temperatures and levels of degeneracy. Properties of the fully degenerate plasma, described within the preceding framework, will now be explored.
If the thermal energy of the particles (electrons and positrons) is much lower than their Fermi energy, the plasma may be treated as cold, i.e., having zero temperature; this can be done even if the thermal energy is near relativistic 10 9 K.19 In this limit, the integral (16) simplifies to:18,
J n m = 0 ϑ F d ϑ sinh n ϑ cosh m ϑ ,
(22)
where ϑ F is related to the Fermi momentum by relation ϑ F = sinh 1 ( p F / m c ). This simplification yields J 21 = sinh 3 ϑ F / 3, and Eq. (13) gives an explicit expression for density
n = 4 π 3 ( m c ) 3 g h 3 x 3 ,
(23)
where x = p F / m c. Similar manipulations reproduce the well know results of Ref. 18:
n ε = π 6 m 4 c 5 g h 3 ( 3 x ( 1 + 2 x 2 ) 1 + x 2 3 sinh 1 x ) ,
(24)
p = π 6 m 4 c 5 g h 3 ( x ( 2 x 2 3 ) 1 + x 2 + 3 sinh 1 x ) ,
(25)
and a remarkably simple expression for enthalpy
W = m c 2 n γ F .
(26)
For A1 and A2, we get
A 1 = m 2 n ( 1 + 6 5 x 2 ) ,
(27)
A 2 = 1 5 m 2 c 2 n x 2 .
(28)
By substituting A1 and A2 into Eq. (21), we obtain
F α = 2 e 3 3 m 2 c 4 F α β x γ T β γ + σ n [ T ¯ α β 1 c U β ( 1 + 2 5 x 2 ) ( 1 + 6 5 x 2 ) T ¯ β γ 1 c 3 U β U γ U α ] .
(29)
Equation (28) constitutes the primary result of the current letter. Note that the first term in the right hand side of (29) is smaller than Lorentz force and can be neglected without loss of generality.
Using this equation along with (26) and (6), the equation of motion of electron fluid (7) can be written as
n m γ F U β U α x β ( g α β 1 c 2 U α U β ) P x β = e c F α β n U β + R α ,
(30)
where the four radiation force is given by
R α = 1 c σ n ( 1 + 2 5 x 2 ) [ T ¯ α β U β T ¯ γ β 1 c 2 U β U γ U α ] .
(31)
In spatiotemporal coordinates, this equation reduces to
m γ F d d t ( γ v V ) + 1 n γ v ( p γ v 2 c 2 V d p d t ) = e ( E + 1 c ( V × B ) ) + R ,
(32)
R = σ γ v ( 1 + 2 5 x 2 ) [ T ¯ α β 1 c U β T ¯ γ β 1 c 3 U β U γ U α ] spat .
(33)
Here d / d t = / t + V · , and E and B are the generated low frequency electric and magnetic fields, respectively. Note that in the right hand side of (33), the index α pertains only to spatial components.

In weakly degenerate case where x = p F / m c 1, the expression (33) for RRF simplifies to the one obtained for a classical Maxwell distribution in the nonrelativistic regime ( T m c 2). Since p F = h ( 3 n / 8 π ) 1 / 3, the condition of weak degeneracy can be written in terms of electron density n n c = 5.9 × 10 29 cm 3. However, it is noteworthy that the gas (fluid) approximation for plasma remains valid only if the average potential energy of particles is less than the Fermi energy, implying that the density falls within the solid density range 10 23 cm 3 n 5.9 × 10 29 cm 3.

For a relativistic degenerate plasma ( n 5.9 × 10 29 cm 3), with Fermi momentum greater than the rest mass, i.e., x 2 1 (i.e., p F m c ), the RRF is strongly boosted up, scaling quadratically with x; it is a most spectacular manifestation of the Compton Rocket effect.

Just for perspective, we remind the reader that in a relativistic dilute (non-degenerate) plasma, the enhancement of RRF would occur only when the kinematic temperature is relativistic ( T m c 2). However, it would be nowhere near as strong as in the strongly degenerate case.

For further clarity, let us consider a scenario where plasma is subjected to a field of incoherent high-frequency photon fluxes. For a plane-parallel (or point source) configuration propagating along e 1, the flux is described by a stress-energy tensor with all components zero, except T 00 = T 01 = T 10 = T 11 = F. The resulting radiation reaction force, then, takes the form
R = σ F ( 1 + 2 5 x 2 ) ( 1 n · V c ) [ n γ v 2 ( 1 n · V c ) V c ] ,
(34)
which, for isotropic radiation with energy density Urad, becomes
R = 4 3 γ v 2 σ U rad ( 1 + 2 5 x 2 ) V c ,
(35)
a very revealing simple expression strongly announcing the enhancement of RRF for strong degeneracy.

We would like to emphasize that the model, described above, is valid in the Thomson limit. The dynamics of the Compton scattering regime, pertaining a cold classical plasma embedded in the x-ray photon fluxes, have been studied for some time.20–22 More recently, the importance of photon plasma interaction in the Compton scattering regime was demonstrated23,24 for a variety of astrophysical conditions. To extend our work to the Compton regime, one should include quantum effects in radiation reaction force as well as introduce photon particle collisions in the relativistic kinetic equation. Such a work will constitute a much more involved study, which is beyond the scope of the current letter.

By invoking a local Fermi-Dirac (thermodynamic) closure, we derived hydrodynamic equations that include the radiation reaction force, RRF. These equations are valid for arbitrary (physically allowed) temperature and densities of electron (positron) plasmas. It is shown that RRF is strongly amplified in strongly degenerate plasmas—it is, perhaps, one of the strongest expressions of the Compton Rocket effect in a well-defined physical system.

Working out the consequences of this phenomenon for compact astrophysical objects is under way.

This research was supported by the Shota Rustaveli National Science Foundation of Georgia under Grant No. FR22-8273. The work of S.M.M. was supported by USDOE under Contract No. DE-FG 03–96ER-54366.

The authors have no conflicts to disclose.

V. I. Berezhiani: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). S. M. Mahajan: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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