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Physical Interpretation of the Poynting–Robertson Effect
Icarus 140, 231–234 (1999) Article ID icar.1999.6136, available online at http://www.idealibrary.com on
Physical Interpretation of the Poynting–Robertson Effect R. Srikanth Indian Institute of Astrophysics, Bangalore 34 and Indian Institute of Science, Bangalore 12, India E-mail: [email protected] Received April 6, 1998; revised March 3, 1999
point. We highlight these points in this article, after a simple general relativistic derivation of the effect. It is sufficient for our purpose to consider a simple model of a spherical, completely absorbing dust, with the added generalization that the absorption and reemission parameters of the dust are mutually distinct. Although for this problem Robertson’s (1937) equations are definitive, unfortunately his treatment was unnecessarily abstruse because it did not exploit the simplicity of a completely covariant appraoch, but instead interrupts its arguements with appeals to certain physical considerations. A more straightforward derivation of the drag equations from a special relativistic viewpoint was given by Burns et al. (1979), but they include gravitation as a nonrelativistic scalar field. A similar special relavitistic treatment of the problem is given also by Robertson and Noonan (1968). There is nothing wrong with this treatment as such since the field is weak and moreover dust velocity in the heliosphere is much smaller than that of light. However, the essential structure of the relevant equation of motion in the PR drag achieves its untmost simplicity and clarity in a general relativistic treatment. In this formulation, the gravitational field is implicit in the covariant derivative, leaving only the absorption and emission terms explicitly in the energy balance. Thus, the fact that PR effect is essentially the outcome of the interplay of absorption and reemission processes, to which gravitation is not central, is brought out in a straightforward manner. Furthermore, it allows a simple extension to the treatment of dust motion in stronger gravitational systems such as accretion disks around neutron stars.
Although the prevalent mathematical description of the Poynting– Robertson effect is correct, its physical interpretation is sometimes problematic. By means of a two-parameter model, we revisit the effect in order to get a better physical understanding of it. The principal conclusion is that the motion of a dust in circumsolar orbit is governed only by solar radiation absorption and not by the asymmetry of reemission, even when viewed in the rest-frame of the Sun. c 1999 Academic Press °
Key Words: Poynting-Robertson effect; Solar System, dust.
I. INTRODUCTION
As is well known, Poynting-Robertson (PR) effect causes small bodies in circumsolar orbit, such as dust and boulders, assumed to totally absorb intercepted radiation and re-emit isotropically in the bodies’ rest-frame (“rest-isotropically”), to inspiral with orbits of decreasing eccentricity (Robertson 1937, Robertson and Noonan 1968). Since then the effect has been generalized to include asymmetric scattering (Burns et al. 1979), dust rotations and nonspherical dusts (Lyttleton 1976), charged particles (Consolmagno and Jokipii 1977), and for a photograviational system of more than one primary (Rogos and Zafiropoulos 1995). PR effect can play a role in the motion of comets, and the formation of circumsolar dust rings (Lamy 1974) and interstellar magnetic fields (Harwit 1982). Since its discovery by Poynting in 1903 until its treatment by Robertson (1937), PR effect remained controversial. The controversy centered on the origin of the drag. In his original paper, Poynting attributed the drag to dust reemission by reasoning that in the Sun’s rest-frame (“heliocentric frame”), the front–back asymmetry in the dust reemission produces a recoil. The inconsistency of this attribution with relativity theory (propounded after Poynting’s paper, in 1905) was pointed out by Page in 1913 and later Robertson (1937). The latter has given historical details of the discovery and understanding of the effect. However, to judge from modern literature on the subject, some conceptual problems still remain. Although these problems do not affect quantitative calculations involving the effect, they are important from a pedagogical and phyical interpretation view-
II. THE EQUATIONS OF MOTION
The generally covariant equations governing the motion of the dust can be written as Dp µ Du µ dm µ ≡ u +m = f ext , Dτ dτ Dτ
(2.1)
where p µ = mu µ is four-momentum of the dust, τ is proper time, f ext is the external four-force, and the operator D/Dτ is the “total” covariant derivative in the general relativistic sense and includes gravitational effects. Thus gravitation is not part of f ext . The mass change term accounts for possible change in the
231 0019-1035/99 $30.00 c 1999 by Academic Press Copyright ° All rights of reproduction in any form reserved.
232
R. SRIKANTH
dust’s internal energy due to heating or cooling. Greek indices run from 0 to 3, with 0 standing for time component, and 1, 2, 3 for spatial components. Spacetime signature is taken as (+ − − −). In the description of PR effect that we adopt, the effect of absorption of sunlight and reemission are characterized by two distinct parameters, respectively. The primary radiation may be considered as a plane-parallel beam flowing radially outward. µ It is represented by the force f rad = ²l µ , where l µ is a dimensionless null vector, with its spatial part being purely radial (Robertson 1937). The scalar ² (>0) is the rest-rate of absorption of momentum from the solar radiation by the dust, with dimension momentum over time. We assume that the dust absorbs all the incident radiation and reemits radiation isotropically in its rest-frame. The relativistic four-force associated with a restµ isotropic (generally: symmetric) emission is f emit = −(ξ/c2 )u µ , where u µ is four-velocity, and ξ (>0) is the rest-frame energy emission rate. Accordingly, Eq. (2.1) becomes dm µ Du µ µ µ u +m = f rad + f emit , dτ Dτ = ²l µ − (ξ/c2 ) u µ .
(2.2)
Contracting Eq. (2.2) by u µ , we get the equations for the internal energy change c2
dm = ²l α u α − ξ. dτ
If we denote by L and σ the rest-luminosity of the primary and cross section of the dust, respectively, then it follows that ²=
mα lµ u µ , r2
(2.7)
where α = Lσ/4π mc. We retain terms upto only first order in the spatial components of Eq. (2.4), and for consistency, terms upto second order in time component, since kinetic energy goes as ∼v 2 . We note that α will be a function of time if m varies with time. With substitution of Eqs. (2.6) and (2.7), and the evaluation of the Christoffel symbols, Eq. (2.4) yields in the weak field limit µ 2 ¶ cα r˙ (2˙r 2 + r 2 θ˙ ) dE = 2 − , dt r c c2 ¸ ¶ µ· dv α v r˙ = 2 1 − rˆ − − ∇φ, dt r c c
(2.8)
where E is the kinetic energy per unit mass of the dust, v is its velocity and rˆ is unit radius vector multiplied by c. Assuming isothermality, in which case α is not a function of time, Eqs. (2.8) are sufficient to derive the PR drag and circularization of orbits (Robertson 1937).
(2.3)
Substituting this back into Eq. (2.2), we get the true equations of motion µ ¶ l α uα Du µ = ² l µ − 2 uµ , (2.4) m Dτ c where the bracketed term is just the part of radiation orthogonal to u µ . The first term on the right-hand side is the radiation pressure term, and the second is the drag term. In spherical coordinates, the Schwarzschild metric gµν due to a nonrotating primary taken in the weak field limit is (Landau and Lifshitz 1962) g00 = 1 + 2φ/c2 , g11 = −1 + 2φ/c2 , g22 = g33 = −1, (2.5) with other components vanishing. Here φ is Newtonian gravitational potential. Since l µlµ = 0 and u µ u µ = c2 , we have
III. CONCEPTUAL ISSUES
A confusion in the physical interpretation of PR effect can be detected even in the relevant modern literature. It is probably due to the effect’s somewhat counterintuitive nature: (a) the reemission possesses an asymmetry in the heliocentric frame, but this produces no drag, as might have been expected; (b) yet, this does not mean there is no drag; and (c) in addition, the role of relativity seems to be misunderstood. As a result, one frequently encounters, explicitly or implicitly, the following beliefs: 1. that dust reemission is a necessary condition for PR drag as seen in the heliocentric frame; 2. that isothermality condition dm/dτ = 0 implies that the dust reemits as much as it absorbs; and 3. that the factor l µ u µ /c in Eq. (2.7) represents red-shift. These three points are dealt with in the subsections below. A. Role of Absorption and Re-emission in PR Drag
l µ ≈ (1 − φ/c2 , 1 + φ/c2 , 0, 0), ˙ 0), u µ ≈ (c + v 2 /2c − φ/c, r˙ , r θ,
µ
(2.6)
where v is dust’s three-velocity. Thus, l µ u µ /c = (1 + v 2 /2c2 − r˙ /c), upto second order in v/c.
In Eq. (2.1), let f ext = f emit alone. Clearly then f emit would cancel out the mass-loss term on the left-hand side, thereby leaving the spacelike Du µ /Dτ term to vanish. In this way, Robertson (1937) argues (rightly) that the reemission should leave the dust motion unaffected because of the mass change term which balances it. However, he further argues that in the presence of solar
233
POYNTING–ROBERTSON EFFECT
irradiation, the rest mass could be replenished and then there would be no term to compensate the back-force due to reemission, resulting in the drag. Robertson’s train of argument suggests that dust reemission is a necessary condition for PR drag as seen in the heliocentric frame. This argument has been accepted in all the modern literature on PR effect that I am aware of. It differs from the original view of Poynting in that it does not treat the forward–backward asymmetry in the dust’s reemission as a sufficient condition for PR drag, as supposed at first by Poynting, but as a necessary condition, the other necessary condition being dm/dτ = 0 (Burns et al. 1979). However, the drag as seen in the dust’s rest-frame is (rightly) attributed to aberrated sunlight, since here the asymmetry in reemission is constrained by relativity to vanish (Harwit 1982). That in one frame the drag is ascribed to reemission but not in the other is seen as a relativistic manifestation of the same force being viewed in two different frames (Robertson 1937). However, in fact reemission, assumed rest-isotropic, plays no role in the motion of the dust. Since ξ does not appear in Eq. (2.4), it follows that the instantaneous motion of the dust is independent of reemission (though, over a finite period of time, if mass-loss due to reemission is nonvanishing, it will play a role). By the principle of covariance, it is clear that there exists no frame, not even the heliocentric frame, in which reemission can be seen to affect instantaneous motion of the dust. The dust’s reemission is generally neither necessary nor sufficient for the drag. On the other hand, absorption is clearly a necessary condition for PR drag, as evidenced by the appearance of ² in Eq. (2.4). One possible reason as to why this result was not often realized is that both absorption and reemission are often merged into a single parameter by the assumption that the dust is isothermal. Indeed, the motivation for treating PR effect as a two-parameter (ξ, ²) phenomenon in the present article has been to disambiguate the roles of absorption and reemission. To physically visualize what causes the drag as seen from the heliocentric reference frame, we adduce the following thought experiment. The behavior of the dust can be broken into a twostep model: absorption, followed be rest-isotropic reemission. We consider a dust moving in a circular orbit about the Sun. A photon of energy e, moving radially outward from the primary, impinges upon the dust and is fully absorbed. From Eq. (2.3), we find that dust mass changes by an amount c2 1m 1 = e(1 − r˙ /c + E/c2 ), which for the assumed circular orbit reduces to e(1 + E/c2 ). From angular momentum conservation, we have ˙ Rearranging terms, we have mr 2 θ˙ = (m + 1m 1 )(r 2 θ˙ + 1(r 2 θ)). upto first order ˙ ≈− 1(r 2 θ)
e 2˙ r θ. mc2
(3.1)
This angular deceleration, in the continuous limit, becomes the θ equation in Eq. (2.8). Thus, the mass increase due to radiation absorption coupled with angular momentum conservation is sufficient to produce PR drag. The subsequent reemission, being rest-isotropic, does not affect the dust’s motion.
Similarly, we find that even though energy of the dust increases by e because of photon absorption, kinetic energy per unit mass diminishes, implying infall. From energy conservation, we have E0 =
µ ¶ m E + e − 1m 1 c2 2e ≈ E 1− < E, m + 1m 1 mc2
(3.2)
where the prime denotes the dust state after photon absorption. Subsequently, let the dust reemit mass −1m 2 (>0) rest˙ which isotropically. The reduction in momentum is −1m 2r θ, equals the reaction due to excess emission in the forward direction as seen in the heliocentric frame. Thus, the subsequent reemission does not alter the instantaneous inertial state of the dust, independently of whether we adopt the isothermal condition 1m 1 + 1m 2 = 0. B. Isothermality The notion is sometimes held that dm/dτ = 0 (the isothermality condition) implies that the dust emits as much as it absorbs. From Eq. (2.2), we find that assuming isothermality the excess of absorbed energy over reemission is balanced by kinetic energy. In an arbitrary frame, this excess is nonvanishing, as seen from Eq. (2.4). Only in the dust’s rest-frame, where d E/dt vanishes, does reemitted energy equal absorbed energy under the isothermal condition. C. Red-shift In Eq. (2.7), the quantity cα/r 2 is the amount of incident radiation that would be incident on the dust if it were at rest in the heliocentric frame. If the dust is moving, the quantity takes the additional factor lµ u µ /c ≈ (1 − r˙ /c), sometimes thought to represent red-shift. Now, if we visualize flux incident on the dust per unit time as being the energy contained in a column of space of length c, cross section σ , and an energy density of say ρ, then the lµ u µ /c factor modifies the length of this column; a dust moving toward or away from the primary will traverse a longer or shorter column of radiation by the factor lµ u µ /c. True red-shift will invovle change in ρ. This is indeed the case if, for example, we transform to the dust’s rest-frame. Thus, lµ u µ /c in Eq. (2.7) is a flux-modification factor and not redshift factor.
IV. CONCLUSIONS
By studying PR effect as a two-parameter process, it is concluded that dust absorption is a necessary condition for drag, whereas reemission (assumed isotropic in dust’s rest-frame) plays no role, even when viewed in the heliocentric frame. Some comments on the physical signficance of isothermality and the l µ u µ /c factor are made.
234
R. SRIKANTH
ACKNOWLEDGMENTS
Lamy, Ph. H. 1974. The dynamics of circumsolar dust grains. Astron. Astrophys. 33, 191–194.
I thank Camilla B. Mantir for useful discussions. I am grateful to Dr. F. Mignard for pointing out a numerical error, and for his helpful suggestions.
Landau, L., and H. Lifshitz 1962. The Classical Theory of Fields. Pergamon, New York. Lyttleton, R. A. 1976. Effects of solar radiation on the orbits of small particles. Astrophys. Space Sci. 44, 119–140.
REFERENCES Burns, J. A., Ph. H. Lamy, and S. Soter 1979. Radiation forces on small particles in the Solar System. Icarus 40, 1–48. Consolmagno, G., and J. R. Jokipi 1977. Lorentz scattering of interplanetary dust. Bull. Am. Astron. Soc. 9, 519–520. Harwit, M. 1982. Astrophysical Concepts, second ed., Wiley, New York.
Robertson, H. P. 1937. The dynamical effects of radiation in the Solar System. Mon. Not. R. Astron. Soc. 97, 423–438. Robertson, H. P., and T. W. Noonan 1968. Relativity and Cosmology. Saunders, Philadelphia. Rogos, O., and F. A. Zafiropoulos 1995. A numerical study of the influence of the Poynting–Robertson effect on the equilibrium points of the photograviational restricted three body problem. Astron. Astrophys. 300, 568–578, 579–590.
Physical Interpretation of the Poynting–Robertson Effect R. Srikanth Indian Institute of Astrophysics, Bangalore 34 and Indian Institute of Science, Bangalore 12, India E-mail: [email protected] Received April 6, 1998; revised March 3, 1999
point. We highlight these points in this article, after a simple general relativistic derivation of the effect. It is sufficient for our purpose to consider a simple model of a spherical, completely absorbing dust, with the added generalization that the absorption and reemission parameters of the dust are mutually distinct. Although for this problem Robertson’s (1937) equations are definitive, unfortunately his treatment was unnecessarily abstruse because it did not exploit the simplicity of a completely covariant appraoch, but instead interrupts its arguements with appeals to certain physical considerations. A more straightforward derivation of the drag equations from a special relativistic viewpoint was given by Burns et al. (1979), but they include gravitation as a nonrelativistic scalar field. A similar special relavitistic treatment of the problem is given also by Robertson and Noonan (1968). There is nothing wrong with this treatment as such since the field is weak and moreover dust velocity in the heliosphere is much smaller than that of light. However, the essential structure of the relevant equation of motion in the PR drag achieves its untmost simplicity and clarity in a general relativistic treatment. In this formulation, the gravitational field is implicit in the covariant derivative, leaving only the absorption and emission terms explicitly in the energy balance. Thus, the fact that PR effect is essentially the outcome of the interplay of absorption and reemission processes, to which gravitation is not central, is brought out in a straightforward manner. Furthermore, it allows a simple extension to the treatment of dust motion in stronger gravitational systems such as accretion disks around neutron stars.
Although the prevalent mathematical description of the Poynting– Robertson effect is correct, its physical interpretation is sometimes problematic. By means of a two-parameter model, we revisit the effect in order to get a better physical understanding of it. The principal conclusion is that the motion of a dust in circumsolar orbit is governed only by solar radiation absorption and not by the asymmetry of reemission, even when viewed in the rest-frame of the Sun. c 1999 Academic Press °
Key Words: Poynting-Robertson effect; Solar System, dust.
I. INTRODUCTION
As is well known, Poynting-Robertson (PR) effect causes small bodies in circumsolar orbit, such as dust and boulders, assumed to totally absorb intercepted radiation and re-emit isotropically in the bodies’ rest-frame (“rest-isotropically”), to inspiral with orbits of decreasing eccentricity (Robertson 1937, Robertson and Noonan 1968). Since then the effect has been generalized to include asymmetric scattering (Burns et al. 1979), dust rotations and nonspherical dusts (Lyttleton 1976), charged particles (Consolmagno and Jokipii 1977), and for a photograviational system of more than one primary (Rogos and Zafiropoulos 1995). PR effect can play a role in the motion of comets, and the formation of circumsolar dust rings (Lamy 1974) and interstellar magnetic fields (Harwit 1982). Since its discovery by Poynting in 1903 until its treatment by Robertson (1937), PR effect remained controversial. The controversy centered on the origin of the drag. In his original paper, Poynting attributed the drag to dust reemission by reasoning that in the Sun’s rest-frame (“heliocentric frame”), the front–back asymmetry in the dust reemission produces a recoil. The inconsistency of this attribution with relativity theory (propounded after Poynting’s paper, in 1905) was pointed out by Page in 1913 and later Robertson (1937). The latter has given historical details of the discovery and understanding of the effect. However, to judge from modern literature on the subject, some conceptual problems still remain. Although these problems do not affect quantitative calculations involving the effect, they are important from a pedagogical and phyical interpretation view-
II. THE EQUATIONS OF MOTION
The generally covariant equations governing the motion of the dust can be written as Dp µ Du µ dm µ ≡ u +m = f ext , Dτ dτ Dτ
(2.1)
where p µ = mu µ is four-momentum of the dust, τ is proper time, f ext is the external four-force, and the operator D/Dτ is the “total” covariant derivative in the general relativistic sense and includes gravitational effects. Thus gravitation is not part of f ext . The mass change term accounts for possible change in the
231 0019-1035/99 $30.00 c 1999 by Academic Press Copyright ° All rights of reproduction in any form reserved.
232
R. SRIKANTH
dust’s internal energy due to heating or cooling. Greek indices run from 0 to 3, with 0 standing for time component, and 1, 2, 3 for spatial components. Spacetime signature is taken as (+ − − −). In the description of PR effect that we adopt, the effect of absorption of sunlight and reemission are characterized by two distinct parameters, respectively. The primary radiation may be considered as a plane-parallel beam flowing radially outward. µ It is represented by the force f rad = ²l µ , where l µ is a dimensionless null vector, with its spatial part being purely radial (Robertson 1937). The scalar ² (>0) is the rest-rate of absorption of momentum from the solar radiation by the dust, with dimension momentum over time. We assume that the dust absorbs all the incident radiation and reemits radiation isotropically in its rest-frame. The relativistic four-force associated with a restµ isotropic (generally: symmetric) emission is f emit = −(ξ/c2 )u µ , where u µ is four-velocity, and ξ (>0) is the rest-frame energy emission rate. Accordingly, Eq. (2.1) becomes dm µ Du µ µ µ u +m = f rad + f emit , dτ Dτ = ²l µ − (ξ/c2 ) u µ .
(2.2)
Contracting Eq. (2.2) by u µ , we get the equations for the internal energy change c2
dm = ²l α u α − ξ. dτ
If we denote by L and σ the rest-luminosity of the primary and cross section of the dust, respectively, then it follows that ²=
mα lµ u µ , r2
(2.7)
where α = Lσ/4π mc. We retain terms upto only first order in the spatial components of Eq. (2.4), and for consistency, terms upto second order in time component, since kinetic energy goes as ∼v 2 . We note that α will be a function of time if m varies with time. With substitution of Eqs. (2.6) and (2.7), and the evaluation of the Christoffel symbols, Eq. (2.4) yields in the weak field limit µ 2 ¶ cα r˙ (2˙r 2 + r 2 θ˙ ) dE = 2 − , dt r c c2 ¸ ¶ µ· dv α v r˙ = 2 1 − rˆ − − ∇φ, dt r c c
(2.8)
where E is the kinetic energy per unit mass of the dust, v is its velocity and rˆ is unit radius vector multiplied by c. Assuming isothermality, in which case α is not a function of time, Eqs. (2.8) are sufficient to derive the PR drag and circularization of orbits (Robertson 1937).
(2.3)
Substituting this back into Eq. (2.2), we get the true equations of motion µ ¶ l α uα Du µ = ² l µ − 2 uµ , (2.4) m Dτ c where the bracketed term is just the part of radiation orthogonal to u µ . The first term on the right-hand side is the radiation pressure term, and the second is the drag term. In spherical coordinates, the Schwarzschild metric gµν due to a nonrotating primary taken in the weak field limit is (Landau and Lifshitz 1962) g00 = 1 + 2φ/c2 , g11 = −1 + 2φ/c2 , g22 = g33 = −1, (2.5) with other components vanishing. Here φ is Newtonian gravitational potential. Since l µlµ = 0 and u µ u µ = c2 , we have
III. CONCEPTUAL ISSUES
A confusion in the physical interpretation of PR effect can be detected even in the relevant modern literature. It is probably due to the effect’s somewhat counterintuitive nature: (a) the reemission possesses an asymmetry in the heliocentric frame, but this produces no drag, as might have been expected; (b) yet, this does not mean there is no drag; and (c) in addition, the role of relativity seems to be misunderstood. As a result, one frequently encounters, explicitly or implicitly, the following beliefs: 1. that dust reemission is a necessary condition for PR drag as seen in the heliocentric frame; 2. that isothermality condition dm/dτ = 0 implies that the dust reemits as much as it absorbs; and 3. that the factor l µ u µ /c in Eq. (2.7) represents red-shift. These three points are dealt with in the subsections below. A. Role of Absorption and Re-emission in PR Drag
l µ ≈ (1 − φ/c2 , 1 + φ/c2 , 0, 0), ˙ 0), u µ ≈ (c + v 2 /2c − φ/c, r˙ , r θ,
µ
(2.6)
where v is dust’s three-velocity. Thus, l µ u µ /c = (1 + v 2 /2c2 − r˙ /c), upto second order in v/c.
In Eq. (2.1), let f ext = f emit alone. Clearly then f emit would cancel out the mass-loss term on the left-hand side, thereby leaving the spacelike Du µ /Dτ term to vanish. In this way, Robertson (1937) argues (rightly) that the reemission should leave the dust motion unaffected because of the mass change term which balances it. However, he further argues that in the presence of solar
233
POYNTING–ROBERTSON EFFECT
irradiation, the rest mass could be replenished and then there would be no term to compensate the back-force due to reemission, resulting in the drag. Robertson’s train of argument suggests that dust reemission is a necessary condition for PR drag as seen in the heliocentric frame. This argument has been accepted in all the modern literature on PR effect that I am aware of. It differs from the original view of Poynting in that it does not treat the forward–backward asymmetry in the dust’s reemission as a sufficient condition for PR drag, as supposed at first by Poynting, but as a necessary condition, the other necessary condition being dm/dτ = 0 (Burns et al. 1979). However, the drag as seen in the dust’s rest-frame is (rightly) attributed to aberrated sunlight, since here the asymmetry in reemission is constrained by relativity to vanish (Harwit 1982). That in one frame the drag is ascribed to reemission but not in the other is seen as a relativistic manifestation of the same force being viewed in two different frames (Robertson 1937). However, in fact reemission, assumed rest-isotropic, plays no role in the motion of the dust. Since ξ does not appear in Eq. (2.4), it follows that the instantaneous motion of the dust is independent of reemission (though, over a finite period of time, if mass-loss due to reemission is nonvanishing, it will play a role). By the principle of covariance, it is clear that there exists no frame, not even the heliocentric frame, in which reemission can be seen to affect instantaneous motion of the dust. The dust’s reemission is generally neither necessary nor sufficient for the drag. On the other hand, absorption is clearly a necessary condition for PR drag, as evidenced by the appearance of ² in Eq. (2.4). One possible reason as to why this result was not often realized is that both absorption and reemission are often merged into a single parameter by the assumption that the dust is isothermal. Indeed, the motivation for treating PR effect as a two-parameter (ξ, ²) phenomenon in the present article has been to disambiguate the roles of absorption and reemission. To physically visualize what causes the drag as seen from the heliocentric reference frame, we adduce the following thought experiment. The behavior of the dust can be broken into a twostep model: absorption, followed be rest-isotropic reemission. We consider a dust moving in a circular orbit about the Sun. A photon of energy e, moving radially outward from the primary, impinges upon the dust and is fully absorbed. From Eq. (2.3), we find that dust mass changes by an amount c2 1m 1 = e(1 − r˙ /c + E/c2 ), which for the assumed circular orbit reduces to e(1 + E/c2 ). From angular momentum conservation, we have ˙ Rearranging terms, we have mr 2 θ˙ = (m + 1m 1 )(r 2 θ˙ + 1(r 2 θ)). upto first order ˙ ≈− 1(r 2 θ)
e 2˙ r θ. mc2
(3.1)
This angular deceleration, in the continuous limit, becomes the θ equation in Eq. (2.8). Thus, the mass increase due to radiation absorption coupled with angular momentum conservation is sufficient to produce PR drag. The subsequent reemission, being rest-isotropic, does not affect the dust’s motion.
Similarly, we find that even though energy of the dust increases by e because of photon absorption, kinetic energy per unit mass diminishes, implying infall. From energy conservation, we have E0 =
µ ¶ m E + e − 1m 1 c2 2e ≈ E 1− < E, m + 1m 1 mc2
(3.2)
where the prime denotes the dust state after photon absorption. Subsequently, let the dust reemit mass −1m 2 (>0) rest˙ which isotropically. The reduction in momentum is −1m 2r θ, equals the reaction due to excess emission in the forward direction as seen in the heliocentric frame. Thus, the subsequent reemission does not alter the instantaneous inertial state of the dust, independently of whether we adopt the isothermal condition 1m 1 + 1m 2 = 0. B. Isothermality The notion is sometimes held that dm/dτ = 0 (the isothermality condition) implies that the dust emits as much as it absorbs. From Eq. (2.2), we find that assuming isothermality the excess of absorbed energy over reemission is balanced by kinetic energy. In an arbitrary frame, this excess is nonvanishing, as seen from Eq. (2.4). Only in the dust’s rest-frame, where d E/dt vanishes, does reemitted energy equal absorbed energy under the isothermal condition. C. Red-shift In Eq. (2.7), the quantity cα/r 2 is the amount of incident radiation that would be incident on the dust if it were at rest in the heliocentric frame. If the dust is moving, the quantity takes the additional factor lµ u µ /c ≈ (1 − r˙ /c), sometimes thought to represent red-shift. Now, if we visualize flux incident on the dust per unit time as being the energy contained in a column of space of length c, cross section σ , and an energy density of say ρ, then the lµ u µ /c factor modifies the length of this column; a dust moving toward or away from the primary will traverse a longer or shorter column of radiation by the factor lµ u µ /c. True red-shift will invovle change in ρ. This is indeed the case if, for example, we transform to the dust’s rest-frame. Thus, lµ u µ /c in Eq. (2.7) is a flux-modification factor and not redshift factor.
IV. CONCLUSIONS
By studying PR effect as a two-parameter process, it is concluded that dust absorption is a necessary condition for drag, whereas reemission (assumed isotropic in dust’s rest-frame) plays no role, even when viewed in the heliocentric frame. Some comments on the physical signficance of isothermality and the l µ u µ /c factor are made.
234
R. SRIKANTH
ACKNOWLEDGMENTS
Lamy, Ph. H. 1974. The dynamics of circumsolar dust grains. Astron. Astrophys. 33, 191–194.
I thank Camilla B. Mantir for useful discussions. I am grateful to Dr. F. Mignard for pointing out a numerical error, and for his helpful suggestions.
Landau, L., and H. Lifshitz 1962. The Classical Theory of Fields. Pergamon, New York. Lyttleton, R. A. 1976. Effects of solar radiation on the orbits of small particles. Astrophys. Space Sci. 44, 119–140.
REFERENCES Burns, J. A., Ph. H. Lamy, and S. Soter 1979. Radiation forces on small particles in the Solar System. Icarus 40, 1–48. Consolmagno, G., and J. R. Jokipi 1977. Lorentz scattering of interplanetary dust. Bull. Am. Astron. Soc. 9, 519–520. Harwit, M. 1982. Astrophysical Concepts, second ed., Wiley, New York.
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