DYNAMICAL EFFECTS OF ESOTROPK RABIATION Thesis for the Degrea of N1. D. MICHQGAN SLATE UNWERSITY David Francis Savickas W966 blit'iugdn Rate {a . _ “’9 Umvez‘sxt‘y This is to certify that the thesis entitled DYNAMICAL EFFECTS OF ISOTROPIC RADIATIOI‘J presented by David Francis Savickas has been accepted towards fulfillment of the requirements for PhoDo degree in PlgSiCS Major professor é Date November 181 1966 0-169 ABSTRACT DYNAMICAL EFFECTS OF ISOTROPIC RADIATION by David Francis Savickas A small spherical body traveling through an isotrOpic radiation field will generally experience a force caused by the momentum it absorbs from the radiation. Due to the DOppler shift, the radiation striking the front of the par— ticle will be of higher frequency than the radiation striking the back of the particle. Also, because of diffraction ef- fects, a small particle will absorb amounts of momentum that differ with the radiation's wavelength. These two factors generally combine to exert a force on the particle and change its velocity. The effect of this force on the particle's motion was investigated. By the use of relativistic mechanics a general ex- pression for the force on the particle due to a plane-parallel beam of radiation was obtained and then integrated over space to obtain the total force of the whole radiation field. The diffraction effects were taken into account by use of either Debye's approximations for the pressure efficiency factors of very small particles, or by the use of graphs of this factor as a function of frequency. D. F. Savickas For approximations at low velocities it was found that the ratio of the initial velocity to the velocity at a later time is independent of the initial velocity. Gener- ally the radiation exerts a retarding force on the motion of the particle. However, for special cases where an essen- tially monochromatic and isotropic field exists, it was found that the particle could be accelerated to higher velo- cities by the radiation. Numerical estimates for the time it would take a particle in interstellar space to come to l/e its initial velocity were found to be of the order of 10 11 from 10 to 10 years. DYNAMICAL EFFECTS OF ISOTROPIC RADIATION By David Francis Savickas A THESIS , Submitted To Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1966 ACKNOWLEDGMENT I wish to express my gratitude to Professor Richard Schlegel for his help throughout the course of this research and also for the incentive he has given me through his stimu- lating interest in radiation pressure and other problems of physics as well. ii TABLE OF CONTENTS ACKNOWLEDGMENT . . . . . . . . . . . . . . . . . LIST OF FIGURES . . . . . . . . . . . . INTRODUCTION CHAPTER I. HISTORICAL BACKGROUND II. THE RELATIVISTIC FORCE OF ISOTROPIC RADIATION . . . . . . . . . Calculation by Use of the Stress-Energy Tensor . Agreement with the Calculation from Robertson's Equation Exact Expression for Qpr III. DYNAMICAL EFFECTS OF THE RADIATION FIELD Exact Solutions and Useful Approxima- tions . The Relativistic Motion of a Large Perfectly Absorbing Sphere Linear Approximations for Low Velocity Particles . . Retarding and Accelerating Forces Very Small Particle Approximations IV. EFFECTS OF RADIATION ON PARTICLES IN INTER- STELLAR SPACE . . . . . . . . . . . The Time Constant t Numerical Results V. SUMMARY REFERENCES iii Page ii iv 16 18 2O Figure LIST OF FIGURES Linear Approximation for Qpr Supplementary Ray Diagram Graph of Cubic Approximation for Qpr Irvine's Graph of Q pr for n = 1.33 Debye' 8 Graph of Qpr for the Perfect Reflector . . . . . . . . iv Page 36 37 Al 58 58 INTRODUCTION A plane wave of energy density u normally incident upon a perfectly absorbing plane surface of area A will ex— ert a force F = u A. The radiation's pressure is seen then to be equal to its energy density. The effects of radiation pressure on the motion of particles in space has been pre- viously investigated for places in space where a single source, such as the sun or a star, exerts a strong influence on a particle's motion. However the effects of an isotropic radiation field on a particle's motion have not been pre- viously investigated. When a particle is at rest in such a field it obviously experiences no radiation force by reason of symmetry. If, however, it is in motion then the radiation incident on the particle in the direction Opposite to its velocity will have a higher frequency than the radiation incident on it which is in the direction of its velocity. This difference is due to the Doppler shift and it is the effect of this,frequency shift which we wish to investigate. Since the particles in space are small, radiation diffraction effects are important and also will be considered. CHAPTER I HISTORICAL BACKGROUND The existence of radiation pressure was first de- duced as a consequence of the electromagnetic theory by Maxwell, and later experimentally confirmed by Lebedew. The effects of radiation pressure exerted by the sun's radiation on spherical particles in space was first considered by J. H. Poyntingl in 1903. His calculations led to the in— correct result that a body moving through space and receiving no radiation from other sources would suffer a decrease in velocity due to its own emission of radiation as its tempera- ture cooled. This effect was presumed to exist because the radiation was not emitted isotropically about the particle, but rather was concentrated in the forward direction and thinned out in the backward direction. Hence, since radiation carries momentum in the direction emitted, more momentum is lost by the particle in the forward direction than in the back- ward direction. Thus Poynting's results show that the particle's velocity decreases. However a detailed investigation by L. Page was made years later showing that, to the order of accuracy questioned, a moving body does not experience a decrease in velocity as a consequence of its own radiation. Using Page's work J. Larmor2 pointed out the error in Poynting's results in 1918. He noted that the loss of momentum did not necessi- tate a decrease in velocity but rather, by the relation m = E/cZ, a decrease in the mass of the particle and con- cluded that the particle "will move on with constant velocity, .g but with diminishing momentum so long as it has energy to radiate." However, Larmor's own corrections for the radia- e ‘YJ-i tion pressure were also incorrect. 3 It was not until 1937, when H. P. Robertson3a con- sidered the problem, that the matter was cleared up. He made a rigorous relativistic derivation of the radiation pressure caused by a plane-parallel beam of light and then used the simpler form of the classical approximation of this relativistic result to obtain the equation of motion for a particle in the field of the sun. Robertson's results showed the existence of a drag of the same kind which Poynting and Larmor had predicted. His expressions for this force on a spherical particle differed from theirs, but had the same effect of causing the particle to spiral into the sun. He showed the cause of this effect to be the particle's loss of angular momentum. The relativistic effects of radiation pressure were considered again in 1960 when Richard Schlegel“ calculated the force on a plane surface moving through an isotropic radiation field. The results obtained were in error because although the expression given for energy density was trans- formed, the expression for the solid angle element d9 was not; hence in the transformation from the laboratory frame 2 v to the particle frame a factor of l - — was lost. 2 c (l + % cos ¢')2 The correction for this angular transformation was subse- quently pointed out by W. Rindler, D. w. Sciama5 and J. 6 Terrell. CHAPTER II THE RELATIVISTIC FORCE OF ISOTROPIC RADIATION Calculation bnyse of the Stress—Energy Tensor The purpose of my investigation is to calculate the effects of radiation pressure on a spherical particle moving through an isotrOpic radiation field. Of particular interest is the radiation field in interstellar space at points distant from any one particular star. According to present day know- ledge "the particles responsible for interstellar absorption are definitely tiny solid grains."7a Beyond their existence as small particles not much is known about them. They may be either metallic or dielectric, and their size is probably on the order of the wavelength of light or much smaller. Much of the interpretation of astronomical observation of parti- cles in space has been based on the theories developed by G. Mie8 in 1908 and P. Debye9a in 1909. Their theories cal- culated the scattering of radiation by spherical particles and the pressure exerted by the scattered and absorbed radia- ‘tion, but with no regard to the effects on the motion of the particles. The radiation pressure on particles depends strongly on their size and they can be divided into three major groups: first, large sized particles (radius much larger than the wavelength of radiation); second, particles on the order of the wavelength; and third, particles much smaller than the wavelength. For the last two groups diffraction effects are important. The objective now is to obtain a general expression for the force exerted by an isotrOpic radiation field on a spherical particle. Consider such a particle moving along the positive direction of the z axis of a coordinate system in an isotropic radiation field. The coordinate system at rest relative to the stars, and in which the radiation is isotrOpic, will henceforth be referred to as the star system. The coordinate system which moves with the particle will be called the particle system. Because of symmetry no forces perpendicular to the z axis will be exerted on the particle. Hence in using the stress-energy tensor we need only calcu— late those elements related to the 2 component of the force. This will be done by dividing the isotropic radiation of energy density u into pencils of radiation (plane waves) of intensity u %% where d9 is the infinitesimally small solid arugle of the pencil of radiation we are considering. This assumes that the energy density u is composed (If an infinite number of plane waves whose directions have Ehl isotropic distribution about any point in space. The amount of energy density moving in a particular direction, speci- fied by the normal vector to the surface element dQ of a unit sphere surrounding a point, is the fractional area dQ/An where An is the total area of the sphere. The Carte— sian components of the electric and magnetic components of a wave in a nonconducting medium satisfy the well known £1 2 equation V2w - l5 i_g = 0. And E. T. Whittaker10 v at mathematically proved that solutions of this equation can in 1903 be broken up into plane waves. Thus the isotropic radiation ‘1‘ r. ‘ can be represented as the sum of plane waves. Consider the tensor 3T“V G“ = - dV . . . . . . . . . . . . . . . (l) \) 3x where the components of x11 are defined as x1 = x, x2 = y, x3 = z, x“ = ct. The quantity dV is interpreted as a four dimensional volume dV = dx dy dz cdt. Gu is a first rank tensor because 3Tuv V ax is the covariant derivative in flat space-time and dV is an invariant. The invariance of dV can be seen by comparing dV in the star system (x“) to dV in the particle system (in). Now d2 = l — 32 dz, di = dx, dt = dt ’ and d§ = dy. l - B iMhere B is the velocity of the particle in the star system diAJided by the speed of light. Therefore we see Q. < ll di o§ d'z' ch 2 cdt dx dy — 8 dz = dV 1 - B The components of G11 physically represent: G1 = ch , G2 = ch , G3 = ch , G“ = dE, x y z where P and E are momentum and energy respectively. The value of these components follow from the evaluation11 of aTuv 3X dV for each value of u. Hence the first three com— ponents of Gu represent c times the amount of momentum contained in the volume dV, and G” represents the amount of energy contained in this volume. The definition of TUV is as follows: let gi repre- sent the density of momentum in the direction i. Then F T14 = cgi, Tu“ = energy density, and T13 = —$ + gi wJ, A J where F1 is the force in direction i exerted by the medium (in this case light) through the area A normal to the J J direction, and wj is the velocity of the medium in the direction 3. Using these definitions we can write the T1“) com- ponents for a pencil of radiation. The relation between the energy and momentum of radiation is E = Pc Therefore energy density and momentum density are related by uf = go, where u.f is the fractional amount of energy density u contained in the solid angle d9. This plane wave has the direction cosines cos a, cos 8, cos y. Hence: _ uf _ u dQ g - —— - — ——, c c An = 3 d9 cos a = 3 d9 cos B = 3 d9 COS g1 c E? ’ g2 c E? ’ g3 c E? Y’ wl = 0 cos 0, W2 = c cos 8, and W3 = c cos y. Since we are concerned with radiation in empty space Fi = 0. Let c be the angle with the z axis and e be the angle with x axis in spherical coordinates. Then cos d = sin c cos 9, cos B = sin c sin 9, cos y = COS ¢. Now using the above expressions the Tuv components may be written as follows: T11 = u %% cos 2d = u %% sin 2c cos 26, T22 = u %% cos 2B = u %% sin 2¢ sin 29, T33 = u %% cos 2y = u %% cos 2c, TMl = u %9, n T1“ = TAll = u %% cos a = u %% sin ¢ cos a, T2“ = T”2 = u %% cos B = u %% sin ¢ sin a, T3“ = T“3 = u %% cos y = u %% cos o, T13 = T31 - u %% cos a cos y = u %%-sin c cos 6 cos d, T12 = T21 u %% cos 8 cos a d9 . 2 . u 3? Sin ¢ cos 6 Sin 6, T32 d9 d9 and T23 u 3? cos 8 cos y u 5? sin ¢ sin 9 cos d. In order to calculate the force acting on the parti- cle we go to the particle system (denoted by bars) and evaluate G3: -32 -33 —34 3T 3T 3T dV. -31 ELL—4. + + 3% ay 3% cat '53:- To find the total value of G3 associated with an object of volume V, the right side of the above equation must be inte- grated over the object's volume. —31 -32 -33 —3 _ 3T + 3T + 3T dV . . . . . . . (2) 3% By 32 (I) I I 34 The last term has dropped out since T is not an explicit function of time. The values of the tensor components T31, T32 , and T33 are now needed. To obtain these we use the tensor prOperty Tu“: 3;“ 3§v TaB 3x“ 3x8 where the relations between in and xu are given by i1 = x1, i2 = x2, i3 =y (x3 - Bx”), and in =y (xu - 8x3) 1 where B = K and y = 1 - B For the particular values of p and v we see that 10 T31 = yT31 - BYTul’ ‘ T32 = YT32 _ BYTu2, and F. o o o o (3) T33 = Y2 T33 - 2y2 6T3” + 8272 Tun. 1 Substituting the values for T31 and TM1 into the first of these last three expressions, we have T31 = y %; d9 sin ¢ cos 6 cos c - By%? d9 sin c cos 9. Now use is made of the following relations between the co- ordinates of the two frames of reference cos ¢ = cos ¢ + B , (l + 8 cos 6) ¢l - B2 sinA§ (l + 8 cos 5), sin ¢ = = (l - 82) d5 (1 + 8 008 $)2 The plane angles 5 and 6 and the solid angle dfi = sin 3 d3 d5 are measured in the particle system and correspond to the angles ¢, 6 and d9 in the star system. The expression for cos d in terms of cos 3 is well known. The expression for sin ¢ can be obtained by simply using the relation sin 2o = l - cos 2o. The expression for da can be obtained »by differentiating cos d and remembering that 5 = 6. Using these eXpressions relating the angles in the two coordinate systems we can write the tensor components in the star system in terms of the angles 5, o, and d5. Then 11 substitute them into equation (3) to obtain: 2 2 _ _ _ T31 = u_ (l - B ) cos 6 sin c cos ¢ d5, u" (1 + 8 cos E)“ —32 u 2 2 sin 5 sin 5 cos 3 — T =—(l"8) -14 (in, An (1 + 8 cos c) 2 2 2- and T33 = 3— (l ‘ B ) COS *1 d6. An (1 + 8 cos 5)“ 'These eXpressions for the components of the stress-energy tensor are those of a pencil of radiation whose direction is specified by the angles 5 and E in the particle system sand are independent of any absorbing or scattering surface. In order to evaluate the integral for G3 the deriva— ‘tives of the tensor components must be specified. Suppose we kmye a perfectly absorbing particle, i.e. one that ab- scurbs all radiation incident upon it. Consider the first ternn in equation (2)1 {3T_ di d§ d2 cdt. ax V ASSUIne that the particle is large compared to the wavelength of raufliation and note that the particle is a sphere in the parthzle system. An incident plane wave in this case illum- inat€355'the front half of the sphere and does not illuminate the t>aick half. Assume that there are no electromagnetic Waves; inside the sphere, so that they do not penetrate the squYic3e but are instead completely absorbed at the surface. 12 Thus Tuv is zero everywhere inside the sphere. Also, since this sphere does not scatter any waves which strike its sur- face, the stress-energy tensor immediately outside the front half of the Sphere is the same as that for the plane wave. The stress-energy tensor is zero in the region Just outside the back half of the sphere. Thus: -3l _ _ 3T_ dX = 1T31 on the front hemisphere, 3X BT31 - _ dx = O on the back hemisphere. , 3X We are now left with a surface integral over the area S of the front hemisphere ; T31 d§ d2 S Since T31 is independent of i, y, and 2 this expression becomes: 31 ft d§ d2 S 8| : d§ d2 = I - d8 3 Emit (0% + Where I is a unit vector in the x direction and d8 is a pOr”tion of the sphere's area. Now consider the volume in- closed by the surface of a hemisphere of radius a and its flat: side of area F = na2. By the divergence theorem 3 I - ds + I - aE = o. s; TheI’efore —> —> JI-dS=-II-dS=-na2cosd 13 where E is the angle between the direction of the incoming radiation and the x axis. _3l _ _ - _ Hence SBT_ dV = -na2 T31 cos a cdt. 3x V Similarly it can be shown that j _32 - _ - _ 3T_ dV = -na2 T32 cos B cdt, 3y 7' -32 _ _ _ - and 3T_ dV = -na2 T33 cos y cdt. 82 V Substituting these values into equation (2) we have G3 = na2 cdt (cos 5 T31 + cos E T32 + cos § T33). Remembering that the value of G3 is chZ we can now write ch F = Z = flaz (cos 5 T31 + cos E T 2 + cos F T33). z cdt Sulrstituting into the above equation the values for the Coexine angles and the tensor components in terms of E and 5 We have 2 2 - .Fz == “32 (l - B ) don 2— (sin 25 Cos 25+sin 25 sin 25 (l + 8 cos F) A" + cos 25) cos 3. 1“ Since the velocity of the particle is in the z direction FZ = Fz and therefore 2 2 - — Fz = FZ = n&2 u (1 ' B ) Cfsui 99 . . . . . (u) (l + B COS ¢) UN This expression gives the 2 component of force in the star system exerted on the particle by the portion of radiation contained in the solid angle dfi. One of the factors (1 — 82)/(l + 8 cos $)2comesfkmm1the transformation of the energy density and the other comes from the transformation for the solid angle. Since u is the magnitude of the pres- sure, a2nu is the total magnitude of force which would be exerted on the particle if B were zero. Thus the effective absorbing area of the sphere is equal to the cross-sectional area of the sphere as expected. An alternative way of look- ing at this result is to consider the absorption of momentum. 'The sphere can be replaced by a perfectly absorbing disk of aarea na2 whose normal is parallel to the path of the incident Inadiation. This disk then absorbs the same amount of momen- thn in the same direction per unit time as does the sphere, sirice both objects absorb all radiation incident on their Suarface. Hence they both experience the same force. Equation (4) can be generalized to include diffrac- tiCDn effects. In the particle system the incident wave is Stfiill.a.plane wave carrying momentum in its direction of motion. In this reference frame, which is electromagnetically 15 equivalent to any other frame, we have a plane wave incident upon a stationary spherical particle and classical electro- magnetic theory is applicable. When diffraction occurs the amount of momentum absorbed may be more or less than that of the perfectly absorbing sphere. The increase or decrease of absorbed momentum due to this diffraction effect must be in the direction of the incoming wave since the wave and sphere are symmetrical about this direction in the particle system. This increase or decrease in absorbed momentum can thus be accounted for by simply multiplying equation (A) by factor Q. Since we are considering the particle in its rest frame, Q in that frame must be independent of B and can only be a function of the particle's radius a and the incident wavelength I. Hence Q = Q(a, I). The ratio of force exerted on a spherical particle to the force on a perfectly absorb- ing disk of the same radius was calculated by Debye in 1909 using classical electromagnetic theory. This ratio is the Q factor and is usually designated Qpr' Hence 2 2 .— Fzr = a2 E Q r (a, X) (l ' B ) 03849 at . . . (5) A p (l + 8 cos c) The "r" subscript serves to remind us that this is the force due to a single ray. !W.,. . W , .- nun-P 16 Agreement with the Calculation from RobertsonTE'Equation This result can also he arrived at through use of H. P. Robertson's paper "Dynamical Effects of Radiation in the Solar System." He studied the mechanism by which a par- ticle, orbiting the sun, would eventually fall into the sun by the loss of angular momentum. These particles absorb radiation from the sun but re-radiate it with a loss of their own angular momentum. At the beginning of his paper Robertson derived a relativistic expression for the force exerted on a spherical particle by a pencil of radiation. He then made a classical approximation and derived results for the motion of a particle orbiting the sun. His relati- vistic expression for the force was derived by transforming the force vector, a first rank tensor. He3b wrote it in its final form, in the star system, as dm uu fw o T=r” u - wu“). . . . . . . . . . . . (6) where m0 is the proper mass, s the prOper time, up is a com- ponent of unit velocity, and c is the speed of light. For n = l, 2, or 3 the value of t“ is 0 times the cosine angle of incident radiation in the star system and to is equal to unity. Also w is equal to AD, the transformed component of 20 in the particle system. The eXpression f is the product of the energydensitycicfiTtheincoming ray in the star system and the particle's effective cross sectional area A. 17 To change equation (6) to an expression for our case of interest let d = u %%. Evaluate lo by the relation -o A0 = 3x8 28, 3x where i0 = t, x0 = t, 23 - E, x3 = 2 Since 20 = 1, 23 = 0 cos ¢, and t = (t - BZ)Y, we find A0 = y(l - 8 cos d). The particle's velocity v is in the z direction only, so up = VY. Substituting these relations into equation (6) we have d(moyv) u T=AY (1-8008 4)) COS¢fidQ - Ay2 (l -8 cos ¢)2 1% %— d9. 1T let m = moy, divide both sides of the above equation by y, and make the substitution for cos c and d9 in terms of cos 6 and d5. The result is 2 2 Q£EX1 = fig- (1 - B )_ cos 5 d5. dt An (1 + 8 cos ¢) Since A is the effective area A = «a2 Qpr' We have then 22 - Fzr = a2 a Q r (1 ' B ) Cfsu¢ ad . . . . . . (7) A p (1 + 8 cos ¢) which agrees with expression (5). . “ml—A _..‘_E— m: '7 I?" 18 The Exact Expression for Qpr Robertson used the classical approximation of equation (6) to study the effect that radiation from the sun would have on the motion of a particle under the gravi- tational influence of the sun. No calculations have been made of the effect on the motion of particles resulting from a Doppler shift of frequency. My purpose is to determine the effect of this fre— S ‘l’r .. quency shift on a particle in an isotrOpic radiation field. Equation (5) gives the force exerted by a single pencil of light. In order to obtain the total force due to the whole radiation field, equation (5) must be integrated over the unit sphere. To do this Qpr (a,X) must be known. Defining Qpr by the equation F = d Qpr U82 where d is the energy density, na2 the particle's cross-sectional area and F the force exerted on the particle, the value for Qpr was derived by Debye9b in rationalized Heaviside units and was found to be: 1 2 ‘2 2 °° a + (X °° * Q r = A H2 Re 2 (2n + 1) n n - 2 _§fl_i_l_ a i oi p n a n=1 2 n=1 n(n + l) °° n(n + 2) *l l *2 2 _ nil n + l (a n oLn+l + a n OLn+1) ’ ' (8) The amplitude of the wave is represented by H and its wavelength is I in the particle system. The alpha terms 19 are complicated functions defined as follows: a ' ' i a1 = N1 wn (kaa) ki wn (kia) — ka wn (kaa) N1 "’11 (kia) n a I l i 9 N1 Cn (kaa) ki wn (kia) - ka Cn (kaa) N1 l"n (kia) and - - . (9) a ' ' 1 a2 = N2 In (kaa) ki q’n (kia) ’ ka wn (kaa) N2 wn (kia) .m n a ' N c (k a) k w' c' i w - 2 n a i n (kia) - ka n (kaa) N2 n (kia) E The functions of ka are defined by ’ flka 1/2 FF wn (ka) = 2 Jn+l/2 (ka) and 1/2 _ nka 2 Cn (ka) ‘ 2 Hn+l/2 (ka)° 2 The functions Jn+l/2 (ka) and Hn+l/2 (ka) are respectively Bessel and Hankel functions. The constants in equations (9) are defined as a a a a 2 52 Nl=i—, N2=i—, ka —2, C C C NI = 1 ea + a, N: = 1 HE, C C C 2 ’2 ‘ k _ cum _ i now i 2 2 C C The angular velocity, speed of light, dielectric constant, mag- netic permeability, and conductivity are respectively represented by 5, c, e, u, and o. The bars placed on w and A again denote quantities measured in the particle system. CHAPTER III DYNAMICAL EFFECTS OF THE RADIATION FIELD Exact Solutions and Useful Approximations The dependence of the frequency 3 on the angle 3 is given by the well known relativistic expression / 2 v1-8 ...............(10) l + 8 cos d v: where v is the frequency of radiation in the star system. Since the terms in equation (8) are functions of 3 equation (5) is seen to be difficult to integrate because of the complicated dependence on 5. Because of the complexity of the integration I had tried an alternate approach in order to obtain an exact solution for F2. This method is rather long and complicated and was not completed. The basic idea of it is to avoid complicated integrations.by the use of boundary conditions. The rariiation pressure is independent of the direction of pOlarization of the waves so any convenient direction of polaJszation can be assumed. This makes it possible to write and expression for the electric and magnetic components 20 21 of the whole isotrOpic and monochromatic radiation field and not just a single ray of radiation. The components are then transformed relativistically from the star system to the par- ticle system. My experience with this method ends at this point. I have obtained by use of a method developed by E. T. Whittaker in a paper cited above, the expressions for L” the electromagnetic components in the particle system. At n this point these components must be matched with a general wt . .« 0'. {A'V‘Lll'n- ‘ expression for the scattered waves by using boundary condi- tions on the surface of the spherical particle. This is difficult since the frequency of the incident radiation varies with the angle 5. If the scattered waves could be. found a stress-energy tensor could be written and probably integrated over any arbitrary surface containing the particle. For many situations however, approximations are easy to obtain, and they simplify Qpr to make equation (5) possi- ble to integrate by the use of restrictions on the values of I or a. Also computed graphs of Qpr as a function of Zia/X can be used to make approximations. The Relativistic Motion of a Large Perfectly Absorbing Sphere Consider the case of a perfectly absorbing sphere whose ruadius a is large compared to the wavelength of inci- dent raxiiation. The value for Qpr is then unity as it is also irl the case of a large completely reflecting sphere. 22 When Qpr is unity we have from equation (5) 2 2 2 - ZI,=&'u(1'8) COS‘Pdfi ........(ll) ’4 (1+80085)u The total force is then n21! 2 ' 2 - F=a—3 g3 2(l+8)2 3(1+e)3 which simplifies to Q ——§——§. 31-8 Hence Fz = -na2 u i ——§——§ . . . . . . . . . . . . (12) 31-8 The force exerted on the particle of rest mass m0 is F=Cg-—-—O-'——-............o(l3) Equatirug the right sides of equations (12) and (13) we write m =-nau3———§......(lu) 3 if 23 It is important to note at this point that equation (IA) assumes that all absorbed radiation is re-radiated isotropically in the particle system. If it were not re- radiated the increase of absorbed energy would cause an increase in the mass of the particle other than that due to the relativistic increase of mass with velocity. It might L_ be thought that an expression should be added to the right side of equation (1A) since it was derived on the basis of absorbtion and scattering only and no stress-energy tensor L terms were added to account for re-radiation. But these additional terms are not necessary. The sum of any such additional terms is zero since the net loss of re—radiated momentum in the particle system is zero. Now using equation (1A) the particle's velocity can be found as a function of time. Differentiating the left hand side of equation (1A) yields d__.e_ 1 ii cmO dt - moc 2 3/2 t . . . . . (15) 1 _ B2 (1 - B ) Substituting this expression into equation (14) and rearrang- ing ternns we have 2 m C U. dt = _ __g_§___— a O {av/1 - 82 NOW ifflzegrate the left side of the above equation with re- um: SpeCt tn) t from t = o to t = t and integrate the right side 2A with respect to B from so to B where so is the initial veloc- ity VO divided by c and B is v/c at a later time t. We obtain: 5 ha ut = ln 1 + 1 - B2 8O \ 3 moc s 1 + J1 - cb%[ Rearranging terms we find _ fl n32ut B B 3 m c = 0 e 0.....(16) l + Jl - e2 l + V1 - 3 2 I When 8 is small (low velocities), we have a classical approxi- mation: _ 5 na2ut %_ =6? 3 mbc’ 0 Thus whenv=évo, the time constant te is found to be m c te=i ‘2’ (17) A ha u This result is interesting because it shows that the time taken for the particle to slow down to g its initial velocity is independent of its initial velocity, but is pro- portional to its rest mass and inversely prOportional to both the energy density u and its cross-sectional area na2. Solving equation (16) for B we have 28 3 11321.11: 0 e 3 mOc l + vi - 302 e = 2 . . . . . . (18) B 2 _ 8 1T8. ut 1 3 moc + O 2 E} (1 + J1 - 802) Consider equation (18) when t is very large. This expres- sion reduces to : m s: d \f— If so is large then the maximum effect of the relativistic correction is that B is almost twice as large as the classi- cal approximation. We see the retarding effect is not as great in the relativistic case. Looking at equation (12) this might seem paradoxical. If 82 is small equation (12) reduced to the classical result _ 2 A FZ - —fla u g 8 . . . . . . . . . . . . . . . . (19) as opposed to its relativistic form in which F2 is greater than the classical value by a factor of l/(l - 82). Hence the relativistic force is greater than the classical force. But now look at equation (15). It shows that the relativis- tic mass increase changes the expression g3?! by a factor of l/(l - 82)3/2. The relativistic increase of force due to radiation is not as great as the relativistic increase 26 in mass. The net result is that the particle's velocity is retarded at a slower rate at higher relativistic velocities. The cause of the retardation of velocity for the perfectly absorbing sphere is similar to the cause of the loss of velocity of a particle in the neighborhood of the sun. The only difference is that instead of losing angular momentum the particle is losing linear momentum by re-radiat- ing the absorbed radiation. The radiation striking the particle on the front side has an increased energy density due to the Doppler shift. For the same reason radiation striking the particle on the back has smaller energy density. This effect combined with the crowding of radiation in front and the thinning out in back exerts a net retarding force on the particle. In the derivation of equation (18) it was assumed that the particle was large compared to the radiation's wavelength. This assumption breaks down when velocities close to that of light occur. This is because at large enough velocities the wavelength of radiation striking the back of the particle is increased. Hence for certain veloc- ities it can take on values of the same order as the radius of the particle and thus diffraction effects can become im- portant. At this point Qpr is no longer equal to unity. 'Phe value for so at which Qpr is no longer equal to unity depends on the ratio of the radius a to the wavelength A. If a/A is large then the values of so which are valid in 27 equation (18) are large. Similarly if a/A is small the values of so in this equation must also be small. Linear Approximations for Low Velocity Particles Cases will now be considered in which Qpr is not a constant but varies as a function of the wavelength of in- cident radiation. If we consider dielectrics where o = O, and look at the results of Debye quoted earlier (equation (8)) we see that Qpr varies only as a function of the ratio a/X. However, for conducting particles this is not the case. A comprehensive study of Mie scattering theory and Debye's radiation pressure is presented by H. C. Van de Hulst12 in his book Light Scattering by Small Particles. He states that if d is the energy density of incident radiation then the total energy of scattered radiation ES is defined by ca use of the cross section AS defined by the relation C8. Esca = d As ca° Similar definitions are used to define the cross sections A and A8 for the absorption and extinction cross sections. abs xt Conservation of energy requires that Aext = Asca + Aabs' Non-absorbing particles will have Aext = Asca' If cos ¢ de- fines the average cosine angle at which radiation is scattered then the radiation pressure cross section Apr can be defined by the relation 1r Tum \ 28 Apr = Aext - cos ¢ A sea: Discussions in light scattering theory use effi- ciency factors Q rather than cross sections and are defined by dividing the cross sections by the cross sectional area of the particle na2. Then the above equation becomes 5 Q = Q - cos ¢ Qsca (20) { pr ext It is by the use of this definition of the efficiency factor for the radiation pressure and Debye's expression for Qpr war that calculations have been made. Since the efficiency factors are complicated, much use has been made of computers to calculate them. One such case of particular interest is given by W. M. Irvine13a who computed Qpr by use of equation (20) for different values of the index of refraction for both dielectric and absorb- ing spheres. Since the index of refraction for metals varies as an explicit function of frequency, Qpr is for them a more complicated function than it is for a dielectric, for which the index of refraction is a function of e and u only. Since Irvine's results show graphs of Qpr as functions of ‘the wavelength for fixed indexes of refraction, they can be lised for computing the force on dielectrics. The dielectric CCDnstant s will be assumed to be essentially constant for diJelectrics when a small range of frequencies are involved. 311108 the temperature of particles in interstellar space area estimated7b to be from 100K to 300K, or up to 1000K,the 29 variation of s with frequency will be less than the usually small variation at room temperatures.lu In order to understand what kind of approximations of Qpr are important, the extreme values of frequency in the particle system must be considered. The frequency 3 seen by the particle is given in equation (10). The two extreme values of frequency which it sees are the higher frequency sf of the radiation in front at angle E = n and the lower frequency 3b of radiation in back at an angle 5 = 0. Thus ‘ B “Fifi; —\) ————--:L_B=\)l-B b 1+8 fie Notice that for small 8, 3f and 3b differ from v by the same amount v8 but the values are shifted in Opposite directions. 1+3= 1+3 Vl- Cl p—b >. . . . . . . . . . (21) Cl l The difference between these frequencies is generally A3: 3f (22) Now square both sides of this equation and solve for B. We find A3 VQA3)2 + sz It: is seen from equation (23) that B can be small and still 8 . . . . (23) égfisve rise to large frequency shifts if v is large. Since 3O figure (5) gives Qpr as a function of x = 2na/X, we write 2na - AX = v. C Equation (23) can then be written B = . . . . . . . . . . . . . . . . . (2“) 13b Now we look at Figure (A). The function Q is ext seen to be made up of major oscillations with minor oscilla- tions, usually called "ripples: superimposed on them. The major oscillations are caused by the interference between diffracted radiation and transmitted radiation. When the diffracted radiation and transmitted radiation constructively interfere a maxima occurs. When they destructively interfere a minima occurs. Since there is no transmitted radiation in the case of the perfect reflector no such maxima or minima occur in that case. This is seen in Figure (5).90 Finer features of these curves are not so easily explained. Be- cause of the complementary nature of the Eds—3 Qsca term in equation (20) the major oscillations do not occur as strongly in the graph of Qpr as they do in the graph of Qext' To make an approximation for the Qpr function we replace the curves in the graphs by simpler curves. There are two ways of doing this. The whole range of values of x from zero to infinity can be considered and a curve fit can be made which ignores the minor features of Qpr (such as 1F 31 "ripples"), and yet still has the general form of the graph of Qpr' But if the minor features are important an alter- nate approach must be used. A small section of the curve will have to be considered and a curve fit made for it. For practical considerations it is these minor features that are important. To illustrate this we use equation (2A). This equation gives the velocity for a particle which encounters values for x extending over an interval Ax. Consider in Figure (A) the region around x = 1A. If Ax is measured from the peak of the ripple occurring at this point to the first trough on the right, then Ax E %. The result of substituting these values into equation (2A) - is that 8 : 85' Since many dust particles in space have 15a velocities averaging about 7 km/sec, these particles tra- vel at speeds much smaller than one-sixtieth the speed of light and hence intervals smaller than one-half are important. Suppose a particle moves in a monochromatic and isotropic radiation field and has a radius a such that x0 = 2na/Ao where A0 is the wavelength in the star system. Assume also that xO falls half way between a peak and trough of a "ripple". As the particle's velocity increases the value of the frequencies of radiation striking it on the front and back diverge. For small 8 it is seen from equation (21) that the values for x for radiation incident on the front and back diverge equal amounts from the value x0. The divergence of values of Qpr reaches a maximum when the 32 difference between x for the radiation incident on the front and back differ by Ax equal to the distance between a peak and adjacent trough. At this velocity the maximum difference occurs between the Qpr value for the radiation on the front and the radiation on the back. As the velocity increases beyond this point the difference between the front and back l values of Qpr will generally go to zero but increase again . at still higher velocities. Because the curve between the peaks and troughs of the "ripples" are very close to straight lines we will approx- imate the Qpr function as a straight line in these regions. Therefore we write Q = mx + b = m 2na pr + b’ on” where m and b are constants whose values can be taken from the graph. Now using equation (10) the above expression for Qpr can be written 21/2 Q=m2—"iu(1‘8) +b........(25) pr c (l + 8 cos F) Substituting equation (25) into equation (5) we obtain under integration 5/2 3 2 FZ = é—El mu (1 - B ) _ 5 cos 5 d5 + 2c (1 + 8 cos c) 2 2 2 - +IJ—aub(l‘8)coffid§. A (l + 8 008 ¢) 1! ~ I . ‘ ' Vxfivg‘.. . l 33 Designate the first integral on the right hand side of the above equation I1 and the second integral I2. Since I2 is the same as b times the integral in equation (11), we have 2 A s I = - b a nu — ———————— (26) 2 2 3 (l - B ) As before, using the relations 3 = cos E and l ds = - sin 5 d 5 the first integral Il can be integrated {m with respect to 5 and written 1 ‘~ I _. a3u1T2mV 2 5/2 8 d8 . L l - ———-—-— (l - 8 ) 5 = C . (1 + 83) -1 Evaluating the integral we find 5/2 I _ a3uw2mv (1 - B2) -1 + l 1 1 ' 2 3 A + 3 c 8 3(1 + B) A(l + 8) 3(1 - 8) - 1 ] A O, . . . . . . . . . (33) mxO > - Uh for F2 < 0. . . . . . . . . . (3h) The interesting point of equations (32), (33) and (34) is that they are independent of velocity (at low veloc- ities). The factor determining whether the force a particle <3Xperiences is positive, negative, or zero is determined by 37 the relative size of mxO and h. Imagine if h were to in- crease significantly while the value of mxO remained the same. The fractional difference of the value of Qpr for the radia- tion in front from that in the back of the particle would diminish. Thus the DOppler shift would dominate and the force would be negative for a large enough value for h. On the other hand if h is held constant and we imagine that mxO becomes more negative, then the values of Qpr will differ more widely for the incident rays. This can be seen by re- writing equation (25) in the form 1/2 2 Q r = mxO (l - B ) _ — l + h. p (l + 8 cos ¢) As mxO increases the fractional difference of values for Qpr for the two angles 51 and 32 increases and enables the Qpr effect to dominate. It is interesting to look at the force exerted on the particle by two rays whose angles with the z axis are supplementary in the particle system as shown in Figure (2). FIGURE 2. Supplementary ray diagram. NI 38 When 3 is small, higher orders of B can be neglected and the DOppler effect is a linear function of cos $. Since the Qpr function is also linear over a small range of frequencies, and since both the Doppler function and the Qpr function are anti—symmetric about 3 = n/2 it is eXpected that the condi- tions stated by equations (32), (33), and (34) can be gotten by splitting up the isotrOpic radiation into pairs of rays which are anti—symmetric about 3 = n/2. For the pair of rays in Figure (2) then 3 F = F 2 z r (51) + Fzr (52), Substituting the force values given by equations (5) and (25), we have _ cos 5 b cos 3 FZ = a2 3 d9 mxO l _ 5 + l_ u + H (l + 8 cos ¢l) (l + 8 cos ¢l) + mxO cos ¢2 + b cos ¢2 (l + 8 cos $2)5 (1 + 8 cos $2)“ From Figure (2) $1 = n — $2 so cos $2 = - cos $1. Using this relation and expanding the denominators of the above equation to exclude terms of higher order than B we have F = a2 % d5 [mxO cos $1 (1 - 58 cos $1) + b cos $1 (1 — “8 cos $1) — mxO cos $1 (I + 58 cos $1) - b cos $1 (1 + “B COS 31%] F = - a2 % d5 cos 2¢l (5me + Mb) 8, OP xur 'Therefore FZ 0 when m = _ g 0 39 This result agrees with equation (30). Equations (32), (33), and (34) also obviously agree. Thus when the net force on the particle is zero the particle absorbs equal amounts of momentum from the radiation at angles 51 and $2. The radia— tion at angle 51 has a larger frequency and higher density than the radiation at angle $2, but it has a smaller Qpr F factor which cancels out the increased momentum effect of the higher frequency and energy density. Consider a dielectric particle with an index of re- I“ fraction n = 1.33 as shown in Figure (4). If it is moving in a monochromatic and isotrOpic radiation field, and the ratio of its size to the wavelength of radiation is such graph where the slope that xO occurs at a pOlnt on its Qpr has a large negative value, then at low velocities it will experience an accelerating force. However,as its velocity increases,the range of wavelengths of radiation striking it also increases, and at a large enough velocity the linear approximation for Qpr (x) will no longer hold. Then the force on the particle will decrease and eventually reach zero and the velocity will remain constant. By making a non-linear approximation for Qpr (x) this velocity can be calculated. As an example we will assume Qpr = fd3 + md + h, where d = x — x h = Qpr (x0) and f and m are constants. O, Approximating equation (5) to the first order in B we find 2 u Fzr = a 4 Qpr (l — 48 cos E) cos 5 d 5. 40 Using the same approximation for equation (lO) we find 3 = v (1 - 8 cos 5). Therefore d = x - x0 = — xOBs where s = cos 5. Substituting this expression for d into the above expression for Qpr and then substituting this Qpr function into the eXpression for Fzr yields under integration F r l 3 3 4 2 3 4 5 J/ (-fxO B s - meOs + hs + 4fxO B s -l _ 4mx082s3 - 48h32) ds E Carrying out the integration we find _ 2 l 3 3 l 4 FZ — -a un (B-fxO B + §m8xo + 38h). As expected the above equation reduces to equation (28) when the term containing 83 is neglected. Setting FZ = O and solving for B we have -mxO - 4h 8' 3 3 .(35) —fx 5 0 Consider a particle of a size such that xO lies half way between a peak and a trough in the Qp graph. One such I" value for X0 would be approximately 12.5 in Figure (4). Qpr can be approximated in this region by setting f = 1.5, m = - .40 and h = .6. A sketch of (Qpr - h) vs. d is con- tained in Figure (3). Setting these parameters into equation 1 2?‘ At this velocity the radiation (35) we find 8 = ,ou = 41 exerts no force. The radiation exerts an accelerating force on the particle for velocities lower than B = .04 and a re- tarding force for larger velocities. This phenomenon depends on monochromatic radiation and can be expected to occur only in cases where the energy density of radiation is concen— trated about a particular wavelength. Since interstellar radiation energy density is generally small a large amount of time would be necessary for the particle to achieve this velocity as a result of radiation pressure alone. FIGURE 3. Graph of Cubic Approximation for Qpr' (Qpr - h) -- l 5 1 t -l d ‘-l Very Small Particle Approximations Now we turn our attention to particles small com- pared to the wavelength of radiation. For these cases 9d Debye made approximations by considering three different sets of values for the conductivity and the dielectric 136' .- - .. _. '1 (11...... 42 For the dielectric with e = w and o = 0, Debye constant. 2n& 4 ) . For finite values A found that Qpr is proportional to 4 2"a} but has a different of e, Qpr is also proportional to A prOportionality constant. If 0 is not zero then Qpr is pro- 2Za. In all cases p = l. A Now we are interested in calculating the effects portional to of the radiation forces on very small particles. First con- sider the case whereo # 0. Debye gives the pressure efficiency factor on the particle due to a plane wave as (9) - 2wa Q = 12 w - . . . . . . . . . (36) pr (8 + 2)2 +(02 X to Rewrite this in the form Qpr = nip (i) (37) 312-) where F (X) = 2 . 2 o (e + 2) + (:) In calculating the total force due to the radiation field a difficulty arises because for metals both a and o ShOW'a strong dependence on wavelength and therefore cannot EH3 treated as constants. However the values of e and o havev tMien experimentally determined for various metals and wave— lengths, and from these values F (X) can be obtained. J- L. Greenstein lists values for F (X) for iron and nickel at various wavelengths. It can be seen from his table that 43 for wavelengths near 5000A, F (X) can be approximated for these metals by the linear function F (X) = rX + t . . . . . . . . . . . . . . . . (38) where r and t are constants. For iron Greenstein states the values of F (X) to be .68, .62, and .55 for the wavelengths of 4410A, 5080A, and 5890A respectively. For nickel he has my, F (X) equal to .33, .24, and .19 for the same respective wavelengths. These values also agree with computations made from the indexes of refraction given by Van de Hulst.12a W The function F (X) is closely approximated over this region by . F (X) — 8.8 x 103? + 1.07 4 and F (5") _ lo 7‘ + .76 >. o o . . . . . . (39) / for iron and nickel respectively. The wavelength is measured in centimeters. Combining equations (37) and (38) we find Qpr = 4tx + 8nar . . . . . . . . . . . . . . . (40) A linear approximation of F (X) in X leads to a linear approximation of Qpr as a function of x. Thus from the values for r and t in equations (39) we have for iron Qpr = 4.3 i — 2.2 x lo5 a, . . . . . . . . . (41) and for nickel Qpr = 3.0 i - 2.5 x lo5 a. . . . . . . . . . (42) Since iron and nickel are thought to be relatively abundant in space, the effect of radiation pressure on them is of ‘1— 44 interest. An estimate of this effect will be made later by use of the above Qpr functions. Now we consider the two other radiation pressure efficiency factors given by Debye. For 0 = O and e = w 4 Q r 2 ii (iii) . . . . . . . . . . . . . . . (u3) pl 3 A For 0 = 0 and finite e 2 4 Q r 2 Q (E ' 1 (2"3) . . . . . . . . . . . (nu) p2 3e+l X Both functions have the same dependence on X, they only differ in their constant coefficients. Writing these functions in terms of U and using equation (10), the resulting functions Q and Qpr are substituted into equation (5) and yield prl 2 4 4 6 4 2 FZ = KI} uuflna V (l - B 3 8 COS 3 d9 c (l + 8 cos ¢) where K represents the constant coefficients in either Qpr l or Qpr . Now as before, using the substitution 5 = cos E, 2 the above equation becomes 4unua6 u V F = K 8' c (l + Bs) “flag! , uu' 45 Integration yields h F = K 8un5a6 V4 (1 - B2) [_ 1 + 1 Z on 82 6 (1 + mg 7(1 + m7 + - . 6(1—8)6 7(1-s)7] By multiplying and rearranging the terms, this equation becomes 2 3 l 5 6 4 (—B + -B -B ) FZ = i-l28Kun5 a X 3 3 2 g c (l - B ) Hence for a particle with o = 0 and v = m the constant K has the value lg so 8u56v“(38*§83+%85) FZ = —7 X 2 -n a _4 2 3 . . . . . (45) l 3 c (l - B ) 2 For 0 = o and e finite, K = 8 (: + 1) so _ _ 5 B + £83 + 78 ) u ( F = _210 u M5 6 v ?(E - l) (46) Z? 3 e + l (l -B 2)3 Both F21 and FZ2 are prOportional to a6 and therefore become smaller much faster than does the mass. These forces are good approximations when the wavelengths are large compared to the particle's size. At higher relativistic velocities this approximation will eventually break down since the wave- length of radiation incident on the front of the particle is shortened. '“fifllv .r’ffl-'uz'l v .3 ‘VJJM‘Y . 46 At low velocities higher order terms may be ignored so equations (45) and (46) reduce to 4 8 u 5 6 v 8 F = -7 x 2 —w a - Z1 3 0—5- 3 ' ° (“7) 10 u 5 6 4 — l B and FZ = —2 —n a £4 (5———— -— (48) 2 3 c e + l 3 It might seem that these forces are large because of the large value of v“. But it must be remembered that a_v these forces are based on the approximation that a/A = c is much less than unity. Hence the radius of the particle will be very small and consequently both forces will be small. CHAPTER IV EFFECTS OF RADIATION ON PARTICLES IN INTERSTELLAR SPACE The Time Constant te In the preceding chapters we obtained expressions for the force exerted by the radiation on particles in terms of the following parameters: the radiation energy density, frequency, the particles radius, dielectric constant, and conductivity. However the actual magnitude of the forces are by themselves not very informative. The forces are small, but another parameter, the particle's mass, is involved and since the mass is also small the effect of the force on the particle's motion may be significant. In this chapter we will make numerical estimates of the effect of the radiation force by computing the time constant te for each case. For practical reasons we are interested in particles traveling at low velocities compared to the speed of light. For this reason we use the first order approximations for the forces which are prOportional to B. In general we have expressed all forces in the form F = k8 ‘l 47 48 where k is a constant which can be either positive (accelerat— ing forces) or negative (retarding forces). In most cases k is negative. Since we neglect relativistic mass changes at low velocities, we have dv m0 at = kB' The integral of this expression yields kt _ m c v - v06? 0 , “TL 9 . “J?“ ' ’9‘ ‘ where V0 is the velocity at t = 0. The value of its time constant is _ o te - TET . . . . . . . . . . . . . . . . . . . (49) If k is negative then te is the time it takes a particle to reach é times its velocity at t = 0. This result is inde- pendent of the initial velocity v0. If k is positive te is the time it takes to increase its velocity to etimes its initial velocity. The time constant is a reasonable indica— tion of the effect of the force on the particle's motion. We will now evaluate it for the various forces previously obtained. Numerical Results If the force on the large perfect absorbing sphere is approximated for small 8, then we have from equation (17) =_3_i‘o__° 4n32 u te 49 This expression, in terms of the particle's density p, is written — BEE te - u . . . . . . . . . . . . . . . . . . . (50) The time constant is seen to be prOportional to the particle's density and radius, and inversely prOportional to the energy ‘. density. In interstellar space it is estimatedle that mm .1131) u = 12 x 10-13 erg/cm3. Consider a metal such as copper, iron or nickel with a den- sity of approximately 8 gm/cm3. We choose a particle of ; radius a = 2 x lo—ucm which,for many cases, is large enough to exclude significant diffraction effects. For these values equation (50) yields te E 4 x 1019 seconds E 1012 years. This is a long time indeed, even by astronomical standards. 17 Astronomers have recently estimated the age of the galaxies to be about 1010 years. This is much too short a time for isotropic radiation to affect the motion of larger inter- stellar particles. The time constant above can be made smaller by decreasing the radius a, but any appreciably smaller value for a will necessitate the consideration of diffraction effects. The above value for te was calculated for particles large compared to the largest wavelength making an important contribution to the energy density u. Now let us go to the "‘1... 50 opposite extreme and consider particles small compared to the smallest wavelength contributing significantly to the radiation energy density. From equations (49) and (29) we find _ c 4ap te -' u (Smxo + “by c o o o o o o o o o o o o (51) Since b is seen in equations (41) and (42) to be prOportional to a, the time constants for small metal spheres are inde- pendent of the particle's size. If we choose A = 5 x 10-5cm and consider iron of density p = 8 gm/cm3, the time constant can be evaluated from equation (51) using the values for m and b in equation (41). At these values we have for iron t8 5 4 x 1017 seconds E 1010 years. For nickel, using the values from equation (42) and a den- sity p = 9 gm/cm3, equation (51) yields 1018 seconds 2 3 x 1010 years ll? te Radiation pressure is generally larger for metal particles than for dielectric particles because metal both scatters and absorbs radiation. However, since the density of metal is also larger, this partly counterbalances the effect of the larger radiation pressure on the particle's motion. Debye's approximation given in equation (36) holds when the ratio 2na/I is on the order of .8 or smaller. It breaks down of course when the particle is so small that it does not have macroscopic qualities. The effect of radiation "'1 g‘w . . r - 51 pressure on metal particles larger than those considered here is complex and I have not made any estimates of te for those cases. For small dielectric particles with e = m (perfect reflector) we have, using equations (47) and (49) 5 t - 9mg 0 - 30ac . . . . . . . . . (52) e 7 x 28u n5a6vu _ 28uxi To find an order of magnitude for this time constant let 1 _ 3 6 x = I0 , p - l gm/cm , and a = 10- cm for A = 6000A. Therefore te E 3 x 1019 seconds 5 1012 years. However if x = % and a = 3 x 10-6 cm, all other parameters remaining the same as above, we find te E 7 x 1017 seconds 2 2 x 1010 years. Again the time constant is large but it is close to the estimated age of the galaxies. For the case of a small dielectric particle with a dielectric constant é we have from equation (48) 9m 05 _e+l 10 5 6 4 ‘ ( TT V e - l 393%, . . . (53) 64ux Comparing this to equation (52) we see that for e = 1.5 the values for te are about twice those given by equation (52) for the perfect reflector. Hence for the same two sets of parameters chosen for equation (52), equation (53) yields 52 2 x 1012 years for the first case and 4 x 1010 years for the second case. t e t e It should be remembered that particles in interstellar space generally have low temperatures and this could be an impor- tant factor in determining the values for the dielectric constant and conductivity of a particle. Particles with a size on the order of the radiation's wavelength are also important to consider. For such cases equation (29) is useful in finding the effects that the radia- tion has on the particle's motion. Since we are interested in low velocities, higher order term of B have been ignored in equation (29) and using equation (31) we write FZ=—‘-3ina2(mxo+uh)c ..........(54) Now consider Figures (4) and (51 Figure (4) is an example of Qpr varying as a function of x for a dielectric (n - 1.33). Figure (5) shows the Qpr for a perfect reflector. From these graphs we can measure the lepes at various values of x and substitute them into equation (54) to determine the force on the particle. In Figure (5) the greatest force exerted on a particle will occur where x < 1 since the slope there is very large. A representative point on this part of the graph can be taken as the point where Qpr = 1. Thus x = .7 and m = 5. Equation (54) then yields F =--una B. 53 If we choose A = 5 x 10-5 cm and assume all energy density u has this wavelength we have then a = 5 x 10-6 cm and F = - 3 x 10—22. To determine the effect of this force on Z the particle's motion we again compute the time constant. Substitute the coefficient of B from equation (54) into the denominator of equation (49) and write the mass in terms of ‘1‘- cu- its density p and radius a. The result is _ 4cao te _ uImXO + 4h! . . . . . . . . . . . . . . (55) Assume p = l and the values of the parameters in the force above, we have then from equation (55) t8 5 5 x 1016 seconds 5 2 x 109 years. In interstellar space however, the radiation density u does not consist of monochromatic radiation, but is made up of a continuum of different wavelengths. If this radiation is concentrated over a region of about %A then it can be seen from Figure (5) that the lepes for values of x in this region are of about the same size as the lepe chosen above at x = .7. For such a case te would be expected to be of the same order of magnitude as computed above. Its exact value would, of course, depend on the distribution of energy density. For cases where x > 1 we see from Figure (5) that the slopes are negative. For a particle of radius 5 x 10-5 cm, again setting A = 5 x 10-5 cm, we find a II! x 6. Therefore m E 0 and h E 1. Here we see that equa- tion (55) reduces to equation (50) as expected for larger 54 particles. It is also seen that for dielectric material of density p= 1 equation (55) yields for these values te = g x 1018 seconds 5 4 x 1010 years. For smaller values of x, where x is still greater than unity, the slope is negative but the value of h increases. In this region we see then that the value for te does not vary } appreciably. Figure (4) shows the behavior of Qpr for n = 1.33 E nus e to be quite different from that for n = w shown in Figure (5). The ripples in Figure (4) make the behavior of the particle's motion much more sensitive to the wavelength than in the case of the perfect reflector. For values of x ranging from about 2 to 5, values of te will be on the same order of magnitude as for the perfect reflector. If we choose x = 4 to repre- 10 years when sent this region we find that te E 5 x 10 A = 5 x 10—5 and p = 1. Of particular interest are the ripples occurring at values of x larger than 10. As was shown before, a particle may experience either a retarding or an accelerating force in this region. If the slope is positive it is a retarding force. If the lepe has a large enough negative value the force accelerates the particle. Consider a typical value for the larger lepes of a particle 'with a value xO between 10 and 15. If the slope is positive ‘then in the linear regions of the graph m = 1. If it is :negative m E — %. Hence for a retarding force in the middle 55 of this region 10 te E 2 x 10 years when A = 5 x 10—5 cm and p = 1. An accelerating force yields t E 8 x 1010 years in the same region, where now the time e constant is the time necessary for the velocity to increase by a factor of e. Again these results hold for monochromatic radiation. Unless the energy density is concentrated at certain wave- lengths an averaging effect would be expected which would reduce the effects of the "ripples". In such a situation some wavelengths would exert a retarding force, while others would exert an accelerating force. If we assume a distribu- tion among the wavelengths such that the value for Qpr can be approximated by a straight line of lepe zero, and h E %; then if p = l the time constant has a value between 1011 and 1012 years in this region. ‘Fimfi . ~ a-. “w.“ CHAPTER V SUMMARY The derivation of the expression for radiation F. ' force was based on relativistic mechanics. The stress- energy tensor for a pencil of radiation was transformed from the star system to the particle system. Then by inte- grating the partial derivatives of its components the radiation force was obtained. This result was also shown to be in agreement with a result calculated from Robertson's equation. The effects of the DOppler shift and the dependence of Qpr on wavelength in an isotropic, and often monochroma- tic, radiation field have been investigated. We have shown that the total resulting force on a particle moving through the field will generally retard the motion of the particle; although it can in special cases accelerate a particle to a velocity where the total radiation force will become zero. It has also been shown that at non—relativistic speeds, the ratio of the initial velocity to the velocity at a later time will be independent of the initial velocity. Finally, numerical estimates were made to determine the order of magnitude of the time constants for this effect 56 57 on particles in interstellar space. The results showed that the time constant has a value on the order of from 1010 to 1011 years for many cases. This value is comparable to some present day estimates of the age of the galaxies. 58 FIGURE (4) Irvine's graph of Qpr for n = 1.33. ' ' l A? 1 ‘r-T'M"‘1 ' l T I F‘T ' I ‘ l ' 17—T I' 7 1 P L on! 4 3 2 P J 5' Wfidfi - 2 8 (Co: 9'- 2.4 on " out " // Cos 9) .8'- WLG -5 "' (I ‘ V. ' W12 . o, -_ 4” ' ‘ lice A14111414_1_1_l PJLLLI+1¥1AL 0 1 A O 2 4 6 8 '0 l2 I4 l6 IS 20 22 24 26 28 30 l FIGURE (5) Debye's graph of Qpr for the perfect reflector pl" 1. 2. \lONUT 10. ll. l2. 13. REFERENCES J. H. Poynting, Phil. Trans. 202A, 525 (1903). J. H. Poynting, Collected Scientific Papers, p. 754. Cambridge University Press, Cambridge, 1920. H. P. Robertson, Monthly Notices Roy,_Astron. Soc. 91, (1937) a) page 423-U28. b) page 428. R. Schlegel, Am; J; Phys. 28, 687 (1960). w. Rindler and D. w. Sciama, Am. J; Phys. 29, 643 (1961). J. Terrell, gm;_l; Phys. 29, 6ND (1961). J. Dufay, Galactic Nebulae and Interstellar Matter, Philosophical Library, New York, 1957. a) page 223. b) page 237—243. G. Mie, Ann. d; Physik 23, 377 (1908). P. Debye, Ann. g; Physik. 19, 57 (1909). a) page 57-136. b) page 91. c) photographically reproduced from Figure (6) page 108. d) page 105, 112, 116. E. T. Whittaker, Mathematische Annalen 51, 347 (1903). For the definition of the stress-energy tensor components see R. C. Tolman, Relativity, Thermodynamics and Cosmology, p. 72. Oxford University Press, Oxford, 1934. H. C. Van de Hulst, Light Scatteringby Small Particles, Wiley and Sons, New York, 1957. a) page 273. w. M. Irvine, J. Opt. Soc. Am. 9;, (1965). a) page 16- 21. b) photographically reproduced from Figure (2) page 18. 59 1U. 15. 16. 17. 60 The variations of 6 with changes of frequency at low temperatures was investigated by Smyth and Hitchcock, J; Am. Chem. Soc. 55, 4631 (1932). C. w. Allen, AstrOphysical Quantities, University of London, Athlone Press, London, 1955. a) page 225. b) page 228. J. L. Greenstein, Harvard Obs. Circular, No. “22, 10 (1937) . F. Hoyle, Galaxies, Nuclei and Quasars, p. 61. Harper and Row, New York, 1965.