Galactic Parameters Derived from Open-Cluster Data T

Astronomy Reports, Vol. 48, No. 2, 2004, pp. 103–107. Translated from Astronomicheski˘ı Zhurnal, Vol. 81, No. 2, 2004, pp. 124–128. Original Russian Text Copyright c 2004 by Gerasimenko.

Galactic Parameters Derived from Open-Cluster Data T. P. Gerasimenko Astronomical Observatory, Ural State University, Yekaterinburg, Russia Received September 20, 2001; in ﬁnal form, July 26, 2003

Abstract—The components U0 and V0 of the solar motion and the Oort constant A0 are determined using the data of a homogeneous open-cluster catalog with corrected distance moduli. The results are based on a sample of 146 open clusters with known radial velocities located in the Galactic plane (b<7◦)within 4 kpc of the Sun. The solar Galactocentric distance R0 is determined usingtwo kinematic methods. The followingresults are obtained: A0 =17.0 ± 0.9 km/s kpc, U0 =10.5 ± 1.0 km/s, V0 =11.5 ± 1.1 km/s, R0 =8.3 ± 0.3 pc. c 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION distance-modulus estimates from the Uniform Cata- logand those determined usingHipparcos trigono- One of the traditional areas of kinematic studies metric parallaxes [12]; the distance scale changed is the determination of the main Galactic parameters by −0.153m, independent of the cluster age. The required to construct theoretical equilibrium models: clusters span a fairly wide age interval. Clusters the gradient of the angular velocity (the Oort constant in different age groups are known to have different AO), the Oort constant B0, and the principal scale motions. Analysis of the rotation of the open-cluster parameter—the distance of the Sun from the Galactic subsystem revealed a small group among the young center, RO. These parameters are difficult to measure, clusters whose motion differs from that of the cluster first and foremost, because of the presence of local sample as a whole [13]. To ensure uniformity, we group motions, which hinder selection of a model to excluded these objects from the sample of young describe the motion of these objects and degrade the open clusters when deriving final estimates of the accuracy of the derived parameters. These parameters Galactic parameters. We introduced weights to allow should be determined by comparingresults obtained for variations in the quality of the initial data. for various methods usingtrustworthy, homogeneous data sets. One such data set is our continuously 3. ANALYSIS METHODS updated, uniform catalogof photometric parameters for 425 open clusters, which have been redetermined 3.1. Components of the Solar Motion and the Oort usingpublished UBV, uvbyj, and DDO photometric Constant A0 data [1]. (Hereafter, we refer to this as the Uniform We determined the components of the solar motion Catalog.) The large size of the open-cluster sample, with respect to the centroid of the open clusters and uniformity of the parameter estimates, which were the Oort constant A0 by solvingthe followingsystem obtained usinga singletechnique, and availability of of equations: radial-velocity data for a large number of the clusters make this cataloga useful tool for the analysis of the Vr = Vr + U0 cos l cos b + V0 sin l cos b + W0 sin b structure of the Galaxy and the kinematics of disk (1) objects. In the current paper, we use only the radial = −2A0(R − R0)sinl cos b, velocities and, therefore, do not estimate the Oort V U V constant B0. where r is the observed radial velocity; 0, 0,and W0 are the components of the solar motion (the X, Y ,andZ axes point toward the Galactic center, in 2. CLUSTER SAMPLE the direction of the Galactic rotation, and toward the North Galactic Pole, respectively); l and b are Galac- Radial velocities are available for 205 of the tic coordinates; and R and R0 are the Galactocentric 425 clusters listed in the Uniform Catalog. We distances of the cluster and the Sun, respectively, so adopted the radial velocities from sources listed in [2– that 11]. We corrected the distance moduli of the clusters 2 2 − 1/2 by subtractingthe mean di fference between the R =[(R0) +(r cos b) 2R0r cos l cos b] . (2)

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Some previous determinations of the solar Galactocentric distance

No. Reference Year R0,kpc Objects 1 Weaver [14] 1954 8.2 Open clusters, Cepheids 2 Feast and Shuttleworth [15] 1965 9.9±0.9 В stars, open clusters, Cepheids 3 Barkhatova and Gerasimenko [16, 17] 1973, 1985 8.2±0.5 Open clusters 4 Merriﬁeld [18] 1992 7.9±0.8 НI 5 Reid [19] 1993 8.0±0.5 6 Feast and Whitelock [20] 1997 8.5 Cepheids 7 Ollingand Merri ﬁeld [21] 1998 7.1±0.4 Cepheids 8 Metzger et al. [22] 1998 7.66±0.32 Cepheids 9 Glushkova et al. [23] 1999 7.4±0.3 Open clusters, Cepheids, supergiants 10 McNamara et al. [24] 2000 7.9±0.3 RR Lyr, Miras 11 Gerasimenko (this paper) 2003 8.2±0.3 Open clusters

In the calculations, we weighted Eqs. (1) in accor- result, only a few of all clusters with known radial ve- dance with the errors in both the radial velocity and locities can be used in these analyses. Weaver himself 2 2 −1/2 used 13 clusters, while Barkhatova and Gerasimenko the heliocentric distance: P =(σr + σ ) ,where Vr used 19 [16] and 21 [17] clusters. Here, we estimate σ2 and σ2 are the errors in the distance and radial r Vr R by applyingWeaver’s method to 28 clusters. The velocity, respectively. O objects used to determine RO must satisfy the follow- ingcriteria. 3.2. Solar Galactocentric Distance (1)Theymustbefairlydistant,r>1 kpc. (2) They must be located in the ring 0.5 < |R − The question of the correct value of the scale R0| < 1.5 kpc. parameter R0 remains open, and available estimates range from 6 to 10 kpc (see table). Moreover, some (3) Their orbits must be close to circular. estimates show a decreasingtrend [23]. Reid [19] (4) The dependence of Vr on C(R, R0),deﬁned compiled a list of RO estimates datingfrom the 1970s below by formula (6), must be nonlinear. onward and plotted R as a function of date. It is evi- We found a total of 28 clusters meetingthese dent from his plot that, on average, RO decreased un- criteria. Figure 1 shows their distribution. til the early 1990s, then began to increase. However, the mean R remains near 8.0 kpc. The table gives a We can see four groups of clusters located in two 0 (inner and outer) critical domains in Fig. 1. The inner short list of RO determinations, includingsome early estimates of this parameter. clusters make up two groups located closer to the Galactic center (a total of 20 clusters), while the Several kinematic methods for determiningthe outer clusters comprise two groups located farther Galactic scale parameter are known. Almost all re- from the center (a total of eight clusters). All the quire that the input data be selected in accordance clusters have ages log T<8.0 (where T is in years). with stringent criteria and assume that the objects We assume the clusters move in circular orbits in the studied move in circular orbits in the Galactic plane. Galactic plane. We varied the preliminary RO dis- In this paper, we use two methods. ∗ tances (denoted R0) from 6 to 11 kpc. The angular- velocity gradient, ω (R0), is negative in all domains. ∗ METHOD OF WEAVER It can easily be shown that, if R0 >R0 (which cor- | ∗ | responds to the outer critical domain), ω (RO) < This method for estimatingthe Galactic scale pa- |ω (R )| R∗ ω (RO) . The two monotonic functions fairly severe restrictions that must be imposed on the have diﬀerent slopes and, therefore, intersect. The initial sample in terms of the locations of the sample point of intersection corresponds to the true value of objects in the Galactic plane relative to the Sun. As a R0.

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Y, kpc Vr, km/s 3 100 inner 80 outer 2 60 1 40

0 20 Toward Galactic center 0 –1 –20

–2 –40 –60 –3 –80 –1.0 –0.5 0 0.5 1.0 1.5 –100 X, kpc –8 –6 –4 –2 0 2 4 6 8 10 12 2 C(R, R0), kpc Fig. 1. Location of clusters used for Weaver’s method projected onto the Galactic plane. Fig. 2. Dependence of cluster radial velocities Vr on the parameter C(R, R0).

For objects located in the domain |R − R0| < 1 kpc, the angular velocity ω(R) depends linearly on ∆V = Vobs − Vmod,whereVobs is the observed radial the Galactocentric distance and can be represented velocity corrected for the Sun’s motion toward the by the ﬁrst term of the expansion apex (or Vr )andVmod is the model radial velocity calculated usingthe formula ω(R)=ω(R0)+(R − R0)ω (R0). (3) − − It follows that Vmod = 2A0(R R0)sinl cos b. (7) − Vr = R0(R R0)ω (R0)sinl cos b. (4) The clusters used must meet the followingconditions: The main relation used in Weaver’s method is | − | (1) R R0 < 1 kpc, where ω(R) depends linearly ω (RO)=Vr /C(R, R0), (5) on R; where (2) 1.5

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ω '(R0), km/s/kpc V0 =11.5 ± 1.1 km/s; we assumed that W0 is known –3 and equal to 8.2 km/s [13]. We used these values to correct the observed radial velocities for the solar –4 motion. Note that the adopted W0 value has little effect on the components U0 and V0.Wefoundthe –5 Galactic-rotation rate at the solar Galactocentric distance to be A0 =17.5 ± 0.9 km/s kpc. –6 Weaver’s method yields a scale parameter of 8.3 ± 0.3 –7 kpc. We determined the error by propagating the errors in the correspondinginput quantities [25]. Ap- –8 plyingthe same method to the data of the catalog inner described by Dambis [27] yields the same result, R0 = –9 outer 8.3 ± 0.3 kpc. Estimating R0 by applyingFokker’s method to 34 clusters meetingthe corresponding –10 conditions yields R0 =8.0 kpc. 5 6 7 8 9 10 11 12 R0, kpc ACKNOWLEDGMENTS Fig. 3. Dependence of the angular-velocity gradient ω (R0) on the scale parameter R0. This work was supported by the Russian Foun- dation for Basic Research (project code no. 00-02- 16217) and the State Research and Technology Pro- 〈∆V2〉, (km/s)2 gram “Astronomy.” 90 89 REFERENCES 88 1. A. V.Loktin, T. P.Gerasimenko, and L. K. Malisheva, Astron. Astrophys. Trans. 20, 605 (2001). 87 2. E. H. Geyer, Astron. Astrophys., Suppl. Ser. 62, 301 86 (1985). 3. J. Hron, Astron. Astrophys., Suppl. Ser. 60, 355 85 (1985). 84 4. L. M. Cameron, Mon. Not. R. Astron. Soc. 224, 822 (1987). 83 5. D. A. Klinglesmith and J. M. Hollis, Astrophys. J., 82 Suppl. Ser. 64, 127 (1987). 6. Garsia, Astrophys. Space Sci. 148, 163 (1988). 81 7. D. Gaisler, Publ. Astron. Soc. Pac. 100, 338 (1988). 8. D. Leisawitz, Astrophys. J., Suppl. Ser. 70, 731 4 567 89101112 (1989). R , kpc 0 9. T. Liu, Astron. J. 98, 626 (1989). 10. E. D. Fril, Publ. Astron. Soc. Pac. 101, 1105 (1989). Fig. 4. Dependence of the weighted mean squared resid- 11. J.-C. Mermilliod, Astron. Astrophys. 237, 61 (1990). ual radial velocity on the scale parameter R . 0 12. A. V. Loktin and G. V. Beshenov, Pis’ma Astron. Zh. 27, 450 (2001) [Astron. Lett. 27, 386 (2001)]. 4. RESULTS 13. T. P. Gerasimenko, Astron. Zh. 75, 840 (1998) [As- tron. Rep. 42, 742 (1998)]. We calculated the components of the solar mo- 14. H. P.Weaver, Astrophys. J. 59 (10), 375 (1954). tion usingclusters distributed about the Sun fairly 15. M. W. Feast and M. Shuttleworth, Mon. Not. R. uniformly in the Galactic plane (b<7.2◦) within he- Astron. Soc. 130, 245 (1965). liocentric distances r =4kpc and with |R − R0| < 16. K. A. Barkhatova and T. P. Gerasimenko, Astron. 1 kpc. We used the same sample as the one employed Tsirk., No. 743, 3 (1973). to estimate the components of the solar motion to 17. K. A. Barkhatova and T. P.Gerasimenko, Byull. Abas- determine the Oort constant. tuman. Astrofiz. Obs., No. 59, 169 (1985). 18. M. R. Merrifield, Astron. J. 103 (5), 1552 (1992). Our final estimate of the components of the solar 19. J. M. Reid, Ann. Rev. Astron. Astrophys. 31, 345 motion is based on a sample consistingof 149 open (1993). clusters. A least-squares solution of Eqs. (1) yielded 20. M. Feast and P.Whitelock, Mon. Not. R. Astron. Soc. the velocity components U0 =10.5 ± 1.0 km/s and 291, 683 (1997).

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21. R. P. Ollingand M. R. Merri ﬁeld, Mon. Not. R. As- 25. D. Hudson, Statistics. Lectures on Elementary tron. Soc. 297, 943 (1998). Statistics and Probability (Geneva, 1964; Mir, 22. M.R.Metzger,J.A.R.Carwell,andP.L.Schechter, Moscow, 1970). Astron. J. 115, 635 (1998). 26. Fokker, Bull. Astron. Inst. Netherlands 20, 29 (1968). 23. E. V.Glushkova, A. K. Dambis, and A. S. Rastorguev, Astron. Astrophys. Trans. 18, 349 (1999). 27. A. K. Dambis, Pis’ma Astron. Zh. 25, 10 (1999) [As- 24. D. H. McNamara, J. B. Madsen, J. Barness, and tron. Lett. 25, 7 (1999)]. B. F. Ericksen, Publ. Astron. Soc. Pac. 112, 202 (2000). Translated by A. Dambis

ASTRONOMY REPORTS Vol. 48 No. 2 2004 Astronomy Reports, Vol. 48, No. 2, 2004, pp. 108–111. Translated from Astronomicheski˘ıZhurnal, Vol. 81, No. 2, 2004, pp. 129–133. Original Russian Text Copyright c 2004 by Kudryashova.

Dynamical Evolution of Gaseous Clouds in the Atmospheres of Wolf–Rayet Stars N. A. Kudryashova St. Petersburg State University (Petrodvorets Branch), Universitetski ˘ı pr. 2, Petrodvorets, 198904Russia Received March 8, 2002; in ﬁnal form, August 8, 2003

Abstract—We have computed the dynamical evolution of homogeneous, spherical gaseous condensations in the atmosphere of a Wolf–Rayet star. The physical conditions in the condensations vary substantially in the course of their motion in the stellar wind, which should result in variations in the observed spectrum of the star. The condensations also move at velocities of up to 1000 km/s relative to the surrounding stellar wind. Variations of the physical conditions in these condensations should be taken into account in models of the stellar winds of Wolf–Rayet stars. c 2004MAIK “Nauka/Interperiodica”.

1. INTRODUCTION the pressure in the cloud that result in its expansion The suggestion that individual condensations (or contraction), the cloud should move relative to (clouds) occur in the expanding atmospheres of the surrounding gas with a velocity that can reach ∼ Wolf–Rayet stars (WR) [1] is based on specific 1000 km/s. When the volume of a gaseous cloud features observed in spectra of the eclipsing binary with constant mass varies, so must its temperature. WR+OBV 444 Cyg. A model with an expanding In addition, the temperature of the cloud should be af- atmosphere containing clouds was developed in [2] fected by the external medium, which should heat the to explain the simultaneous presence of lines with ap- cloud, since it will usually have a higher temperature. preciably different ionization potentials in the spectra Thus, a gas condensation should be substantially of WR stars. It was suggested that the gas that flows nonsteady-state, with its radiation also varying with from the star contains clouds of various sizes. The time. narrow small peaks observed in the spectra of some Here, we estimate the impact of these factors on early-type stars, which shift along the profiles of broad the dynamical evolution of gaseous clouds in the at- emission lines, were explained in [3, 4] as the radiation mospheres of WR stars. In the absence of a theory of of gaseous clouds moving through a more rarefied cloudformation,wehaveusedasimplephenomeno- medium. logical model with homogeneous, spherical clouds. Later, Brown et al. [5] attempted to derive the A detailed solution for the gas dynamical interaction physical parameters of the clouds from variability of between real clouds and the surrounding gas and, theemission-lineprofiles in the spectra of WR stars. accordingly, for the influence of the cloud radiation Such variability was also calculated theoretically for on the observed spectrum of the star can be obtained early-type stars, assuming a stochastic ensemble of only after the cloud-formation mechanism has been clouds in the atmosphere [6]. Modeling of the spectral determined, so that the correct initial conditions for features due to the clouds did not taken into account the computations can be specified. the dynamical interactions of the clouds with the surrounding medium. It was assumed that the clouds do not move relative to the surrounding gas and that 2. THE CLOUD MODEL their density, temperature, and, hence, emitted radi- AND FORMULATION OF THE PROBLEM ation remain constant. The models are obviously not self-consistent under these assumptions. Let a homogeneous spherical condensation have a radius of R (r) and a constant mass of M .Theradius If the velocities of a cloud and the surrounding c c of the condensation is then gas flowing from the star are the same, the physical 1 conditions in the cloud cannot remain the same as 3M 3 R (r)= c , (1) they were when it formed, since both the temperature c 4πρ (r) and density in the surrounding gas vary. The pres- c sure and temperature in the cloud also vary, as was where r is the distance from the center of the star and pointed out in [7]. Taking into account variations of ρc(r) is the density of the condensation.

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The condensation is also described by its pressure Model parameters independent of the presence of a con- Pc(r) and temperature Tc(r). The heating of the condensation densation by the hot external medium and the star’s radiation, and also radiative energy losses, lead to M 10M nonadiabaticity of the variations in the physical con- R 3R ditions in the condensation. It is impossible to take 5 this effect into account rigorously, since we do not L 10 L −5 have complete information about the internal com- M˙ 10 M/year position of the cloud. Let us represent the pressure × 3 inside the condensation by the polytropic dependence Tw 40 10 K n v∞ 2000 km/s Pc(r)=Kρc , (2) v 400 km/s with the index 1 ≤ n<5/3. This form of the equa- 0 tion of state simplifies the problem dramatically and β 1 makes it possible to partially take radiation processes τ 0–1 into account. The equation describes an adiabatic process when n =5/3. At the time of its formation, a cloud should be in a which, if Tw does not depend on r, yields state of equilibrium; the pressure in the condensation 1 2 n Pc(r) should be equal to the pressure in the surround- r0vw(r0) ρc(r)=ρc(r0) 2 . (5) ing gas Pw(r). Further, this approximation remains r vw(r) valid if the condensation has time to adjust to variable external conditions. The parameters of the wind, labeled with the index w, are related as follows: 3. EQUATION OF MOTION OF A CONDENSATION k Pw(r)= ρw(r)Tw(r). (3) µwmH A condensation in the stellar wind is affected by the force of gravity, the radiation pressure, and the The continuity condition for the homogeneous gas-pressure gradient. When calculating the velocity component of the wind implies that of the condensation, the resistance of the gas consti- ˙ 1 tuting the wind should also be taken into account. ρw(r)=M 2 , (4) 4πr vw(r) The turbulent resistance of the medium is pro- ˙ portional to the square of the relative velocity of the where M is the mass-loss rate of the star and vw(r) is − thevelocityoftheoutflow as a function of the distance condensation, vc(r) vw(r). The corresponding term from the center of the star. in the equation of motion should be positive for a negative relative velocity of the condensation. To take We adopt a parametric law for vw(r),whichcanbe into account the absorption of the stellar radiation in written in the generally adopted form [8] ≤ the medium, we introduce the factor τ 1 in the term β describing the radiation pressure. R vw(r)=v0 +(v∞ − v0) 1 − . r The equation of motion of the condensation can then be written in the form Here, v0 is the wind velocity near the boundary of the photosphere; v∞, the terminal velocity of the wind; dvc − 1 dPw − GM vc = 2 R, the distance from the center of the star to the dr ρc dr r lower boundary of the wind; and β, a parameter de- | − | − cx 2 − 2 vc vw(r) scribing the degree of acceleration of the material. ρwπRc [vc vw(r)] − When the outflow of gas from hot stars is considered, 2Mc (vc vw(r)) it is generally assumed that β 1. τL 2 1 + 2 πRc , We obtain from the pressure-equality condition 4πr Mc c Pc(r)=Pw(r) and Eqs. (2) and (3) the equality where M is the mass of the star and c is the co- x 1 efficient of turbulent resistance; we will adopt cx =1. Pw(r) n ρc(r)=ρc(r0) , The last term is associated with the radiation pressure Pw(r0) of a star with luminosity L.

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α ρ ρ ρ ρ = c/ w c/ 0 170 1.0 n = 5/3 n = 5/3 n = 1.4 0.8 130 n = 1.2 n = 1.0 0.6 90 n = 1.4 0.4

50 0.2 n = 1.2 0 n = 1.0 10 1.0 4.5 8.0 11.5 15.0 19135 r/R* r/R*

Fig. 1. Variation of the ratio of the densities of the con- Fig. 2. Variation of the density of the condensation with ρ (r) distance from the center of the star as a function of n.The densation and surrounding gas α = c with distance ρ (r) curves were obtained for an initial density ρ0 =10ρw(r0) w −3 from the center of the star as a function of n.Thecurves and condensation radius R0 =10 R. were obtained for an initial density ρ0 =10ρw(r0) and −3 condensation radius R0 =10 R. 4. CALCULATIONS AND RESULTS The table presents selected values of the exter- Using the obtained relations (1) and (3)–(5), we nal parameters that are characteristic of Wolf–Rayet transform the equation of motion as follows: stars. dv 2 1 dv (r) 1−n Equation (6) was solved numerically for a broad v c = A + w r2v (r) n c dr 1 r v (r) dr w range of initial densities and condensation radii. We w did not consider condensations with initial radii ex- (6) −2 ceeding 10 R, since they should substantially af- 1 2−3n − A − A r2v (r) 3n fect the state of the surrounding medium, making the 2 r2 3 w problem much more complicated. Solutions were ob- |v (r) − v (r)| tained for various initial distances of the condensation × [v (r) − v (r)]2 c w c w (v (r) − v (r)) from the center of the star. We restrict the computa- c w tional domain to the zone between the surface of the 2−3n 2 3n star and 15R . + A4 r vw(r) vw(r), Independent of the initial density of the cloud, the where the factors A1,A2,A3, and A4 depend on the ratio of the densities of the condensation and sur- parameters of the star, the initial conditions, and rounding gas α = ρc(r)/ρw(r) increases as the con- physical constants. densation moves in the medium, whose density decreases outwards. This is due to the chosen polytropic Thus, computing the evolution of the condensa- form of the equation of state for the condensation with tion reduces to solving (6) with the initial conditions polytrope index n ≥ 1. α remains constant when n = r0, ρ(r0)=ρ0, Rc(r0)=R0,andvc(r0)=vw(r0). 1. Figure 1 presents the variation of α with distance These initial conditions are specified so that a con- from the center of the star for various n. densation with radius R0 and density ρ0 is initially The density of the cloud itself decreases substan- located a distance r0 from the center of the star and is tially (by an order of magnitude) relative to its initial at rest relative to the moving medium. value. The radius of the cloud increases as it moves into the more rarefied outer layers of the atmosphere. The numerical solution of (6) also requires that Figures 2 and 3 present the variations of the conden- external parameters that are independent of the pres- sation density and radius as a function of n.Wecan ence of the condensation be specified: v∞, v0, β, R, see that both parameters vary especially strongly at L, M, M˙ , Tw,andτ. small distances from the stellar surface. This is due to

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Rc/R0 can be both higher and lower than the velocity of the homogeneous stellar wind. Note that the velocity 0.009 n = 1.0 of a cloud relative to the surrounding gas can reach substantial values. The influence of the polytropic index n (2) in the n = 1.2 0.007 dependence of the velocity of the condensation on distance from the center of the star is small. This n = 1.4 dependence is basically specified by the initial radius 0.005 and density of the cloud. n = 5/3 5. CONCLUSIONS 0.003 We have considered a model with homogeneous, spherical condensations and a polytropic equation of state. The physical conditions in the condensations 0.001 vary substantially as they move in an accelerated medium with non-constant density and pressure. The 1.0 4.5 8.0 11.5 15.0 r/R density and radius of the condensation vary by several * orders of magnitude compared to their initial values. Fig. 3. Variation of the radius of the condensation with The variations of the physical conditions in the con- distance from the center of the star as a function of n.The densation affect its parameters and should be taken curves were obtained for an initial density ρ0 =10ρw(r0) into account in stellar-wind models and calculations −3 and condensation radius R0 =10 R. of spectra. The condensations can move with substantial velocities relative to the surrounding gaseous Vc(r)/V∞ flows. Thus, the clouds cannot be treated like static 1.3 formations with constant characteristics that are at rest relative to the expanding atmosphere of the star. 1.1 These results can be considered a first step toward clarifying the mechanisms that give rise to the formation of inhomogeneities in the atmospheres of hot 0.9 stars.

0.7 ACKNOWLEDGMENTS TheauthorisgratefultoV.G.Gorbatskiı˘ for useful ρ ρ 0.5 0 = 10 w(r0), R0 = 0.001R+ comments. This work was supported by the Russian ρ ρ 0 = 10 w(r0), R0 = 0.01R+ Foundation for Basic Research (project code 01-02- ρ ρ 0.3 0 = 150 w(r0), R0 = 0.001R+ 16858). ρ ρ 0 = 200 w(r0), R0 = 0.001R+ ρ ρ 0.1 0 = 50 w(r0), R0 = 0.005R+ REFERENCES 1. A. M. Cherepashchuk, Kh. F. Khaliullin, and 0.1 4.5 8.0 11.5 15.0 J. A. Eaton, Astrophys. J. 281, 774 (1984). 2. I. I. Antokhin, A. F. Kholtygin, and A. M. Cherepa- r/R* shchyuk, Astron. Zh. 65, 558 (1988) [Sov. Astron. 32, Fig. 4. Velocities of clouds as a function of distance from 285 (1988)]. the center of the star. Each curve is marked with the 3. A. F. J. Moﬀat, S. Lepine,´ R. N. Henriksen, and corresponding initial radius and density of the cloud. All C. Robert, Astrophys. Space Sci. 216, 55 (1994). curves were calculated with n =1.2. The dotted curve 4. T. Eversberg, S. Lepine,´ and A. F.J. Moﬀat, Astrophys. indicates the dependence between the cloud velocity and J. 494, 799 (1998). distance for the homogeneous component of the wind. 5. J. C. Brown, L. L. Richardson, I. Antochin, et al., Astron. Astrophys. 295, 725 (1995). the law for variations of the velocity of the homoge- 6. N. A. Kudryasheva and A. F.Kholtygin, Astron. Zh. 78, neous wind vw(r) and, accordingly, its density, which 1 (2001) [Astron. Rep. 45, 287 (2001)]. undergo their strongest variations in this region of the 7.J.C.Howk,J.P.Cassinelli,J.E.Bjorkman,and atmosphere. H. J. G. L. M. Lamers, Astrophys. J. 534, 348 (2000). To illustrate this, Fig. 4 presents examples of the 8. H. J. G. L. M. Lamers and J. P. Cassinelli, Intro- duction to Stellar Winds (Cambridge Univ. Press, computed dependences of the condensation velocity Cambridge, 1999), p. 9. vc(r). The general form of these curves is consistent with the results of [7]. The velocity of a condensation Translated by K. Maslennikov

ASTRONOMY REPORTS Vol. 48 No. 2 2004 Astronomy Reports, Vol. 48, No. 2, 2004, pp. 112–120. Translated from Astronomicheski˘ıZhurnal, Vol. 81, No. 2, 2004, pp. 134–142. Original Russian Text Copyright c 2004 by Savokhin.

Axially Symmetrical Models of the Products of a Conservative Merger of a Binary White Dwarf with Similar-Mass Components D. P. Savokhin Astronomical Observatory, Ural State University, Yekaterinburg, Russia Received June 28, 2003; in ﬁnal form, August 8, 2003

Abstract—The possibility of a conservative merger of a binary white dwarf whose components have similar masses is studied.Axially symmetrical models for single, rapidly rotating white dwarfs that are possible products of such mergers are constructed and their physical characteristics investigated.The merger products must be turbulent, and the viscosity of the electron gas is not suﬃcient to support the observed luminosities of massive, bright white dwarfs.The amount of dissipative energy and the timescale for its release are estimated. c 2004 MAIK “Nauka/Interperiodica”.

1.INTRODUCTION (mass, angular momentum, energy) are close to those The recent discovery of close binary white dwarfs of the initial binary system.This paper is one such [1] has stimulated numerous theoretical studies of study. the evolution of these systems.This is related, first In our paper [7], we focused on conservative and foremost, to the possibility of explaining obser- mergers of binary components with mass ratios of q = vations of type Ia supernovas in old elliptical galaxies (0.35 – 0.55). However, according to population syn- (whose stars have masses lower than ∼0.8M)as thesis studies [1, 9], most binary white dwarfs should being associated with mergers of the components of have components with similar masses.Therefore, the close binary white dwarfs [2–4].The origin of the main channel for the formation of the progenitors recently discovered massive and very bright white of type Ia supernovas is mergers of carbon–oxygen dwarfs PG 0136+251, PG 1658+441, and GD 50 [5] white dwarfs with mass ratios q close to unity.Here, with masses of 1.2, 1.3, and 1.2M and luminosities we turn our attention to binary white dwarfs with such −2 L =(6.7, 1.2, 9.7) × 10 L, respectively, also must mass ratios, q =(1– 0.6). be explained.This may be possible in a scenario with the merger of two low-mass dwarfs.Mergers of de- We have computed axially symmetric models for generate stars have also been considered as a possible zero-temperature, strongly distorted, steady-state, origin for gamma-ray bursts (see, e.g., [6]). rapidly differentially rotating white dwarfs for four rotation laws.The temperature corrections are small, The merger of the components of a binary white ∼ 8 dwarf is accompanied by a number of complex phys- even for very hot white dwarfs with T 10 K [10], ical processes: the radiation of gravitational waves, and can be neglected (they cannot be neglected in formation of a common envelope, accretion of matter, studies of the dynamics of coalescence accompanied formation of a massive disk (torus), viscous transport by the formation of shock waves and carbon burning of angular momentum in the merger product, release at the dwarf surface, when the temperature can rise 9 of the energy of viscous friction, nuclear burning of to ∼10 K).We found the total energy and angular helium, carbon, and oxygen, and other processes.All momentum of a semidetached white-dwarf binary of these play equally important roles.The process of based on computations of models of the internal coalescence and the properties of the merger product structure of the white dwarfs and a standard binary and its further evolution depend most strongly on the model with two point masses, assuming the rotation component-mass ratio before coalescence [7].Only of the dwarfs to be synchronous with their orbital two dynamical computations of mergers of two white motion.By comparing the total energy and angular dwarfs have been carried out [5, 8] and have provided momentum of a single, massive, differentially rotating models for this process for a narrow range of param- white dwarf with the energy and momentum of a eters of the binary systems.Due to the complexity semidetached white-dwarf binary with similar total of the accompanying phenomena, mergers are often mass, we can identify those binary systems that studied in another way: a steady-state model is com- can produce single, massive, rapidly rotating dwarfs puted for a merged configuration whose parameters during their coalescence.

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50 2 50 J, 10 g Òm /s Etotal, 10 erg 9 (a) 0 (b) 6 0.6 10 8 0.8 –2 107

7 8 –4 10 9 6 1 10 –6 5 1010 0.6 –8 4 0.8 –10 3 0.3 7 J0 –12 2 10 108 106 Energy of the binary for J = 0.3 J0 1 1 0.2 J0 –14 Energy of the binary for J = 0.2 J0 109 0 –16 0.4 0.8 1.2 1.6 2.0 2.4 0.4 0.8 1.2 1.6 2.0 2.4 Mtotal

Fig. 1. (a) Angular momenta and (b) total energies of binary white dwarfs (dotted curves) and massive single white dwarfs (dashed curves) obeying rotation law (1) as a function of the mass of the system or dwarf.The numbers next to the dashed curves indicate the component-mass ratios q for the binary systems; the numbers by the dotted curves label the maximum 3 density in the models (in g/cm ).The dotted curves with the labels “0.3 J0” and “0.2 J0” show the momenta of binaries with q =1after the loss of 70% and 80% of their initial momentum J0 via gravitational-wave radiation.The energy of these systems after the loss of these fractions of their initial momenta are shown by the two lower dotted curves at the right side of the plot.

50 2 50 J, 10 g Òm /s Etotal, 10 erg 9 (a) 0 (b) 0.8 –1 106 0.6 7 2 × 106 8 –2 10 –3 7 108 1 –4 6 –5 109 –6 5 –7 0.6 2 × 107 4 –8 107 –9 0.8

3 2 × 107 –10 0.3 J 0 –11 Energy of the binary for J = 0.3 J0 2 –12 Energy of the binary for J = 0.2 J0 1 × 6 0.2 J0 2 10 –13 1 106 108 109 –14 0 –15 0.4 0.8 1.2 1.6 2.0 2.4 0.4 0.8 1.2 1.6 2.0 2.4 Mtotal

Fig. 2. Same as Fig.1 for single dwarfs obeying rotation law (2).The dash –dot curve represents the upper boundary of the region of binary systems whose conservative mergers can produce single dwarfs obeying law (2).The thin dashed curve shows models of single dwarfs with the same total energy as that of semidetached binaries with component-mass ratios q =0.6.

2.FORMULATION OF THE PROBLEM energy as contact or semidetached white-dwarf bina- AND COMPUTATIONAL METHOD ries with the same total masses.A rigorous math- We computed models for axially symmetric, stron- ematical formulation of this problem for such de- gly ﬂattened white dwarfs in a state of stationary rapid generate conﬁgurations and the method used in the rotation having the same angular momenta and total computations are presented in detail in [7, 11].Here,

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50 2 50 J, 10 g cm /s Etotal, 10 erg 9 0 (a)6 (b) 0.8 10 0.6 –1 6 8 –2 2 × 10 –3 7 1 2 × 106 –4 6 –5 9 –6 10 107 5 6 –7 0.6 10 107 2 × 107 × 7 4 –8 2 10 –9 0.8 3 –10 8 10 0.3 J0 –11 2 Energy of the binary for J = 0.3 J0 –12 Energy of the binary for J = 0.2 J 0.2 J0 0 –13 1 1 108 109 –14 0 –15 1.0 1.4 1.8 2.2 2.6 1.0 1.4 1.8 2.2 2.6 Mtotal

Fig. 3. Same as Fig.1 for white dwarfs obeying rotation law (3).In this case, the dash –dot curve coincides with the dotted curve for q =1.This curve and the thin dashed curve have the same meaning as the corresponding curves in Fig.2 (see the text).

50 2 50 J, 10 g Òm /s Etotal, 10 erg 9 (a)0 (b) 0.8 0.6 –1 106 8 –2 2 × 106 7 –3 1 –4 6 –5 109 –6 107 0.6 5 –7 2 × 107 4 –8 –9 0.8 –10 3 7 2 × 107 8 9 10 10 10 108 6 0.3 J –11 2 10 0 2 × 106 –12 0.2 J0 Energy of the binary for J = 0.3 J0 1 –13 1 Energy of the binary for J = 0.2 J0 –14 0 –15 1.0 1.4 1.8 2.2 2.6 1.0 1.4 1.8 2.2 2.6 Mtotal

Fig. 4. Same as Fig.1 for single white dwarfs obeying rotation law (4). we are interested in the properties of dwarfs obeying cylindrical rotation laws: Ψ(ω) ∼ 1/ω, (4) ∼− Ψ(ω) ω, (1) where Ψ is the potential of the centrifugal forces, which is related to the angular velocity Ω(ω) as Ψ(ω) ∼−ω0.4, (2) Ψ(ω)=− Ω2(ω)ωdω, (5) Ψ(ω) ∼−ln(ω), (3)

ASTRONOMY REPORTS Vol.48 No.2 2004 MERGER OF A BINARY WHITE DWARF 115 where ω is the distance to the rotational axis.Law (3) ry corresponds to rotation for which the linear velocity 0.5 inside the star is constant, while law (4) corresponds Model 1 to rotation for which the angular velocity depends on 0.4 the distance in a way similar to Keplerian rotation. 0.3 The accuracy of the computations was monitored using a virial test [7, 11].For all the computed models, 0.2 − − VT ∼ 10 4–10 5. 0.1 0 3.MODELS OF THE WHITE-DWARF 0.5 Model 2 MERGER PRODUCTS 0.4 The white-dwarf models obeying rotation law (1) differ from those obeying laws (2)–(4) in the con- 0.3 centration of their specific angular momenta predom- 0.2 inantly in outer layers, close to the equator.Therefore, as the angular momentum is increased, there comes 0.1 a time when the effective acceleration at the equator becomes positive (this is qualitatively similar to rigid- 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 body rotation).The solid broken curve in Fig.1a rx roughly bounds from above the region containing models having geff < 0 that do not lose matter from Fig. 5. Density distribution in two selected models.The their equator.In this case, the maximum possible density at two neighboring constant-density surfaces differs by a factor of ten.The properties of the models are angular momentum of the computed configurations is given in Table 1. much lower than that of the binary system (Fig.1a). Thus, it is not possible to produce a merged object that obeys law (1) or rotates according to a law with the formation of a merged dwarf from a binary with a centrifugal potential that decreases with distance q<0.98 is possible only in the nonconservative case. from the rotational axis more slowly than (1) via the Single white dwarfs obeying rotation law (4) completely conservative coalescence of the compo- and having masses and angular momenta similar to nents of a semidetached binary. semidetached white-dwarf binaries have higher total The white-dwarf models obeying rotation law (2) energy (Fig.4).Therefore, a completely conservative can be the products of conservative mergers of two scenario for the formation of merger products obeying white dwarfs with total masses of 1.6M ∼1.1M (Fig.6). (Fig.3).If q<0.98, single dwarfs with total masses Let us first consider the case M1 = M2.The ap- and momenta similar to those of the corresponding proach of the components due to the radiation of binaries have higher total energies than those for the gravitational waves turns a detached binary into a corresponding pair of white dwarfs (Fig.3b), and contact system.Since gravitational-wave radiation their formation is energetically unfavorable.Thus, remains the only mechanism for the loss of angular

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Table 1. Models of massive, diﬀerentially rotating dwarfs

Rotation 9 50 2 Etotal, 3 Lvisc, Model M, M R , 10 cm J, 10 gcm /s ρ , g/cm ε law e 1050 erg max L

1 2 2.1747 1.0645 6.8557 –5.362 1.01 × 107 0.06 0.0625 2 2 2.1747 0.2950 2.1071 –15.967 1.08 × 109 12.75 0.3025

Note: The columns present the model number, rotation-law number, mass M, equatorial radius Re, angular momentum J, total energy Etotal, maximum density ρmax, dissipative luminosity Lvisc, and ratio of polar and equatorial radii ε.The models were computed using agridwith(r, θ) = 100 × 80 nodes and had virial tests with VT 10−5–10−6. momentum (the magnetic stellar wind may play an An efficient, stable disk can exist only if it is located important role, but we do not consider this mecha- inside the inner critical surface of the primary.In nism here), the subsequent merger of the components the case of semidetached systems, the dimensions of will proceed, as before, on a timescale that is much the primary and its inner critical surface gradually longer than the dynamical timescale, τdyn ∼ porb ∼ become more similar as q increases from 0.75 to 1. 20 s, and is equal to the timescale for the loss of angu- Thus, the space available for the disk is diminished. lar momentum via gravitational-wave radiation, τgr, On the other hand, the inner Lagrangian point (L1) which is ∼103 yr for such systems [12].During this approaches the rotational axis of the system, so that time, the binary will initially have a dumbbell shape, the specific angular momentum of the matter lost by then form a dwarf with a binary core and common the secondary through the inner Lagrangian point is envelope.The two cores will continue to approach one decreased.Analytical ballistic estimates show that another due to the radiation of gravitational waves, the radius of the disk, which is determined by the which carry away most of the angular momentum and angular momentum at L1, very rapidly decreases as q energy over a time of ∼103 yr.It is possible that, at approaches unity and becomes much smaller than the some point, the cores will merge into a single axially radius of the primary.As a consequence, the stream of symmetric object.If the angular momentum of the matter from the inner Lagrangian point must impact the surface of the primary.The dynamical computa- system is ∼30% of the initial momentum of the binary tions of [8] confirm this conclusion.Hence, the disk J at the moment of coalescence (Figs.1а, 2а, 3а, 0 does not form when q becomes large enough.Thus, 4а), the merged, axially symmetric object will have the there must exist some q above which the dynam- structure shown in Fig.5 (model 2).The maximum cr ically conservative merger regime does not occur. density in such axially symmetric merger products The value of q is between 0.75 and 1. When q> (model 2 in Table 1) is very close to the density for cr qcr, the coalescence must occur in the gravitational- which pulsational carbon burning can be initiated q ∼ × 9 3 wave regime.In Fig.6, cr is represented by the line ( 2 10 g/cm [13]); this density can be attained corresponding to binaries with q =0.95.The pre- as more angular momentum is lost. cise value of qcr must be determined via dynamical Thus, the merger of binary components with equal computations of the mergers.When 0.35 0.9, the dynamically conservative merger not have time to lose momentum, mass, or energy. regime is possible for binaries with 0.9

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M2 1.4 1 CO ONeMg 1.3 ONeMg T = 2 × 108 0.95 1.2 1.1 CO 1.0 Gravitational-wave merger 0.9

0.8 0.55 0.7

0.6 Dinamically conservative merger 0.5 Livne-Arnett 0.35 He 0.4 SN I explosion Dinamically conservative merger with torus formation scenario M + M = M 0.3 1 2 Ch CO 0.2 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 M1

Fig. 6. Regions of various merger regimes in the plane of the masses M1 and M2 of the primary and secondary components of the binary white dwarfs.The dotted curve shows the positions of binaries in which the di fference of the surface potentials is sufficient to heat matter to temperatures of 2 × 108 К.The dash –dot curve bounds from below the region of binaries for which conservative mergers are possible on the dynamical timescale.The dashed lines mark the boundaries between helium, carbon–oxygen, and oxygen–neon–magnesium dwarfs.The filled circles show the positions of binary systems for which the dynamics of the disruption of their less massive components were computed in [4] (right circle) and [5] (left circle).

mass ratio qcr

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Table 2. Dissipative luminosities of diﬀerentially rotating white dwarfs

Rotation 50 2 3 9 Model M, M J, 10 gcm /s L , L ρ , g/cm R , 10 cm ε law visc max e

A 3 1.20 3.70 0.002 1.07 × 106 1.59 0.19 B 3 1.20 0.26 0.013 9.70 × 107 0.44 0.96 C 4 1.20 1.83 0.042 5.03 × 106 0.92 0.00 D 4 1.20 0.17 0.039 1.03 × 108 0.43 0.96

Note: The columns indicate the model designation, rotation law, mass M, angular momentum J, dissipative luminosity Lvisc, maximum density ρmax, equatorial radius Re, and ratio of polar and equatorial radii ε.The models were computed using a grid with (r, θ)=50× 40 nodes and had virial tests with VT ∼ 10−4–10−5. in the total volume of the star per unit time is [5] centers and quasi-keplerian rotation in their massive envelopes (as for the merger product obtained in [8]), 1 dΩ 2 Lvisc = r η(r)dV, (6) the dissipative luminosity of the rigid-body core is 2 dr zero.Therefore, models obeying law (4) will have the where η is the viscosity coefficient.We shall call this largest dissipative luminosity among models obeying quantity the dissipative luminosity.The dissipative the five different rotation laws.If we assume that the luminosity is, thus, determined by two factors: the viscosity is determined by the molecular viscosity rotation curve and the viscosity of the stellar material. of the electron gas alone, it is possible to estimate We computed the dissipative luminosity for models 1 the ability of the electron gas to support the high and 2 in Tables 1 and 2 assuming that η is the molec- luminosity of the white dwarfs.For this purpose, ular viscosity of the electron gas, which is determined we computed four white-dwarf models with masses by the relations [5] 1.2M and found their dissipative luminosities. A comparison of the computed L values with  visc  × −5 5/3 −1 −1 the observed luminosities of bright white dwarfs of 1.2 10 ρ gcm s similar mass (Table 2, see also the Introduction)  − ρ<2 × 106 gcm 3 shows that, if the viscosity is solely due to the electron η = (7) e  −1 −1 0.19ρ gcm s gas and the white dwarf obeys rotation law (3),  the dissipative luminosity is a factor of five to ten ρ> × 6 −3. 2 10 gcm lower than is observed.The dissipative luminosities The high observed luminosities of the three known of models obeying the quasi-keplerian law (4) are bright white dwarfs PG 0136+251, PG 1658+441, consistent with the observed luminosities, but, as we and GD 50 cannot be explained by the nuclear burn- have pointed out above, the conservative formation of ing of carbon in their central regions or by hydrogen or objects obeying this law is not possible (nevertheless, helium burning at their surface layers because of their models C and D in Table 2 display the maximum low temperatures (the observed surface temperatures dissipative power of the electron gas).Thus, we can are ∼(30–40) × 103 К) and the high thermal conduc- conclude that the viscosity of the material comprising tivity of the electron gas that supports the virtually the observed bright white dwarfs is higher than we isothermal structure of the dwarf.The only possi- have assumed, e.g., due to turbulence. A sufficient ble means to support the high luminosities of these condition for the stability of the gas against turbu- dwarfs is via the radiation of dissipative energy.Thus, lence is [16, 17] the discovery of bright, massive white dwarfs, first, Ri > 1/4, (8) provides evidence for mergers of binary white dwarfs and, second, suggests that the merger products rotate where Ri is the Richardson number, computed from differentially. the relation The dissipative luminosity of the white-dwarf g 2 γ dΩ 2 Ri = eff 1 − / r , (9) models increases along the sequence of rotation us Γ dr laws (1)–(4) (with the remaining conditions being the same).In the case of the merger products of where geff is the effective acceleration of the matter, binaries with component-mass ratios considerably us is the sound speed, γ =(d ln P/dln ρ),andΓ= lower than qcr displaying rigid-body rotation at their 5/3.Violation of (8) does not ensure turbulence of the

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Ri the radiation of gravitational waves, ∼103 yr, during 2.75 B D which gravitational waves carry away nearly all the 2.50 angular momentum.As a result, a white dwarf with a 2.25 density sufficient for the ignition of carbon may form. 2.00 Binary white dwarfs with component-mass ratios 1.75 C lower than some qcr merge in the dynamically conser- 1.50 vative regime.The value of qcr is in the range 0.75–1 1.25 2 and must be determined via dynamical computations 1.00 of the coalescence process.The result of a merger in 0.75 1 this regime is a dwarf surrounded by a massive disk 0.50 A that accumulates the total angular momentum of the 0.25 binary.The maximum densities in such dwarfs are not 0 sufficient for the ignition of carbon. –0.25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 If qcr > 0.9, the conservation laws do not rule out r the formation of axially symmetric, differentially rotating white dwarfs from binaries with component-mass ratios 0.9 Stellar Evolution (Nauka, Moscow, 1989) wave regime.The merger occurs on the timescale for [in Russian].

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11. A.G.Aksenov and S.I.Blinnikov, Astron.Astrophys. 14.E.Livne, Astrophys.J. 354, L53 (1990). 290, 674 (1994). 15. E.Livne and D.Arnett, Astrophys.J. 452, 62 (1995). 12. I.Hachisu, Y.Eriguchi, and K.Nomoto, Astrophys.J. 16.R.Mochkovitch and M.Livio, Astron.Astrophys. 308, 161 (1986). 209, 111 (1989). 13.N.V.Dunina-Barkovskaya, V.S.Imshennik, and 17.R.Mochkovitch and M.Livio, Astron.Astrophys. S.I.Blinnikov, Pis’ma Astron.Zh. 27 (6), 412 (2001) 236, 378 (1990). [Astron.Lett. 27, 353 (2001)]. Translated by L. Yungel’son

ASTRONOMY REPORTS Vol.48 No.2 2004 Astronomy Reports, Vol. 48, No. 2, 2004, pp. 121–135. Translated from Astronomicheski˘ı Zhurnal, Vol. 81, No. 2, 2004, pp. 143–158. Original Russian Text Copyright c 2004 by Zheleznyakov, Koryagin, Serber.

Thermal Cyclotron Radiation by Isolated Magnetic White Dwarfs and Constraints on the Parameters of Their Coronas V. V. Zheleznyakov, S. A. Koryagin*, and A. V. Serber Institute of Applied Physics, Russian Academy of Sciences, ul. Ul’yanova 46, Nizhni Novgorod, 603600 Russia *E-mail: [email protected] Received April 22, 2003; in ﬁnal form, August 8, 2003

Abstract—An efficient method for the detection and estimation of the parameters of the coronas of isolated white dwarfs possessing magnetic fields of about 107 G is tested. This method is based on the detection of thermal radiation of the coronal plasma at harmonics of the electron gyrofrequency, which is manifest as a polarized infrared excess. The Stokes parameters for the thermal cyclotron radiation from the hot corona of a white dwarf with a dipolar magnetic field are calculated. A new upper limit for the electron density, 1010 cm−3, in a corona with a temperature of 106 K is found for the white dwarf G99-47 (WD 0553+053). This limit is a factor of 40 lower than the value derived earlier from ROSAT X-rayobservations. Recommendations for subsequent infrared observations of isolated magnetic white dwarfs aimed at detecting their coronas or deriving better constraints on their parameters are presented. c 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION absorption of the Alfven waves can support the high temperature of coronas. Hot outer envelopes—coronas—exist in almost all types of stars. The most well known are the coronas of In their analysis of archival data of the Einstein main-sequence stars. At the same time, it is unclear if Observatoryfor the five nearest cool, isolated, mag- coronas are present in isolated white dwarfs. Coronal netic white dwarfs, Arnaud et al. [9] found with high X-rayemission has been reliablydetected from only (99%) probabilitythat G99-47 (WD 0553 +053) is a one isolated white dwarf, KPD 0005+5106 [1, 2]. This source of X rays with energies of 0.2–3.5 keV. The nonmagnetic white dwarf possesses a veryhot pho- radiation flux derived from the archival data corre- × 27 tosphere with an effective temperature of Teff =1.2 × sponds to an X-rayluminosityof 4.1 10 erg/sand 5 × 12 −3 10 K. The coronal temperature in KPD 0005+5106 a coronal electron densityof ne0 =2.4 10 cm . was estimated to be about 2.6 × 105 K, onlya factor This last estimate was derived assuming that the observed X-ray flux is due to bremsstrahlung in a corona of two greater than Teff. It is believed that the coronal 7 plasma is heated byshock waves formed in super- with low optical depth and temperature Tcor 10 K. sonic plasma flow from the stellar surface [1, 3]. However subsequent observations of G99-47 bythe Along with KPD 0005+5106, about one hundred ROSAT satellite did not confirm the Einstein data and provided onlyan upper limit for the luminosityof veryhot ( T 2.5 × 104 K) isolated white dwarfs eff × 26 / . . have been detected in the X-ray[4], but this radia- 1.03 10 erg sat0 1–2 5 keV. Such a luminosity × 11 −3 tion originates in their photospheres [5, 6]. In cooler corresponds to an electron densityof 4.6 10 cm 4 in a corona with a temperature of 107 K [10]. objects (Teff 2.5 × 10 K), the X rays from deep photospheric layers are absorbed by the upper layers Therefore, the available X-rayobservations do and do not escape from the star. Therefore, detect- not answer the question of whether isolated white ing X-rayradiation from these stars would provide dwarfs possess coronas. In particular, the sensitivity unambiguous evidence for the presence of coronal of modern X-raydetectors is su fficient to detect X-ray envelopes in their atmospheres. bremsstrahlung onlyfrom verydense coronas with The theoretical calculations of [7, 8] show that electron densities several orders of magnitude greater a quite powerful flux of Alfven waves (of the order than the densityin the solar corona. − − of 1010 erg cm 2s 1) is generated in the convective However, we can approach the problem of detect- layers of relatively cool white dwarfs with strong mag- ing the coronas of isolated white dwarfs in a com- 4 4 netic fields (Teff ∼ 2 × 10 K, B 10 G), and this pletelydi fferent way. Due to the presence of mag- flux reaches the upper layers of the photosphere. The netic fields of the order of several tens of millions of

1063-7729/04/4802-0121$26.00 c 2004 MAIK “Nauka/Interperiodica” 122 ZHELEZNYAKOV et al. Gauss in some white dwarfs (in particular, in G99- The electron distribution over the Landau levels 47) [11], emission features can be formed at har- formed byspontaneous and induced radiative transi- monics of the electron gyrofrequency in the optical tions at the first (fundamental) cyclotron harmonic, and infrared spectrum of a magnetic white dwarf as well as bycollisions, was found analyticallyin [13, surrounded bya hot corona with a su fficientlyhigh 19, 20]. Expressions for the radiative power and the particle density[12 –14]. Cyclotron features can be absorption coefficients of the normal waves at all the observed even when the coronal electron densityis cyclotron harmonics used in our work were obtained substantiallyless than is required for the detection of in [20]. X rays. For example, according to [14], a corona with Note that the electron distribution in the velocity an electron densityof 2.4 × 1010 cm−3 and a temper- along the magnetic field, which is less affected byra- ature of 107 K would double the luminosityof G99- diative processes, is taken to be specified both in [13, 47 at wavelengths 4–8 µm, corresponding to the 19, 20] and in our analysis. In other words, some ex- electron gyrofrequency for the magnetic field of this ternal source supports a constant longitudinal (along star. This shows that searching for cyclotron radiation the magnetic field) electron temperature. The energy in the infrared could be a verye fficient and sensitive of this source is transformed into cyclotron emission method for detecting the coronas of isolated white from the white-dwarf corona. Analysis of particular dwarfs possessing magnetic fields of about 107 G. mechanisms for field-aligned heating of electrons is beyond the scope of this paper. The cyclotron emission by a corona with low optical depth was calculated analytically in [15, 16]. The Stokes parameters for coronal thermal cy- The aim was to explain the polarization of the optical clotron emission are calculated below for the particular case of the isolated magnetic white dwarf G99- continua of stronglymagnetized ( B 108 G) white 1 47. The results of these detailed calculations are com- dwarfs. Ingham et al. [16] considered thermal cy- pared with photometric observations of G99-47 in the clotron radiation bya corona with low optical depth infrared [21, 22]. As a result, we obtained an upper in local thermodynamic equilibrium (LTE), when the limit for the electron densityin the corona of this electron velocitydistribution in the entire corona is white dwarf corresponding to a minimum value for the an isotropic Maxwellian distribution with a constant 6 coronal kinetic temperature of Tcor 10 K. This new temperature. The assumption of LTE is applicable upper limit is more than a factor of 40 lower than the onlywithin a limited range of conditions and is often value derived earlier from X-rayobservations [10]. not satisfied bymagnetic white dwarfs. It was noted as earlyas in [16], and later in [12], that the characteristic lifetime of an electron in excited 2. FORMULATION OF THE PROBLEM Landau levels in a magnetic white dwarf provided by AND CALCULATION METHOD cyclotron radiation, G99-47 is a typical cool, magnetic white dwarf 3 2 2 with a hydrogen photosphere. Only the Balmer Hα tc =3mc /4e ω , B absorption line is observed in its optical spectrum is considerablyshorter than the mean free time be- [23]. The effective temperature Teff determining the −1 bolometric luminosityof the star is estimated to be tween successive collisions of the electron, νeff .Here and below, m is the electron mass; e>0, the elemen- 5790 ± 110 K [22, 24, 25]. The free-fall acceleration × 8 2 tarycharge; c, the speed of light; and ωB = eB/mc, at the stellar surface is g =(1.44–1.74) 10 cm/s . the electron gyrofrequency. If νefftc 1, the system is G99-47 is located quite close to the Sun (at a dis- not in LTE, and the distribution of electrons over the tance of 8 pc [26]) and has been observed at various Landau levels is formed for the most part byradiation wavelengths from the radio [27] to the X-ray[9, 10]. at the cyclotron frequency, with collisions resulting The stellar magnetic field results in the Zeeman onlyin small adjustments. The acceptabilityof the splitting of the Hα line, as well as polarization of assumption of LTE (νefftc 1) is obviouslyviolated the optical continuum of the order of 1% [28]. In a in the upper layers of an isothermal corona with a simple approximation, the magnetic field of G99-47 barometric densitypro file [see (1)]. In this case, the can be represented bya dipole located at the center intensityof the cyclotron radiation and the electron 7 of the star and producing a field Bp =2.5 × 10 G distribution over the Landau levels must be calculated at the magnetic poles [29]. The dipole axis is tilted in a self-consistent way. ◦ by Θobs = 120 to the direction toward the Earth. A more complex model of the magnetic field in the form 1 The continuum polarization is usuallythought to be associated with the difference between the absorption coefficients of a dipole shifted from the stellar center was used in (dichroism) of normal waves in the photospheres of the mag- [28]. The best agreement between the observed and netic white dwarfs [17, 18]. calculated polarizations of the radiation was achieved

ASTRONOMY REPORTS Vol. 48 No. 2 2004 THERMAL CYCLOTRON RADIATION 123 when the dipole was shifted from the stellar center by nobs m 9 about 0.14R∗ (where R∗ =10 cm is the radius of the ◦ ◦ star) and inclined by Θobs = 100 ± 5 to the direction yobs toward the Earth. Here, we calculate the cyclotron y radiation of the corona for an arbitraryangle Θobs and x obs z assume that the stellar magnetic field is produced by Θ δ × 7 obs Θ a dipole at the stellar center with Bp =2.5 10 G. * Further, we assume that the corona is composed of Φ x fullyionized hydrogen plasma uniformlycovering the * entire surface of the white dwarf and that the field- B aligned temperature of the electrons Tcor is constant throughout the corona. The distribution of the electron densityin height z will then be described bythe barometric formula Fig. 1. Coordinate systems used to calculate the coronal cyclotron radiation. ne = ne0 exp(−z/Hcor), (1) where n is the densityat the base of the corona, e0 2kBTcor × 6 Tcor Hcor = =1.65 10 6 the photosphere, whose spectral intensityis given by mpg 10 K a Planck law with temperature Teff = 5700 K: −1 g 3 3 2 × cm ∗ ω /4π c 108 cm/s2 Bω(ω; Teff)= , exp(ω/kBTeff) − 1 is the scale height of the isothermal corona, k is B where is Planck’s constant. the Boltzmann constant, mp is the proton mass, and g =108 cm/s2 is the free-fall acceleration. Note that In the adopted model, the observed coronal cyclotron radiation is determined bythe six parameters Hcor is considerablyless than the distance Bp, g, Teff, ne0, Tcor,andΘobs and is described by √ T 1/2 the well-known Stokes parameters F , Q, U,andV . l 2β L =6.12 × 106 cor B Tcor B 106 Here, F is the spectral densityof the energy flux of K the radiation received, and the ratios V/F = ζc and R∗ × cm Q2 + U 2/F = ζ are the degrees of circular and 109 cm l linear polarization, respectively. In general, Q and U at which the change in the electron gyrofrequency due depend on the choice of coordinate system in the to the vertical nonuniformityof the magnetic field is plane of the sky. They also determine the orientation comparable to the Doppler width of the cyclotron line of the polarization ellipse with respect to the axes of 2 this system. As follows from the symmetry of this (here and below, βTcor = kBTcor/mc is the ratio of the thermal velocityof the electrons to the speed of system, the axes of the polarization ellipse are directed along and across the projection of the stellar magnetic light c and LB R∗/3 is the characteristic spatial dipole onto the plane of the sky. Consequently, it scale of the stellar dipolar magnetic field). 2 Accord- ingly, we shall take into account only nonuniformity is convenient to use the plane-of-the-skyCartesian x ,y of the magnetic field over the stellar surface, neglect- coordinate system obs obs showninFig.1.The x ing its variation with height of the corona (or, more obs axis is directed opposite to the projection of the precisely,along anyrayinside the corona). stellar magnetic dipole onto the plane of the sky; the basis vector y◦ =[n , x◦ ] forms a right-handed We shall assume that no radiation is incident at obs obs obs system with the unit vector nobs directed toward the the upper boundaryof the corona and that the lower ◦ observer and the basis vector x . In this system, boundaryis irradiated byunpolarized radiation from obs U =0 and the degree of linear polarization is ζl = 2 | | The quantity lB is of the order of the thickness of the so- Q /F .IfQ is positive and negative, the major axis of called gyroresonant layer in a fairly large region of plasma the polarization ellipse is directed along and perpen- (whose size is much greater than lB ). The gyroresonant dicular to the projection of the stellar magnetic dipole layer is responsible for the cyclotron radiation and absorption onto the plane of the sky. According to the definition of photons with frequency ω propagating at the angle α with respect to the direction of the magnetic field, whose of the Stokes parameters [32, 33], positive values of V characteristic scale for nonuniformityis LB (for more details, correspond to right-hand circular polarization, when see, for example, [30] and [31, § 19.3]). the electric-field vector rotates in the counterclock-

ASTRONOMY REPORTS Vol. 48 No. 2 2004 124 ZHELEZNYAKOV et al. wise direction if the observer looks at the source of is close to ω (here, ωBp = eBp/mc is the electron radiation. gyrofrequency at the magnetic pole). Inside a region The position of a point on the stellar surface can be of efficient radiation, there is a resonant interaction described in the spherical coordinate system shown between the photons and thermal electrons, whose in Fig. 1. The magnetic latitude Θ∗ is measured from the north magnetic pole, and the longitude Φ∗ is mea- normalized momentum along the magnetic field has sured from the plane specified bythe stellar magnetic the magnitude dipole m and the line of sight nobs. The coronal cyclotron radiation at frequency ω is p|| ω − sω formed in narrow regions of latitude on the stellar √ ≡ ξ = √ B (2) surface (in regions of efficient radiation), where one 2mcβTcor 2βTcor ω cos αobs of the harmonics sω of the local gyrofrequency B 2 ωB = ωBp 1+3cos Θ∗/2 less than on the order of unity. Here,

2 − (3 cos Θ∗ 1) cos Θ√obs +3cosΘ∗ sin Θ∗ cos Φ∗ sin Θobs cos αobs = , (3) 1+3cos2 Θ∗

αobs is the angle between the direction of the local F − Fph and the Stokes parameters Q and V in the magnetic field and the direction of propagation of the form radiation (the line of sight n ). Beyond the region of smax ∗ obs F − F Iout + Iout − B efficient radiation, the speed of the resonant electrons ph = s1 s2 ω (4) F πB∗ is substantiallygreater than the thermal speed, and ph s=1 ω the optical depth of the corona at a fixed frequency cos ϑobs>0 × tends to zero. cos ϑobsdΦ∗d cos Θ∗, smax out − out If the frequency ω is in the interval sωBp /2 ω Q Is1 Is2 = ∗ sωBp , there are two regions of efficient radiation at the F πB ph s=1 ω sth harmonic, which are located in the northern and cos ϑobs>0 × ∗ − ∗ southern hemispheres symmetrically about the mag- (esxesx esyesy)cosϑobsdΦ∗d cos Θ∗, netic equator and have the characteristic size in lati- smax out − out ∆Θ∗ ∼ β 1 ω V Is1 Is2 tude Tcor . As the frequency decreases, = i these two regions shift toward the equator and merge F πB∗ ph s=1 ω cos ϑobs>0 at ω = sωBp /2. The magnetic field has a minimum at × ∗ − ∗ the equator and varies more slowlywith latitude, so (esxesy esxesy)cosϑobsdΦ∗d cos Θ∗. that the characteristic width of the equatorial region Here, Iout and Iout are the intensities of the extraor- of efficient radiation increases to ∆Θ∗ ∼ β . As s1 s2 Tcor dinaryand ordinaryradiation at the sth harmonic ω continues to decrease, this region shrinks near the outgoing from the corona, and the summation over ω equator and disappears when the frequency is less s is limited to the harmonic s , above which the sω /2 β sω max than Bp byabout Tcor Bp , which is the Doppler coronal cyclotron radiation is negligible. width of the cyclotron line at the sth harmonic. As The integration in (4) is carried out over the entire ω increases, the regions of efficient radiation shift observable surface of the star, which is defined bythe toward the magnetic poles and, at ω = sω ,they Bp condition that shrink near the poles to spots with a characteristic cos ϑobs =cosΘ∗ cos Θobs +sinΘ∗ cos Φ∗ sin Θobs ∆Θ∗ ∼ β size in latitude Tcor . When the difference (5) − ω sωBp becomes a factor of a few greater than be positive (ϑobs is the angle between the line of sight βTcor sωBp , the spots disappear. ◦ nobs and the normal vector z to the stellar surface at If ∆Θ∗ 1, the regions of efficient radiation of dif- the point of integration). Strictlyspeaking, integrals ferent harmonics s do not intersect: theyare separated (4) for fixed s are calculated not onlyover the regions byextensive regions of the stellar surface where the of efficient radiation at the sth harmonic but also corona is transparent to radiation at the frequency ω. over the corresponding regions for other harmonics; This enables us to represent the excess observed radi- however, this does not change the results appreciably, ation flux with respect to the photospheric continuum since the integrands in (4) fall to zero veryrapidlywith

ASTRONOMY REPORTS Vol. 48 No. 2 2004 THERMAL CYCLOTRON RADIATION 125 − increasing distance of the point of integration from the iKs1hy hx esy = regions of eﬃcient radiation at the sth harmonic. The (1 + K2 )(h2 + h2) integrals are equal to zero when the region of eﬀective s1 x y radiation at the sth harmonic is essentiallyabsent

(i.e., when the frequency ω is greater than sωBp or less than sωBp /2 byseveral βTcor sωBp ). in (4) represent the components of the unit vector The coeﬃcients ◦ ◦ es = esxxobs + esyyobs of the polarization of the ex- hy + iKs1hx esx = , (6) traordinarywave at the sth harmonic, where 2 2 2 (1 + Ks1)(hx + hy)

2 3cosΘ∗ sin Θ∗ cos Φ∗ cos Θobs +(1− 3cos Θ∗)sinΘobs 3cosΘ∗ sin Θ∗ sin Φ∗ hx = √ ,hy = √ (7) 1+3cos2 Θ∗ 1+3cos2 Θ∗

◦ are the components of the unit vector h = hxx + coincide with the corresponding values for the stellar ◦ obs hyyobs +cosαobsnobs along the magnetic field at the dipole field at this point (Fig. 1). The direction of the point (Θ∗, Φ∗) in the coordinate system of the ob- vector nobs at the point (Θ∗, Φ∗) is described bythe server. The coefficient same angles, αobs and ϑobs [see (3) and (5)]. 2s cos αobs Ks1 = (8) If ηsl(τ,ξ,nobs) is the radiative power and 2 4 2 2 sin αobs + sin αobs +4s cos αobs χsl(τ,ξ,nobs) is the absorption coefficient of the normal wave l at the sth harmonic in the direction gives the ratio of the amplitudes E|| and E⊥ of the n ,3 then electric field in the extraordinarywave along and obs across the projection of the magnetic field onto the ∗ − | | | | out Bω(Teff)exp( τsl(τ0)) plane of the sky: E||/E⊥ = Ks1 [31, § 5.2]. The Isl = (9) ∗ ◦ − 2 polarization vector of the ordinarywave, esyxobs τsl(τ0) e∗ y◦ , is orthogonal to the polarization vector e sx obs s − of the extraordinarywave, so that the di fference of + Ss exp( τsl)dτsl, out out the wave intensities Is1 and Is2 appears in the 0 expressions for Q and V in (4). The asterisks in where Ss(τ,ξ)=ηsl(τ,ξ,n)/χsl(τ,ξ,n) and the superscripts of the components esx and esy in (4) and the subsequent expressions denote complex +∞ conjugation, and i is the imaginaryunity. χ dz τ (τ,ξ,n )= sl (10) Using (6) and (7), we can write explicit ex- sl obs cos ϑ ∗ − ∗ obs pressions for the factors esxesx esyesy and z(τ) ∗ ∗ i(e e − e e ) in (4) in terms of the quantities h τ sx sy sx sy x χsl(τ ,ξ,nobs) dz and hy: = dτ cos ϑ dτ (1 − K2 )(h2 − h2 ) obs ∗ − ∗ s1 y x 0 esxesx esyesy = 2 2 2 , (1 + Ks1)(hx + hy) are the corresponding source function and optical 2K depth, i(e e∗ − e∗ e )= s1 . sx sy sx sy 1+K2 s1 √ +∞ 2 2 out out 4 2π e To calculate the intensities Is1 and Is2 of the τ(z)= nedz (11) radiation leaving the corona at the point (Θ∗, Φ∗),we 3mcβTcor ωB z approximate the corona bya plane-parallel plasma − n (z) B 1 layer in a√ uniform magnetic field whose absolute value =1.14 × 103 e 2 10 −3 7 B = Bp 1+3cos Θ∗/2 and inclination to the verti- 10 cm 10 G cal z axis 3 Here and below, the subscripts l =1and l =2denote quan- 2cosΘ∗ tities characterizing the extraordinaryand ordinarywaves, δ = arccos √ 2 respectively. The quantity ξ,defined√ by(2), will be used as 1+3cos Θ∗ − a frequencyvariable: ω = sωB/(1 2βTcor ξ cos αobs).

ASTRONOMY REPORTS Vol. 48 No. 2 2004 126 ZHELEZNYAKOV et al. −1 1/2 levels, which is determined bythe radiation at the first × g Tcor 108 cm/s2 106 K cyclotron harmonic and Coulomb collisions. These quantities were found in [19], taking into account only is the characteristic optical depth to the extraordinary collisions and the radiation of the extraordinarywave wave at the first harmonic, which is used to specifythe at the first cyclotron harmonic. height z of a point in the corona, and τ0 = τ(z =0)is The formulas of [19] can easilybe generalized to the corresponding optical depth of the entire corona. the case when the effect of the ordinarywave of the The radiative-transport coefficients ηsl and χsl de- first cyclotron harmonic on the electron distribution pend on the electron distribution over the Landau is also taken into consideration [20, 34]:

− ω3 s q¯s−kpk s (1 +q ¯)s k (1 + p)k − q¯s−kpk S (τ,ξ)= , (12) s 8π3c2 k! s (¯/ + r) k! s (¯/ + r) k=0 r=k k=0 r=k 2 2 2 s−1 s s−k k − s−k k dz 3(1 + Ksl cos α) s ωB sin α (1 +q ¯) (1 + p) q¯ p χsl(τ,ξ,n) = φ(ξ) . (13) dτ 4(1 + K2 ) |cos α| 2mc2 /¯−1k! s (¯/ + r) sl k=0 r=k √ Here, φ(ξ)=exp(−ξ2)/ π is the Doppler proﬁle of the cyclotron line;

2 2 2π c [Ψ11(α)I11(τ,ξ,n)+2(ξ)Ψ12(α)I12(τ,ξ,n)] dn q¯(τ,ξ)= 4π (14) 3 ωB (1 + 2(ξ)) is the mean number of photons near the first cyclotron is the normalized thermalization parameter, where harmonic characterized bythe same normalized mo- / = ν tc; eff √ √ mentum ξ; the functions 4 8 π( 2+1)e neL νeff = (15) 2 1/2 3/2 3 1+cos α 15m (kBTcor) Ψ11(α)= , 4 is the effective rate of Coulomb collisions, which char- 2 24 sin4 α 1+2cos2 α acterizes the rate at which the field-aligned and trans- Ψ12(α)= (147π − 448) (1 + cos2 α)3 verse electron temperatures are equalized and is determined byboth electron –ion and electron–electron are the normalized (to unity) directivity diagrams for collisions; and L =ln(rB/λB) is the Coulomb loga- radiation byan electron of the extraordinaryand ordi- rithm, where rB = vT /ωB is the Larmor radius and narywaves at the first cyclotron harmonic in a rarefied λB = /mvT is the de Broglie wavelength for an plasma; the parameter electron with the characteristic thermal velocity vT = k T /m.4 (147π − 448) β2 B cor Tcor 2(ξ)= 2 The parameter Ks1 in (13) is determined by(8), 16π |w(ξ)| 2 and Ks2 = −1/Ks1. The factor 3(1 + K12 cos α) /[4 × 2 | | represents the ratio of the powers of spontaneous (1 + K12) cos α ] in the coefficient (13) is formally radiation byan electron in the ordinaryand extraor- zero for the ordinarywave at the first cyclotron dinarywaves at the first cyclotron harmonic, where harmonic (s =1, l =2). In this case, this factor w(ξ) is the Kramp function [35, formula (7.1.3)]; must be replaced bythe more accurate expression 2(ξ)Ψ12(α)/| cos α| [19, 38]. 1 The distribution (14) for the mean number of pho- p(Tcor)= − exp( ωB/kBTcor) 1 tons q¯(τ,ξ) determining the source functions (12) and the absorption coefficients (13) can be found from the is the equilibrium number of photons in one normal solution of the system of integro-differential trans- wave at the first cyclotron harmonic at the tempera- fer equations [34] for the intensities I11(τ,ξ,n) and ture Tcor; 4 Note that expression (15) for the collision rate is valid only /(τ) 5 /¯(τ)= when Tcor 10 K. The collision rate changes radicallyin 1+2(ξ) the photospheres of cool magnetic white dwarfs [31, 36, 37].

ASTRONOMY REPORTS Vol. 48 No. 2 2004 THERMAL CYCLOTRON RADIATION 127

I12(τ,ξ,n): cos Θ∗ and Φ∗ was usuallya few tens of thousand if sωB /2 <ω0) = Bω(ωB; Teff)/2. condition is satisfied when the phase difference The first boundarycondition indicates that there is ∞ ω n no radiation incident at the upper boundaryof the θ = (n − n )dz =3.32 e0 (17) corona. The second boundarycondition corresponds c 2 1 1010 cm−3 to irradiation of the lower boundaryof the corona by 0 photospheric blackbodyradiation with temperature −1 −1 × Tcor B g Teff. 106 107 108 / 2 K G cm s A numerical solution of system of equations (16) − 1.88β 1 Im w(ξ), if s =1, α =0 was obtained using the method of Feautrier with vari- × Tcor obs 5 2 able Eddington factors, as is described in [39, 40]. 3/(s − 1), if s ≥ 2, αobs =0, The optical depth τ was covered using an approx- imatelylogarithmic grid containing about 80 nodes acquired bythe normal waves at the height of the per decade of optical depth at the center of the cy- corona is much greater than unity. In (17), n1 and clotron line (at ξ =0). A special grid whose density n2 are the indices of refraction for the extraordinary increased near the directions n perpendicular to the and ordinarywaves [31, § 5.2] and Im w(ξ) ∼ 1 is magnetic field or the normal vector to the layer was the imaginarypart of the Kramp function [35, formula developed for the numerical integration over the solid (7.1.3)]. We can see that the condition θ 1 is easily 6 angle in (16). This enabled us to take into account satisfied in the region of parameters Tcor ≥ 10 K 10 −3 the sharp angular variations in the radiation intensity and ne0 ≥ 10 cm for harmonics s ≥ 2 that are near these directions. not veryhigh. It follows from (11) and (17) that, at out θ 1 The intensities of the outgoing radiation Is1 and the first harmonic, the phase difference only out when τ0 1. If the optical depth of the corona at the Is2 in (4) were calculated using (9). The numerical method enables us to integrate exactlythe piecewise- frequency ω is high, the normal waves must acquire a phase difference much greater than unityat a distance linear functions Ss of the optical depth τsl (this is analogous to integrating using the trapezoid formula corresponding to an optical depth of the order of unity in the case when the integrands have exponential [42]. This requirement is satisfied for the second and weights). higher harmonics but not the first harmonic [38, 43, § 5.3.1]. The numerical integration over the stellar surface in (4) for the required Stokes parameters was carried Nevertheless, the radiation of an opticallythick out successivelyover the variables cos Θ∗ (outer) and corona at the first harmonic is approximatelythe same Φ∗ (inner) using the adaptive procedure [41], which as the radiation of the mutually-incoherent normal automaticallyincreases the densityof the integra- waves. This is due, first, to the considerable differ- tion grid near the regions of efficient radiation at the ence between the absorption coefficients of the nor- specified frequency ω and minimizes the number of mal waves at the first harmonic and, second, to the calculations of the integrands far from these regions. fact that the polarization of the radiation of a single The total number of grid nodes in the integration over electron nearlycoincides with the polarization of one of the normal waves (namely, the extraordinary wave). 5 The software for the numerical calculations carried out for In this case, the radiation flux with extraordinarypo- this work was prepared byS.A. Koryagin. larization leaving the corona is formed byelectrons

ASTRONOMY REPORTS Vol. 48 No. 2 2004 128 ZHELEZNYAKOV et al.

5 6 7 Tcor = 10 K Tcor = 10 K Tcor = 10 K

102 –3 cm

101 11 = 10

0 e0 10 n

102 –3 cm ph

1 10 F 10 / F = 10

0 e0 10 n

102 –3 cm 101 9 = 10 e0 100 n

1/2 1 2 4 1/2 1 2 4 1/2 1 2 4 ω ω / Bp

Fig. 2. Spectral ﬂux density F of the coronal cyclotron radiation of G99-47 calculated for a pole-on orientation of the star ◦ 5 6 7 (Θobs =0 ). The coronal temperatures are Tcor =10 , 10 ,and10 K for the plots in the left, central, and right columns, 9 10 11 −3 respectively. The electron densities are ne0 =10 , 10 ,and10 cm for the plots in the bottom, middle, and top rows, respectively.

◦ 5 6 located in its upper part, at optical depths τ 1.The (Θobs =0 ), the temperatures Tcor =10 , 10 ,and 7 9 10 11 −3 contribution of these electrons to the radiation of the 10 K, and densities ne0 =10 , 10 ,and10 cm ordinarypolarization is small. The radiation flux of (Figs. 2 and 3). the ordinarypolarization is formed byother electrons, − located at substantiallylarger depths 1 τ β 2 . Tcor Their extraordinaryradiation is absorbed byupper We can see in Fig. 2 that the flux of coronal layers of the corona and does not contribute appre- cyclotron radiation appreciably exceeds the photo- ciablyto the outgoing radiation with extraordinary 6 7 polarization. Since the radiation of the two groups spheric continuum for Tcor =10 and 10 Kand 10 11 −3 of electrons is uncorrelated, the extraordinaryand ne0 =10 and 10 cm .Thefirst and second ordinarywaves in the radiation leaving the corona are cyclotron harmonics, corresponding to frequencies incoherent. ωBp /2 <ω<ωBp and ωBp <ω<2ωBp , are clearly 7 visible in the radiation spectrum. When Tcor =10 K 11 −3 3. DISCUSSION OF THE CALCULATED and ne0 =10 cm , the third harmonic at frequen-

SPECTRA cies 3ωBp /2 <ω<3ωBp also becomes visible. Let us follow the variations in the frequencyspectra of the ﬂux and polarization of the coronal cyclotron The presence of these harmonics in the integrated radiation as functions of Tcor and ne0.Considerour numerical results corresponding to the case where the spectrum of the star is explained byFig. 4, which white dwarf G99-47 is observed from the north pole shows the regions of the Tcor–ne0 plane where the

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5 6 7 Tcor = 10 K Tcor = 10 K Tcor = 10 K 1.0

0.5 –3 cm

0 11 = 10

–0.5 e0 n –1.0 1.0

0.5 –3 cm F / 10

V 0 = 10

–0.5 e0 n –1.0 1.0

0.5 –3 cm 0 9 = 10

–0.5 e0 n

–1.0 1/2 1 2 4 1/2 1 2 4 1/2 1 2 4 ω ω / Bp

Fig. 3. Degree of circular polarization ζc = V/F of the coronal cyclotron radiation of G99-47 calculated for the same conditions as in Fig. 2.

weighted-mean optical depths of the corona at the third harmonic τ31 isclosetounity,sothat this component of the cyclotron radiation becomes cos ϑ |cos α| τsl(τ0,ξ=0, n)dn appreciable in the integrated radiation of the star. cos ϑ>0 Note that the coronal cyclotron radiation in the τ = (18) sl case T =107 Kandn =1011 cm−3 increases the ϑ | α| n cor e0 cos cos d bolometric luminosityof the star byan appreciable cos ϑ>0 factor. To support the powerful cyclotron radiation, 6 a nonradiative energy-transport mechanism would are equal to unityat the stellar pole. In the parameter have to exist in the white dwarf’s photosphere, to region located above a line τsl =1,theopticaldepth provide a flux of energyto the corona exceeding the of the corona becomes high for the normal wave l radiative flux from the photosphere. This essentially at the sth harmonic. As can easilybe seen, when excludes the existence of coronas with such parame- 6 11 −3 7 Tcor =10 Кandne0 =10 cm ,andTcor =10 К ters. and n =1010 cm−3, the optical depth of the corona e0 When there are onlytwo cyclotron harmonics in to the extraordinarywave at the second cyclotronhar- the spectrum (and the third harmonic is absent), the τ ∼1 monic is 21 . Therefore, the second harmonic is ordinarywave dominates in the first harmonic and observable in the coronal radiation in the case of these the extraordinarywave in the second harmonic. This parameters, as well as at the higher values of Tcor = 7 11 −3 follows from the polarization of the radiation calcu- 10 Kandne0 =10 cm . In this last case, the 6 11 −3 lated for Tcor =10 Kandne0 =10 cm , Tcor = optical depth of the corona to the extraordinarywave 6 10 −3 7 10 Kandne0 =10 cm ,andTcor =10 Kand 10 −3 6 To simplifythe calculations, it is assumed in (18) that q¯ = p ne0 =10 cm (Fig. 3). The region of efficient ra- at anyheight in the corona [see (10), (13)]. diation of the first harmonic at frequencies ωBp /2

ASTRONOMY REPORTS Vol. 48 No. 2 2004 130 ZHELEZNYAKOV et al.

1012 〈τ 〉 22 = 1 〈τ 〉 12 = 1

〈τ 〉 21 = 1 〈τ 〉 = 1 1011 31 –3 , cm e0 n ϖ = ε 1010 ετ 0 = 1

109 105 106 107 Tcor, K

Fig. 4. Separation of the Tcor–ne0 parameter plane into characteristic regions with various conditions for the transport of cyclotron radiation in the corona of G99-47 for B =2.5 × 107 Gandδ =0. To the right of the dotted line = ,theparameter (ξ =0) is greater than (τ0), so that the conversion of normal waves via cyclotron scattering is appreciable over the entire height of the corona. The mean optical depths of the corona (18) exceed unityabove the solid straight lines τsl =1.The optical depth of the corona to the extraordinarywave at the first harmonic, τ11, exceeds unityfor all values of Tcor and ne0 plotted in the figure. The thermalization length of the radiation of the extraordinarywave at the first cyclotron harmonic is less than the height of the corona in the region above the dashed line (τ0)τ0 =1. √ ω ωBp / 2 is located near the equator, where, in / 1 and 2 1, the source function at the top of general, the magnetic field is directed awayfrom the the corona is substantiallyless than in the deeper observer. Right-hand circular polarization (V>0)at layers and the characteristic spatial scale for varia- these frequencies corresponds√ to the ordinarywave. tions in the source function is considerablygreater than the distance corresponding to unit optical depth At higher frequencies ωBp / 2 ω ωBp , the region of efficient radiation shifts toward the pole, where to the extraordinaryradiation [19, 44]. Therefore, the the magnetic field is directed toward the observer. extraordinaryradiation is formed in upper coronal lay- The direction of rotation of the electric-field vector ers, where the source function is relativelysmall. For 6 in the normal wave changes, reversing the sign of the parameter combinations Tcor =10 Kandne0 = 11 −3 6 10 −3 the circular polarization√ of the coronal radiation near 10 cm , Tcor =10 Kandne0 =10 cm ,and T =107 Kandn =1010 cm−3,theopticaldepth the frequency ωBp / 2. The same arguments for the cor e0 of the corona to the ordinaryradiation is τ12 1. frequencyrange ωBp ω 2ωBp , corresponding to the second cyclotron harmonic, clarify the dominance As a result, the ordinaryradiation is formed primarily of the extraordinarywave in this case. in central regions of the corona, where the source function is greater. The ratio of the intensities of the To explain the dominance of the ordinarywave in normal waves at the first harmonic depends on the the coronal radiation at the first harmonic, we must carryout a special analysisof the transfer equations degree of scattering of the extraordinaryradiation in taking into account thermalization and mutual con- the upper layers of the corona and the transparency of version of the normal waves during cyclotron scat- the corona to the ordinaryradiation. As follows from tering. The line 2 = / (where the efficiencies of the the calculated frequencypro files of the polarization, above processes are comparable) goes through the the scattering of the extraordinarywave is verystrong. considered range of Tcor and ne0 in the Tcor–ne0 plane On the contrary, the dominant contribution of the (Fig. 4). Accordingly, we limit our discussion here to a extraordinarywave at the second cyclotronharmonic few remarks. The coronal optical depth to the extraor-√ can be explained verysimply.The corresponding op- dinarywave at the first harmonic τ11 =3τ0/ π is tical depths of the corona to the extraordinaryand substantiallygreater than unityfor all combinations ordinarywaves, τ21 and τ22, are less than or of of Tcor and ne0 considered above [see (11)]. When the order of unity. However, the optical depth to the

ASTRONOMY REPORTS Vol. 48 No. 2 2004 THERMAL CYCLOTRON RADIATION 131 extraordinarywave, τ21, is greater (Fig. 4), and so it comparable to the gravitational force for lower elec- dominates in the coronal radiation. tron densities and temperatures in the corona. This Let us speciallynote the flux and polarization of should be taken into account when studying white- 6 dwarf coronas whose cyclotron radiation at the first the radiation at the temperature Tcor =10 Kand 9 −3 harmonic is in the optical. the minimum density ne0 =10 cm for which the calculations were performed. In this case, the radiation flux differs onlyslightlyfrom the photospheric 4. UPPER LIMIT FOR THE CORONAL continuum, and the presence of the corona can be ELECTRON DENSITY OF G99-47 detected onlyfrom variations in the polarization of the Let us compare our calculations with the results radiation in the range ω /2 <ω<ω , correspond- Bp Bp of photometric observations of the white dwarf G99- ingtothefirst cyclotron harmonic. The dominance of 47intheinfrared.Theaboveanalysisshowsthatthe the ordinarywave in this case is due to the fact that coronal cyclotron radiation can be observed only as the photospheric radiation of the ordinarypolariza- broadband emission features in the infrared radiation tion passes through the corona virtuallyunchanged of the white dwarf, with the maximum excess over out ≈ ∗ (I12 Bω(Teff)/2). On the other hand, the intensity the photospheric continuum being in the wavelength of the outgoing radiation in the extraordinarywave interval λ =(0.5–2.0) × 2πc/ωB =2.14–8.57 µm, ∗ p drops below Bω(Teff)/2 (primarilydue to scattering). corresponding to the first and second cyclotron har- As a result, the intensityof the ordinaryradiation monics. Consequently, the emission of the corona will be greater than the intensityof the extraordinary should increase the observed brightness of G99-47 radiation. in the K, L, L,andM photometric bands (with the At lower densities ne0 (for which calculations were effective wavelengths λ 2.2, 3.5, 3.8,and4.8 µm, not carried out), the corona behaves like a scattering respectively[45]) and increase the color indices of layer, and so the observed radiation flux should be these bands with respect to bands with smaller effec- lower than the photospheric flux. However, this can- tive wavelengths, such as V (0.55 µm), J (1.2 µm), or not result in the formation of a prominent absorption H (1.6 µm). feature in the observed spectrum, since radiation at According to the catalog [46], G99-47 has been the given frequency ω is efficientlyscattered onlyin observed several times in the J, H,andK [21, 22, a verysmall region of the corona (within the narrow 24, 47–50] infrared photometric bands and once in ∼ 7 latitude interval ∆Θ∗ βTcor 1). The relative vari- the L and L bands [21]. The B, V , R, I, J, H, ation in the stellar radiation flux should be about βTcor , and K observations of [22, 24] revealed no peculiarity which is virtuallyundetectable. in the K magnitude of G99-47 compared to other At low temperatures T 105 K, the corona also white dwarfs with approximatelythe same photo- cor spheric temperature, 5700 K. The (B − V )–(V − K) affects the observed radiation flux onlyslightly.Even − − at the maximum possible intensityof the coronal and (V I)–(V K) diagrams for white dwarfs with emission B∗ (T ) (inside the region of efficient radi- hydrogen photospheres resemble narrow strips whose ω cor half-widths in B − V , V − I,andV − K are approx- ation), its relative contribution to the observed radia- m m m tion flux from the entire star, imately 0.05 , 0.05 ,and0.12 ; i.e., theyapprox- imatelycoincide with the errors [24, Fig. 13]. The β B∗ (T ) T 3/2 white dwarf G99-47 is located nearlyin the centers Tcor ω cor ∼ cor , B∗ (T ) 5 × 105 K of these diagrams; i.e., no excess infrared radiation ω eff in the K band is observed. This means that the flux 5 becomes negligible when Tcor 10 K. of the coronal cyclotron radiation at the second harmonic in this band is much less than the photospheric Concluding this section, we note that the force due flux. Consequently, there is no appreciable cyclotron to the pressure of the cyclotron radiation in the very radiation in the shorter-wavelength J and H bands dense coronas of magnetic white dwarfs can be com- containing the third and higher cyclotron harmonics. parable to or even greater than the gravitational force − − [34]. In this case, the height profile of the electron Let us compare the observed J K, J L,and J − L color indices from [21, 22] with our calcula- densityin the corona will di ffer from the barometric profile (1). In the corona of G99-47, the radiation- tions. The J and K, L, L bands have fairlysimilar pressure force will exceed the gravitational force only wavelengths, so that the corresponding color indices 11 −3 7 when n > 5 × 10 cm and T > 10 K (i.e., be- e0 cor 7 The aim of the observations of [21] in the J, H, K, L,andL yond the parameter region presented in Figs. 2 and 3). bands was to search for a low-mass, cool companion (planet) However, in other white dwarfs with stronger mag- near G99-47. The blackbodyradiation of such a companion netic fields, the cyclotron-radiation pressure will be should increase the observed infrared flux from G99-47.

ASTRONOMY REPORTS Vol. 48 No. 2 2004 132 ZHELEZNYAKOV et al.

Observed and theoretical color indices for the white dwarf G99-47

Model parameters Color indices Data source −3 Tcor,K ne0,cm Θobs,deg J − K J − L J − L

Observations [21, 22] – – – 0m. 20–0m. 40 0m. 16–0m. 58 0m. 11–0m. 63

Model without corona – 0 – 0m. 34 0m. 51 0m. 53

Model with corona 106 1010 0, 180 0m. 36 0m. 73 0m. 81

Model with corona 106 1010 30, 150 0m. 37 0m. 68 0m. 78

Model with corona 106 1010 60, 120 0m. 37 0m. 7 0m. 78

Model with corona 106 1010 90 0m. 37 0m. 67 0m. 82 The row of observed color indices presents the lower and upper boundaries for these quantities. The J − K values measured in [22] are given in the CIT photometric system, and J − L and J − L values obtained in [21], in the IRTF system. The calculated color indices are presented in the Johnson–Glass photometric system [45]. The small discrepancies in the definitions of the color indices in the different photometric systems are not important for our analysis. The column of Θobs presents the two values Θobs1 and ◦ Θobs2 = 180 − Θobs1 at which the same color indices are observed. The observations at Θobs1 and Θobs2 correspond to opposite directions of the line of sight with respect to the stellar magnetic dipole. are less sensitive to the particular form of the spec- J − K, J − L,andJ − L color indices fall outside trum of the photospheric continuum than are the the confidence intervals for the observations. Such a color indices for photometric bands with larger wave- detailed analysis is beyond the scope of the present length differences.8 The coronal cyclotron radiation work, and we present the upper limit ne0 onlyfor the 6 can change the flux in the K, L,andL bands but does characteristic coronal temperature Tcor 10 K. In not affect the flux in the J band, since it corresponds the case of higher temperatures Tcor, the upper limit to the veryweak third and fourth cyclotronharmon- on ne0 should be less. ics. The transmission coefficients for the J, K, L, Comparison of the calculations and observational and L filters and the constants required to calculate data shows that the electron densityin the corona 10 −3 the color indices in the Johnson–Glass photometric of G99-47 does not exceed ne0 =10 cm if the 6 system were taken from [45]. The J − K, J − L,and temperature is Tcor =10 K (see the table). Cyclotron J − L color indices calculated for G99-47 without radiation of the corona with the above parameters re- a corona (i.e., for a blackbodyspectrum with tem- sults in J − L and J − L being outside the confidence perature 5700 K) proved to be within the confidence intervals for the observed values for anyobservation intervals of their observed values (first and second angle Θobs. This conclusion is also valid for Θobs = ◦ rows in the table). Therefore, the radiation of G99- 120 , which is in best agreement with the polarization 47 in the K, L,andL bands could be formed solely of the radiation in the Zeeman-split Hα line [28, 29]. bythe photosphere (without a corona). Therefore, As the temperature Tcor increases, the coronal cy- the available observational data enable us to derive clotron radiation increases. Therefore, at higher tem- 6 onlyupper limits on the coronal electron density ne0 peratures (Tcor > 10 K), the coronal electron density 9 10 −3 for various temperatures Tcor, i.e., to draw a line ne0 cannot exceed the limit 10 cm established for 6 in the Tcor–ne0 plane above which the calculated Tcor =10 K (in fact, ne0 should be even lower). The frequencydependences of the flux and polar- 8 We did not use H − K, H − L,andH − L, although the ization of the coronal cyclotron radiation for Θobs = H band is closer to the K, L,andL bands, because the ◦ 6 10 −3 H photometric data for G99-47 presented in [21, 22] differ 120 , Tcor =10 K, and ne0 =10 cm are pre- from each other by 0.52m. This discrepancyappears to be sented in Fig. 5. For comparison, the frequency due to a misprint in [21], since it is stated there that the H ◦ dependences of these quantities for Θobs =0 and observations of G99-47 in [21] are in agreement with the ◦ Θobs =90 are also shown. The frequencyspectra earlier data of [48], which coincide with [22] within 5%. ◦ ◦ 9 With fixed B , g, T ,andΘ ,theflux of the coronal cy- of the radiation flux for Θobs = 120 and Θobs =90 p eff obs possess well-defined maxima at frequencies slightly clotron radiation depends on onlytwo parameters: Tcor and ne0. above ωBp /2 at the first harmonic and slightlyabove

ASTRONOMY REPORTS Vol. 48 No. 2 2004 THERMAL CYCLOTRON RADIATION 133

ω ω ω ω / Bp / Bp 0.5 0.7 1.0 1.4 2.0 0.5 0.7 1.0 1.4 2.0 0.75 3.0 2.5 0.50 0.25

ph 2.0 F F / /

F 0 V 1.5 –0.25 –0.50 1.0 –0.75 0.5 0.7 1.0 1.4 2.0 0.5 0.7 1.0 1.4 2.0 0.5 0.7 1.0 1.4 2.0 0.5 0.7 1.0 1.4 2.0 1.0 0.75 0.50 0.8 wK 0.25 0.6 F / , arb. units ' w wL 0 L' Q L

w 0.4 ,

L –0.25 w

, 0.2 K –0.50 w 0 –0.75 8 7 6 5 4 3 2 0.5 0.7 1.0 1.4 2.0 λ µ ω ω , m / Bp

Fig. 5. The spectral flux density F and degrees of circular and linear polarization (ζc = V/F and ζl = |Q|/F )ofthecoronal ◦ ◦ ◦ radiation of G99-47 calculated for observation angles Θobs = 120 (solid curves), Θobs =0 (dotted curves), and Θobs =90 6 10 −3 (dashed curves). The coronal temperature is Tcor =10 K and the electron densityis ne0 =10 cm . The transmission coefficients wK , wL,andwL for the K, L,andL filters [45] are presented in the bottom left panel whose upper and lower

horizontal axes, refer, respectively, to the wavelengths and the ratios ω/ωBp (also used in other panels of this ﬁgure), along the upper axis.

ωBp at the second harmonic. The radiation at these the range ωBp <ω<2ωBp of the second cyclotron frequencies is formed near the equator, where the harmonic, where the radiation of the extraordinary magnetic field is minimum and the region of efficient wave dominates. radiation is substantiallybroader. The planes of the 10 −3 Our new upper limit ne0 =10 cm on the elec- linear polarization, defined bythe sign of the ratio tron densityin the corona of the white dwarf G99- Q/F , coincide at the boundaries of the frequency 6 47 for the temperature Tcor ≥ 10 Kismorethan range ωBp /2 <ω<ωBp of the first cyclotron har- a factor of 40 lower than the value derived in [10] monic. This feature of the linear polarization is due to from ROSAT X-rayobservations. Therefore, the de- the axial symmetry of the stellar dipole magnetic field tection of coronas of isolated magnetic white dwarfs [16]. The magnetic field is directed along the same via their cyclotron radiation has high sensitivity cur- line (but in opposite directions) at the equator and rentlyunattainable with X-rayobservations. Since the pole, where the radiation at the frequencies ωB /2 2 p the X-raybremsstrahlung flux is proportional to ne0, and ωBp is formed. In addition, the radiation of the the sensitivityof these measurements must be en- ordinarywave, whose polarization ellipse is stretched hanced bya factor of 402 ∼ 2000 to be comparable to along the projection of the magnetic field onto the our method based on analyzing the coronal cyclotron plane of the sky, dominates at the first harmonic. As radiation. a result, the plane of linear polarization of the coronal Our upper limit n can be further decreased us- x e0 radiation goes through the same axis obs in the plane ing polarization observations of G99-47 at 4–8 µm, of the skyat frequencies slightlyabove ω /2 and Bp corresponding to the first cyclotron harmonic.10 The slightlybelow ωBp . Such polarization corresponds to positive Q/F , as is demonstrated in Fig. 5. The same 10 Most polarization observations of white dwarfs have been relation for the planes of linear polarization occurs in carried out onlyin the optical [51].

ASTRONOMY REPORTS Vol. 48 No. 2 2004 134 ZHELEZNYAKOV et al. degree of polarization of the radiation at these wave- REFERENCES lengths could be several percent in the case of a 1. T. A. Fleming, K. Werner, and M. A. Barstow, Astro- 6 9 −3 corona with Tcor =10 Kandne0 =10 cm . phys. J. 416, L79 (1993). Note that radiation from the photosphere of an 2. I. J. O’Dwyer, Y.-H. Chu, R. A. Gruendl, et al., isolated magnetic white dwarf is polarized due to the Astron. J. 125, 2239 (2003). dichroism of normal waves in a magnetized plasma 3. E. M. Sion and R. A. Downes, Astrophys. J. 396,L79 (1992). [17, 18]. A characteristic feature of the polarization 4. T. A. Fleming, S. L. Snowden, E. Pfeffermann, et al., associated with the presence of the corona is the Astron. Astrophys. 316, 147 (1996). change in the sign√ of the circular polarization near 5. H. L. Shipman, Astrophys. J. 206, L67 (1976). the frequency ωBp / 2. Both the linear and circular 6. C. Martin, G. Basri, M. Lampton, and S. M. Kahn, polarization of the coronal radiation decrease sharply Astrophys. J. 261, L81 (1982). 7. Z. E. Musielak, Astrophys. J. 322, 234 (1987). to zero below ωBp /2 (where cyclotron radiation is 8. Z. E. Musielak and J. M. Fontenla, Astrophys. J. 346, absent) and above ωB ifthecoronaistransparentto p 435 (1989). cyclotron radiation at the second harmonic. 9. K. A. Arnaud, V. V. Zheleznyakov, and V. Trimble, The choice of the magnetic white dwarf G99- Publ. Astron. Soc. Pac. 104, 239 (1992). 47 for our studywas dictated, first, bythe reported 10.R.Cavallo,K.A.Arnaud,andV.Trimble,J.Astro- detection of coronal X-rayradiation bythe Einstein phys. Astron. 14, 141 (1993). satellite (which was not confirmed later) and, second, 11. D. T. Wickramasinghe and L. Ferrario, Publ. Astron. bythe fact that this star was observed preciselyin Soc. Pac. 112, 873 (2000). the region of the infrared where, in principle, coronal 12. V. V. Zheleznyakov, Astrophys. Space Sci. 97, 229 cyclotron radiation could be detected. Upper limits on (1983). the coronal electron densities of other magnetic white 13. V. V. Zheleznyakov and A. A. Litvinchuk, Astrophys. − × 7 Space Sci. 105, 73 (1984). dwarfs—G99-37 (WD 0548 001, Bp =2 10 G, 14. V.V.Zheleznyakov and A. V.Serber, Adv. Space Res. Teff = 6400 K) and G195-19 (WD 0912+536, Bp = 16, 77 (1995). 8 10 G, Teff = 8000 K) [11]—can be determined in the 1 5 . V. N . S a z o n o v a n d V. V. C h e r n o m o r d i k , A s t r o p h ys . same wayusing their infrared photometry[22, 52]. We Space Sci. 32, 355 (1975). will do this in a separate paper. 16. W. H. Ingham, K. Brecher, and I. Wasserman, Astro- phys. J. 207, 518 (1976). In conclusion, we note that it would be desirable 17. J. C. Kemp, Astrophys. J. 162, 169 (1970). to carryout photometric and polarization observa- 18. J. C. Kemp, Astrophys. J. 213, 794 (1977). tions of the unique magnetic white dwarf GD 356 19. A. V.Serber, Astron. Zh. 67, 582 (1990) [Sov. Astron. × 7 (WD 1639+537, Bp =1.3 10 G, Teff = 7500 K) 34, 291 (1990)]. in the range 4–16 µm, corresponding to the first and 20. A. V. Serber, Izv. Vyssh. Uchebn. Zaved., Ser. Ra- second cyclotron harmonics (for example, using the diofiz. 42, 1035 (1999) [Radiophys. Quant. Electron. SIRTF (Spitzer Space Telescope) satellite). The Hα 42, 911 (1999)]. and Hβ lines in the optical spectrum of this star are 21. C. Krishna Kumar, Publ. Astron. Soc. Pac. 97, 294 observed in emission rather than absorption, as for (1985). other white dwarfs [53]. This suggests an enhanced 22. P. Bergeron, S. K. Leggett, and M. Ruiz, Astrophys. temperature in the upper layers of its photosphere J., Suppl. Ser. 133, 413 (2001). and the probable presence of a chromosphere [54] and 23. F.Wesemael, J. L. Greenstein, J. Liebert, et al.,Publ. Astron. Soc. Pac. 105, 761 (1993). corona in this object. 24. P.Bergeron,M.T.Ruiz,andS.K.Leggett,Astrophys. J., Suppl. Ser. 108, 339 (1997). 25. S. K. Leggett, M. T.Ruiz, and P.Bergeron, Astrophys. ACKNOWLEDGMENTS J. 497, 294 (1998). This work was supported bythe Russian Founda- 26. G. P. McCook and E. M. Sion, Astrophys. J., Suppl. 121 tion for Basic Research (project code 02-02-16236), Ser. , 1 (1999). 27. V. A. Hughes, P. A. Feldman, and A. Woodsworth, the President of the Russian Federation Program of Astrophys. J. 170, L125 (1971). Support of Young Russian Scientists and Leading 28. A. Putneyand S. Jordan, Astrophys.J. 449, 863 Scientific Schools of the Russian Federation (project (1995). nos. NSh-1744.2003.2 and MK-2161.2003.02), the 29. D. T. Wickramasinghe and B. Martin, Mon. Not. R. program “NonstationaryPhenomena in Astronomy ” Astron. Soc. 188, 165 (1979). of the Presidium of the Russian Academyof Sciences, 30. V. V. Zheleznyakov, Itogi Nauki Tekhn., Ser. Astron. and the Federal Science and TechnologyProgram in 22, 135 (1983) [Sov. Sci. Rev. Astrophys. Space Phys. Astronomy(project no. 40.022.1.1.1103). Rev. 3, 157 (1984)].

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3 1 . V. V. Z h e l e z n ya k o v, Radiation in Astrophysi- 42. Yu.N.GnedinandG.G.Pavlov,Zh.Eksp.´ Teor. Fiz. cal Plasmas (Kluwer, Dordrecht, 1996; Yanus-K, 65, 1806 (1973). Moscow, 1997). 43. A. I. Akhiezer, I. A. Akhiezer, R. V. Polovin, et al., 32. Standard Definitions of Terms for Radio Wave Prop- Plasma Electrodynamics (Nauka, Moscow, 1974; agation, IEEE Trans. Antennas Propag. 17, 270 Pergamon, Oxford, 1975). (1969). 44. V. V. Zheleznyakov and A. V. Serber, Astron. Zh. 70, 33. J. P.Hamaker and J. D. Bregman, Astron. Astrophys., 1002 (1993) [Astron. Rep. 37, 507 (1993)]. Suppl. Ser. 117, 161 (1996). 45. M. S. Bessell and J. M. Brett, Publ. Astron. Soc. Pac. 3 4 . V. V. Z h e l e z n ya k o v, S . A . K o r ya g i n , a n d A . V. S e r b e r , 100, 1134 (1988). Pis’ma Astron. Zh. 25, 522 (1999) [Astron. Lett. 25, 46. D. Y. Gezari, P. S. Pitts, and M. Schmitz, Catalog 445 (1999)]. of Infrared Observations (Centre de Donnees´ As- 35. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,Ed.byM.Abramowitz tronomiques de Strasbourg, 1999); http://cdsweb.u- and I. A. Stegun (Dover, New York, 1965, 1971; Nau- strasbg.fr/viz-bin/VizieR?-source=II/225. ka, Moscow, 1979). 47. J. Mould and J. Liebert, Astrophys. J. 226,L29 3 6 . V. V. Z h e l e z n ya k o v, S . A . K o r ya g i n , a n d A . V. S e r b e r , (1978). Pis’ma Astron. Zh. 25, 513 (1999) [Astron. Lett. 25, 48. D. T.Wickramasinghe, D. A. Allen, and M. S. Bessell, 437 (1999)]. Mon. Not. R. Astron. Soc. 198, 473 (1982). 37. S. A. Koryagin, Zh. Eksp.´ Teor. Fiz. 117, 853 (2000) 49. R. G. Probst, Astrophys. J., Suppl. Ser. 53, 335 [JETP 90, 741 (2000)]. (1983). 3 8 . V. V. Z h e l e z n ya k o v, A s t r o fizika 16, 539 (1980) [As- 50. S. K. Leggett, Astron. Astrophys. 208, 141 (1989). trophysics 16, 316 (1981)]. 51. S. C. West, Astrophys. J. 345, 511 (1989). 39. D. Michalas, StellarAtmospheres (Freeman, San 52. C. Krishna Kumar, Astron. J. 94, 158 (1987). Francisco, 1978; Mir, Moscow, 1982). 53. J. L. Greenstein and J. K. McCarthy, Astrophys. J. 40. D. Mihalas, J. Comp. Phys. 57, 1 (1985). 289, 732 (1985). 41. G. Forsythe, M. Malcolm, and C. Moler, Computer Methods forMathematical Computations 54. L. Ferrario, D. T. Wickramasinghe, J. Liebert, et al., (Prentice-Hall, Englewood Cliffs, 1977; Mir, Mon. Not. R. Astron. Soc. 289, 105 (1997). Moscow, 1980). Translated by Yu. Dumin

ASTRONOMY REPORTS Vol. 48 No. 2 2004 Astronomy Reports, Vol. 48, No. 2, 2004, pp. 136–144. Translated from Astronomicheski˘ı Zhurnal, Vol. 81, No. 2, 2004, pp. 159–167. Original Russian Text Copyright c 2004 by Kuz’min, Losovsky.

Two Scales for the Frequency Dependence of the Scattering of Pulsar Radio Emission A. D. Kuz’min and B. Ya. Losovsky Pushchino Radio Astronomy Observatory, Astro Space Center, Lebedev Institute of Physics, Russian Academy of Sciences, Pushchino, Moscow oblast, 142290 Russia Received May 20, 2003; in ﬁnal form, August 8, 2003

Abstract—We have measured the pulse broadening by scattering at 40, 60, and 111 MHz for the pulsars PSR B0809+74, B0950+08, B1919+21, and B2303+30. The frequency dependence of the scatter-broadening parameter is analyzed based on these measurements and data from the literature. The dependence obtained purely from the literature data is not consistent with the theory, and the scattering magnitudes differs considerably from the data of the catalog of 706pulsars of Taylor et al. A two-component model for the frequency dependence of the scattering of the pulsar radio emission in the interstellar medium is proposed. Allowing for the presence of two scattering scales removes both inconsistencies between the observational data for these four pulsars and differences between the observed and theoretical frequency dependences for the scattering, as well as the need to invoke anomalous scattering magnitudes. The data of the catalog of Taylor et al. need to be corrected for the difference in the scattering magnitudes in the two branches of the frequency dependence. c 2004MAIK “Nauka/Interperiodica”.

1. INTRODUCTION Thus, the above results show that the index in the frequency dependence γ is approximately −4, The frequency dependence of the scattering of corresponding to a spectrum with a normal distri- pulsar radio emission is an important characteristic bution of inhomogeneities. However, based on mea- that can identify appropriate models for the scattering surements of the frequency dependence of the decor- interstellar medium. However, this dependence has relation bandwidth ∆ν (ν) for five pulsars, Cordes been studied only for individual pulsars, and the avail- d et al. [8] found the mean value γ = −4.45 ± 0.3 and able sparse data are quite contradictory. concluded that the frequency dependence does not The standard theory predicts the frequency de- agree with a normal inhomogeneity distribution and ∝ γ − pendence τsc(ν) ν ,whereγ = 4 for a normal is instead consistent with a Kolmogorov spectrum, for distribution of the inhomogeneities of the inter- which γ = −4.4. This spectral index γ is adopted in stellar medium and γ = −4.4 for a Kolmogorov the catalog of 706pulsars of Taylor et al. [9], which model [1]. Based on their observations of the pulsar contains the most complete data on pulsar scattering. − PSR B0833 45 at four frequencies from 300 to In addition, for a number of pulsars having many 1410 MHz, Komesaroff et al. [2] found the shape published measurements of the decorrelation band- of the observed profiles to be consistent with fre- width and pulse scatter broadening, the frequency quency dependence for the scatter broadening of the dependence of the scattering differs considerably from ∝ −4 pulses τsc ν . Measurements of the frequency the theoretical expectations. In particular, the pub- dependence of the scatter broadening for the Crab lished data for PSR B0809+74, PSR B0950+08, pulsar by Rankin et al. [3] and Kuzmin and Losovsky PSR B1919+21, and PSR B2303+30 indicate γ = [4] yielded the index for the frequency dependence −2.7 ± 0.7, −2.0 ± 0.4, −1.8 ± 0.5,and−5.8 ± 2.9, γ −4. The measurements for this same pulsar of respectively. Kuzmin et al. [5] over a very broad frequency interval, These observational results demonstrate that the −3.8±0.2 from 40 to 2228 MHz, yielded τsc ∝ ν . Based scatter broadening for these pulsars at 1 GHz differs on an analysis of their measurements and other from the value given in the catalog of Taylor et al.[9] published data for nine pulsars, Kuz’min et al.[6] by more than an order of magnitude. determined the mean value γ = −3.0 ± 0.7.From Thus, the question of the frequency dependence of measurements of giant pulses of the millisecond the pulse scattering, and, consequently, of the distri- pulsar PSR B1937+21 and published data, Kuz’min bution of inhomogeneities of the interstellar medium, and Losovskiı˘ [7] derived the frequency dependence remains unsolved. The most complete and widely −4 τsc ∝ ν . used list of magnitudes of scatter broadening τsc is the

1063-7729/04/4802-0136$26.00 c 2004 MAIK “Nauka/Interperiodica” TWO SCALES FOR THE FREQUENCY DEPENDENCE 137

Table 1. Observation conditions and measurement results

−3 PSR DM, pc cm ν,MHz ∆tDM,ms Res, ms τ,ms N τsc,ms δτsc,ms B0809+74 5.751 40 0.93 0.82 1.0 14 5.0 1.0 60 0.25 0.82 1.0 2 0.6 0.4 B0950+08 2.7 40 0.45 0.205 0.3 10 1.0 0.7 40 0.45 0.409 0.5 22 1.0 0.7 B1919+21 12.43 40 1.9 1.228 1.5 10 15 3.5 60 0.56 1.228 1.5 10 4.5 3.0 111 0.09 0.409 0.5 2 0.6 0.4 B2303+30 49.9 40 129 2.56 3.0 2 360 100 111 5.9 2.56 3.0 8 13 3

Note: DM is the dispersion measure; ν, the observing frequency; ∆tDM, the dispersion broadening; Res, the data sampling interval; τ, the time constant for the postdetector filter; τsc, the measured pulse scatter broadening; δτsc, the corresponding error; and N,the number of measurements. catalog of Taylor et al. [9], which is based on selective time constant for the postdetector filter of the receiver, single measurements and the frequency dependence and N is the number of measurements. −4.4 τsc ∝ ν and, therefore, requires refinement and The observations were carried out in an individual- experimental verification. pulse recording regime, with subsequent analysis of We have measured the pulse scatter broadening for the integrated pulse for the entire observing session PSR B0809+74, PSR B0950+08, PSR B1919+21, and of the strongest individual pulses. and PSR B2303+30 at 40, 60, and 111 MHz, The scatter-broadening magnitude was deter- expanding considerably the frequency range of these mined by matching the observed pulsar pulse with data. Based on our measurements and literature data, a model-scattered template representing the intrinsic we have analyzed the frequency dependences of this pulse of the pulsar. The template was specified using parameter and propose a two-scale model for the a pulse consisting of several Gaussian components frequency dependence of the scattering. The model that best fit the shape of the intrinsic pulsar pulse. The removes the inconsistencies indicated above and template pulse was model-scattered by convolving it enables us to determine the parameters for scattering with a truncated exponential function, of the pulsar radio emission in the interstellar medium  more precisely.  exp(−t/τsc) for t ≥ 0 G(t)= (1) 0 for t<0, 2. MEASUREMENTS AND REDUCTION The measurements were carried out at three representing scattering in a thin screen. The pulse frequencies (40, 60, and 111 MHz) on the Large was also dispersion-broadened over the channel Phased Array and decameter radio telescopes of the bandwidth of the receiver, ∆f =1.25 or 20 kHz, and Pushchino Radio Astronomy Observatory (Astro broadened with the time constant of the postdetector Space Center, Lebedev Physics Institute, Russian filter of the receiver. A least-squares fit to the observed Academy of Sciences) from April 2002 to June pulsar pulse was found using the resulting model- 2003. Linear polarization was received. We used a scattered pulse. The magnitude of the scatter broad- 128-channel receiver with channel bandwidth ∆f = ening, τsc, and the amplitudes, positions, and widths 1.25 kHz. For PSR B2303+30, a 128-channel re- of the Gaussian components were found, as well as ceiver with the channel bandwidth ∆f =20kHz was the time delay of the template pulse. Figure 1 shows used. samples of recordings and the processed pulses. The conditions for the observations are summa- To eliminate the effect of the intrinsic shape of rized in Table 1, where DM is the dispersion measure, the initial pulsar pulse, we measured τsc at sev- ν is the observing frequency, ∆tDM is the dispersion eral frequencies. The value of τsc at 102 MHz for broadening, Res is the data sampling interval, τ is the PSR B0950+08 was taken from Kuzmin et al.[5].

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Table 2. Literature data on scattering measurements

PSR ν,GHz τsc,ms δτsc,ms ∆νd,kHz δ∆νd,kHz References B0809+74 0.063 8.5 5 Shitov [10] 0.102 2.2 0.5 Kuz’min et al.[5] 0.105 130 130 Shitov [10] 0.15 1200 Rickett [11] 0.408 2000 Lyne and Smith [12] 0.43 4000 Lyne and Smith [13] B0950+08 0.050 30 Phillips and Clegg [14] 0.063 97 22 Shitov [10] 0.102 0.05 0.02 Kuz’min et al.[5] 0.102 2.8 0.7 Kuz’min et al.[5] 200 50 0.105 210 50 Shitov [10] 0.320 792 Cordes et al.[8] 1188 0.430 20000 Lyne and Smith [12] 0.430 1089 Cordes et al.[8] 0.436 2000 Johnston et al. [15] B1919+21 0.102 3.8 Kondratiev et al.[16] 8.6 25 30 51 0.102 2 0.5 Kuz’min et al.[5] 4 1 7 2 0.105 6.4 0.9 Shitov [10] 0.320 800 Rickett [17] 0.320 7.4 Cordes et al.[8] 19 20 25 54 0.335 730 RobertsandAbles[18] 0.410 2400 0.410 3000 Rickett [11] 0.430 393 Boriakoﬀ [19] 0.430 900 Smith and Wright [13] 0.430 100 Lyne and Smith [12] 0.430 99 Cordes et al.[8] 105 119 129 149 347 B2303+30 0.160 9.9 3.6 Alurkar et al. [20] 0.410 87 Armstrong et al. [21] 0.410 190 Wolszczan [22] 0.430 0.77 Cordes et al.[8] 1.7 4.3

Notes: ν is the frequency; τsc, the pulse scatter broadening; δτsc, the corresponding error; ∆νd, the decorrelation bandwidth; and δ∆νd, the corresponding error.

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PSR B0809 + 74 PSR B0950 + 08 1.0 40 MHz 40 MHz

0.5

0 50 100 20 40 60 80

PSR B1919 + 21 PSR B2303 + 30 Flux, arb. units 1.0 40 MHz 111 MHz

0.5

50 100 150 150 200 250 Time, ms

Fig. 1. Sample recordings and processed data. The solid curve shows the observed pulse, and the dashed curve, the model- scattered template.

−γ 3. RESULTS of the scattering ∆νd(ν) ∝ ν of γ = −2.0 ± 0.4 instead of the theoretical value γ = −4 to −4.4 and the Our results are listed in the last columns of Table1, value adopted by Taylor et al.[9],γ = −4.4. The pulse τ where sc is the measured pulse scatter broadening scatter broadening extrapolated from the observa- and δτsc is the corresponding error in τsc. tional data at 1 GHz, log τsc = −7.77,isthreeorders We used data from the literature to analyze the fre- of magnitude lower than the catalog value (log τsc = quency dependences of this parameter, summarized −10.84). The measurements of Phillips and Clegg in Table 2. To reduce the data to a uniﬁed system and [14] at 50 MHz and of Shitov [10] at 63 and 105 MHz compare them with the data of Taylor et al.[9],we indicate that the scattering is an order of magnitude recalculated the literature values for the decorrelation smaller than for other nearby pulsars. bandwidth ∆νd to the pulse scatter broadening τsc using the relationship used in [9], 2πτsc∆νd =1.53. To explain this anomaly, Phillips and Clegg [14] proposed that the ﬂuctuations of the electron density along the line of sight are an order of magni- 4. ANALYSIS tude smaller toward PSR B0950+08 than toward any other pulsar. They suggest that the anomalously PSR B0950+08 small value of γ = −1.8 derived by comparing their The pulsar showing the largest inconsistencies of data with the results of Cordes et al. [8] is due to the available scattering data and their frequency de- the fact that the receiver bandwidth in the 320 and pendence is PSR B0950+08. In particular, measure- 430 MHz measurements of [8] was considerably nar- ments of the decorrelation bandwidth [8, 10, 12, 14] rower than the decorrelation bandwidth correspond- (Fig. 2a) yield an index for the frequency dependence ing to the data of Phillips and Clegg [14] with γ =

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τ τ sc, ms sc, ms (a) 0 10–1 10 [14] PSR B0950 + 08 PSR B0809 + 74 10–2 10–1 [10] –3 10 [10] [8] 10–2 [10] [15] 10–4 [10] 10–3 –5 [12] 10 [11] [12] 10–4 10–6 [13] 10–7 10–5 10–8 10–6

101 (b) 101 (b) 0 100 10 –1 [5] 10–1 10 [14] [10] –2 10 10–2 –3 [8] 10 [10] –3 [10] [10] [15] 10 10–4 [12] –4 [11] –5 10 10 [12] [13] –5 10–6 10 10–7 10–6 10–8 0.01 0.1 1 ν, GHz 0.01 0.1 1 ν, GHz Fig. 3. Same as Fig. 2 for PSR B0809+74.

Fig. 2. Frequency dependence of the pulse scatter broadening for PSR B0950+08 (a) purely from previously pub- branches in the frequency dependence of scattering in lished data and (b) from our measurements together with the interstellar medium. The basis for this approach is the published data. The filled diamond shows the value as follows. from the catalog of Taylor et al.[9],andthefilled squares, our own results. The filled circles in the lower panel show Popov and Soglasnov [23] have found two charac- the microstructure measurements of Kuzmin et al.[5]. teristic frequency scales for the decorrelation bandwidth of PSR B0329+54. Kondratiev et al.[16] have reported the presence of more than one char- −4.4; this means that Cordes et al. [8] did not mea- acteristic scale for the decorrelation bandwidth of sure the actual decorrelation bandwidth. However, PSR B0329+54, PSR 1508+55, PSR 1919+21, and the data of [8] have been confirmed by the measure- PSR 1641−45. Based on measurements of the pulse ments of Johnston et al. [15]. Moreover, following the microstructure at 102 MHz, Kuzmin et al.[5]have argument of Phillips and Clegg [14], it is also possible detected two scales in the decorrelation bandwidth of that the frequency resolution of the receiver used by PSR B0950+08, ∆νd1 = 200 ± 50 kHz and ∆νd2 = Phillips and Clegg [14] was considerably lower than 2.8 ± 0.5 kHz. the decorrelation bandwidth corresponding to the da- Cordes et al. [8] have proposed a two-component ta of Cordes et al. [8] with γ = −4.4, so that Phillips model for the scattering interstellar medium: a uni- and Clegg [14] likewise did not measure the actual form medium with a characteristic scale H ≥ 0.5 kpc decorrelation bandwidth. and local condensations in compact areas with char- We propose a different approach that removes the acteristic scales H ≤ 100 pc. The existence of two above inconsistencies, the anomalous frequency de- types of inhomogeneities in the interstellar medium is pendence of the scattering, and the need to invoke also noted by Smirnova et al. [24]. anomalously small fluctuations of the electron density Our measurements of pulse scatter broadening at of the interstellar medium toward this pulsar. We the very low frequency of 40 MHz, the microstructure suppose that all the quoted measurements have been measurements at 102 MHz of [5] (∆ν1 =2.8 kHz carried out correctly but are related to two different and τsc =50µs), the data of Cordes et al.[8],and

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τ τ sc, ms sc, ms (‡) 0 10 (a) PSR B2303 + 30 PSR B1919 + 21 101 [20] [16] 10–1 0 [10] 10 [8] 10–2 [16] [12] 10–1 [8] [8] 10–3 [19] [18] 10–2 [13] [21] –4 [11] 10 [11] –3 [18] 10 [22] –5 3 10 10 (b)

2 (b) 10 1 10 101 [20]

0 10 100 [16] [8] 10–1 [5] 10–1 [10] –2 10–2 [16] [8] 10 [8] [21] –3 [22] 10–3 [19] 10 [18] [13] [11] 0.01 0.1 1 10–4 [11] ν, GHz [18] –5 10 Fig. 5. Same as Fig. 2 for PSR B2303+30. 0.01 0.1 1 ν, GHz PSR B0809+74 Fig. 4. Same as Fig. 2 for PSR B1919+21. There are also considerable inconsistencies in the available data on the scattering of PSR B0809+74 and its frequency dependence. For instance, mea- the data of Johnston et al. [15] constitute the up- surements of the decorrelation bandwidth [10–13] γ per branch of the frequency dependence, τsc(ν) ∝ ν (Fig. 3a) indicate the index of the frequency de- ∝ −γ with γ = −4.0 ± 0.3 (Fig. 2b), which agrees with the pendence of the scattering ∆νd(ν) ν to be γ = theoretical relationship for γ = −4. The lower branch −2.7 ± 0.7. of this dependence is formed by the data of Phillips Our measurements and the 102-MHz microstruc- and Clegg [14], Shitov [10], Kuzmin et al.[5](∆ν2 = ture measurements of [5] also reveal a second scale 200 kHz), and Lyne and Smith [12] with γ = −3.0 ± in the frequency dependence of the scattering. The − 0.2.Thedifference from the theoretical value γ = 4 upper branch of the frequency dependence τsc(ν) ∝ is probably due to the stated uncertainty in the data of νγ with γ = −4.6 ± 0.2 (Fig. 3b), which is consistent Lyne and Smith [12]. with the theoretical value γ = −4, is formed by our Extrapolation of the upper branch of the frequency measurements at the very low frequency 40 MHz, dependence to 1 GHz, the frequency adopted by Tay- the microstructure measurements of [5], and the data lor et al. [9], yields log τ 1 GHz [s] = −8.2, consistent of [12, 13]. The lower branch of this dependence is sc formed by the data of Rickett [11] and Shitov [10] with with the scattering data for other pulsars [25]; this − ± removes the need to invoke an anomalous scattering γ = 5.7 0.3.Thedifference from the theoretical value of γ = −4 could be due to the availability of magnitude. The difference of two and a half orders a small number of measurements in a fairly narrow of magnitude between this value and the value of frequency range. 1 GHz − Taylor et al.[9](log τsc [s] = 10.84), which is based only on the 50-MHz observations of Phillips The extrapolation of the upper branch of the fre- and Clegg [14] and a very long extrapolation to 1 GHz quency dependence to 1 GHz, the frequency adopted 1 GHz − with γ = −4.4, demonstrates the need to correct the by Taylor et al. [9], yields log τsc = 8.67, consis- catalog data. tent with the catalog data.

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Table 3. Summary of the frequency dependence of the scattering

PSR γ δγ log τ 0.1 (s) log τ 1.0 (s) Reference B0809+74 –4.7 UP 0.2 –4.11 UP −8.81 UP This work –5.7 LW 0.3 –5.67 LW −11.35 LW This work –2.7 0.7 –5.55 −8.27 Literature data –4.4 –4.14 −8.54 Taylor et al.[9] B0950+08 –4.0 UP 0.3 –4.35 UP −8.35 UP This work –3.0 LW 0.2 –5.99 LW −8.94 LW This work –2.0 0.4 –5.82 −7.77 Literature data –4.4 –6.44 −10.84 Taylor et al.[9] B1919+21 –3.8 UP 0.2 –3.15 UP −6.95 UP This work –3.5 LW 0.4 –4.54 LW −8.05 LW This work –1.8 0.5 –4.84 −6.56 Literature data –4.4 –3.59 −7.99 Taylor et al.[9] B2303+30 –3.6UP 0.6 –1.54 UP −5.15 UP This work –[4] LW –3.27 LW −7.27 LW This work –5.8 2.9 –0.93 −6.77 Literature data –4.4 –1.55 −5.95 Taylor et al.[9] Notes: γ is the spectral index of the frequency dependence; δγ, the error in γ; log τ 0.1 GHz, the logarithm of the pulse scatter broadening at 0.1 GHz; and log τ 1 GHz, the logarithm of the pulse scatter broadening at 1 GHz for the upper (UP) and lower (LW) branches of the frequency dependences τsc(ν).

PSR B1919+21 lower branch, which includes the measurements of The index for the frequency dependence of the [5, 10, 11, 13, 16–19], displays the frequency depen- −γ dence γ = −3.5 ± 0.4, and the scatter broadening, scattering ∆νd(ν) ∝ ν for this pulsar from previous 1 GHz − measurements of the decorrelation bandwidth [8, 10, log τsc = 7.86, is consistent with the catalog 11, 13, 16–19] (Fig. 4a) is γ = −1.8 ± 0.5.Extrapo- values. lation of this dependence to the catalog frequency ν = PSR B2303+30 1 GHz recalculating ∆ν to τsc using 2πτsc∆ν =1.53 There are also considerable inconsistencies in the 1 GHz − yields log τsc = 6.58; this diﬀers from the catalog available data on the scattering of PSR B2303+30 1 GHz − value, log τsc = 7.99, by one and a half orders of and its frequency dependence. Published measure- magnitude. ments of the decorrelation bandwidth [8, 20, 21, 28] Our measurements at 40, 60, and 111 MHz yield the index of the frequency dependence of the − ± show the presence of a second scattering scale scattering γ = 5.8 2.9. In Fig. 5a, the pulse scat- and its frequency dependence (Fig. 4b). The up- ter broadening extrapolated from the observational 1 GHz − per branch, which includes our measurements and data, log τsc = 6.77, is almost an order of mag- 1 GHz those of Cordes et al. [8], displays the frequency nitude lower than the catalog value [9], log τsc = dependence γ = −3.8 ± 0.2, matching the expected −5.95. − theoretical value γ = 4 to 4.4 within the errors. Our measurements also reveal the presence of 0.1 GHz − The scatter broadening, log τsc = 3.15,iscon- a second scale in the frequency dependence of the sistent with the broadening obtained from the sta- scattering. The upper branch with γ = −3.6 ± 0.6 tistical dependence of Kuz’min and Losovskiı˘ [7], (Fig. 5b), which agrees with the theoretical depen- 2.1 0.1 GHz − − τsc(DM) = 40(DM/100) log τsc = 3.3.The dence γ = 4, is formed by our measurements at

ASTRONOMY REPORTS Vol. 48 No. 2 2004 TWO SCALES FOR THE FREQUENCY DEPENDENCE 143 111 MHz and the measurements of [8, 20]. The lower Allowing for two scattering scales removes incon- branch of this relation is formed by the data of [21, 22]. sistencies in the observational data for these pulsars Extrapolation of the upper branch of the frequency and also the discrepancy of the frequency dependence dependence to the frequency of 1 GHz adopted in [9] of the scattering with theoretical expectations. 1 GHz − yields log τsc = 5.15, almost an order of magni- The index in the frequency dependence of the scat- γ ∼ tude greater than the value in [9], likewise indicating tering τsc(ν) ∝ ν for the four pulsars studied, γ = the need to correct the catalog data. −4.0, is in agreement with the theory if the inhomogeneities obey a normal distribution. The scattering data in the catalog of 706pulsars of 5. DISCUSSION Taylor et al. [9] require corrections for the difference in Table 3 summarizes the results of our analysis of the scattering magnitudes in the two branches of the the frequency dependence of the scattering of pulsar frequency dependence. pulses based on our own measurements and data from the literature. This table lists the following parameters derived from least-squares fits of the data: ACKNOWLEDGMENTS the index of the frequency dependence of the pulse The authors are grateful to V.V.Ivanova, K.A. La- scatter broadening γ, logarithm of the pulse scatter paev, and A.S. Aleksandrov for their software and broadening at 0.1 GHz log τ 0.1 GHz, and logarithm technical support of the observations; to A.A. Ershov of the pulse scatter broadening at 1 GHz for the and V.A. Samodurov for the data reduction software; upper (UP) and lower (LW) branches of the frequency and to V.I. Shishov for useful discussions. The au- 1 GHz dependences τsc(ν), log τ . Due to the limited thors thank L.B. Potapova for help with preparation of amount of data, we adopted γ = −4 for the lower the manuscript. This work was partially supported by scattering branch for PSR B2303+30. The table also the Russian Foundation for Basic Research (project lists analogous parameters derived from published code 01-02-16326). data and the scatter broadening values presented in [9]. REFERENCES For all four pulsars studied, allowing for the presence of two scattering scales removes inconsistencies 1. R. N. Manchester and J. H. Taylor, Pulsars (Free- between the observational data and the discrepancy man, San Francisco, 1977; Mir, Moscow, 1980). between the frequency dependence of the scattering 2. M. M. Komesaroff, P. A. Hamilton, and J. G. Ables, Aust. J. Phys. 25, 759 (1972). and the theoretical expectations, as well as the need 3. J. M. Rankin, J. M. Comella, H. D. Craft, et al., to invoke anomalous scattering magnitudes. Astrophys. J. 162, 707 (1970). The average index in the frequency dependence 4. A. D. Kuzmin and B. Ya. Losovsky, Astron. Astro- of the scattering for the upper branches of the four phys. Trans. 18, 179 (1999). pulsars studied (which include a greater number of 5. A. D. Kuzmin, P. A. Hamilton, Yu. P. Shitov, et al., measurements over a broader frequency band) is γ = Mon. Not. R. Astron. Soc. 344, 1187 (2003). −4.0 ± 0.5, in agreement with the theoretical expec- 6. A.D.Kuz’min,V.A.Izvekova,V.M.Malofeev,et al., tation for a normal distribution of inhomogeneities. Pis’ma Astron. Zh. 14, 120 (1988) [Sov. Astron. Lett. 14, 58 (1988)]. Note that the difference between the scattering 7. A. D. Kuz’min and B. Ya. Losovsky, Pis’ma Astron. magnitudes on the upper and lower branches is nearly Zh. 28, 1 (2002) [Astron. Lett. 28, 21 (2002)]. the same for all the pulsars—about one and a half 8. J. M. Cordes, J. M. Weisberg, and V. Boriakoff,As- orders of magnitude. trophys. J. 288, 221 (1985). 9. J. H. Taylor, R. N. Manchester, A. G. Lynn, et al., Catalog of 706 Pulsars (1995, unpublished). 6.CONCLUSIONS 10. Yu.P.Shitov,Astron.Zh.49, 470 (1972) [Sov. Astron. We have measured the scatter broadening of 16, 383 (1972)]. the pulses of PSR B0809+74, PSR B0950+08, 11. B. J. Rickett, Mon. Not. R. Astron. Soc. 150,67 PSR B1919+21, and PSR B2303+30 at 40, 60, and (1970). 111 MHz. 12. A. G. Lyne and F. G. Smith, Nature 298, 825 (1982). 13. F. G. Smith and N. C. Write, Mon. Not. R. Astron. Based on these measurements and data from Soc. 214, 97 (1985). the literature, we have analyzed the frequency de- 14. J. A. Phillips and A. W.Clegg, Nature 60, 137 (1992). pendences of this parameter. We propose a two- 15. S. Johnston, L. Nicastro, and B. Koribalsky, Mon. component model for the frequency dependence of Not. R. Astron. Soc. 297, 108 (1998). the scattering of the pulsar radio emission in the 16. V. I. Kondratiev, M. V. Popov, V. A. Soglasnov, et al., interstellar medium. Astrophys. Space Sci. 278, 43 (2001).

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17. B. J. Rickett, Ann. Rev. Astron. Astrophys. 15, 479 22. A. Wolszczan, Acta Astron. 27, 127 (1977). (1977). 23. M. V.Popov and V.A. Soglasnov, Astron. Zh. 61, 727 18. J. A. Roberts and J. G. Ables, Mon. Not. R. Astron. (1984) [Sov. Astron. 28, 424 (1984)]. Soc. 201, 1119 (1982). 24. T. V. Smirnova, V. I. Shishov, and D. R. Stinebring, 19. V. Boriakoﬀ, Ph. D. Thesis (Cornell Univ., 1973). Astron. Zh. 75, 866 (1998) [Astron. Rep. 42,766 20. S. K. Alurkar, O. B. Slee, and A. D. Borba, Aust. (1998)]. J. Phys. 39, 433 (1986). 25. A. D. Kuz’min and B. Ya. Losovskiı˘ ,Astron.Zh.76, 21. J. W.Armstrong, J. M. Cordes, and B. Rickett, Nature 338 (1999) [Astron. Rep. 43, 288 (1999)]. 292, 561 (1981). Translated by G. Rudnitski ˘ı

ASTRONOMY REPORTS Vol. 48 No. 2 2004 Astronomy Reports, Vol. 48, No. 2, 2004, pp. 145–152. Translated from Astronomicheski˘ı Zhurnal, Vol. 81, No. 2, 2004, pp. 168–175. Original Russian Text Copyright c 2004 by Lozhechkin, Filippov.

Coronal Mass Ejections and Eruptive Prominences: Mutual Spatial Arrangement A. V. Lozhechkin and B. P. Filippov Institute of Terrestrial Magnetism, Ionosphere, and Radiowave Propagation, Russian Academy of Sciences, Troitsk, Moscow oblast, 142090 Russia Received July 10, 2003; in ﬁnal form, August 8, 2003

Abstract—The mutual spatial arrangement of coronal mass ejections and eruptive prominences on the Sun is considered. These phenomena occur on different scales and are observed at different heights above the solar surface. In spite of the presumed causal connection between them, they are often widely separated in position angle at epochs of solar minimum. This means that the motion of a prominence in the corona is not strictly radial and has an appreciable component along the surface. This behavior can be explained in a model of a filament as a magnetic flux rope in equilibrium in the coronal magnetic field. The initial trajectory of the filament is determined by the structure of the global field. c 2004MAIK “Nauka/Interperiodica”.

1. INTRODUCTION In this paper, we study the mutual spatial arrangement of a number of eruptive prominences and CMEs The conditions for observing solar-activity phe- at an epoch of solar minimum and compare the results nomena usually make it impossible to observe the with a model that enables us to interpret phenomena development of a single process continuously, from observed at various levels of the solar atmosphere as its origin in the lower layers of the atmosphere to its parts of a single process associated with the evolution transition into an interplanetary disturbance. This is of a magnetic-flux rope in the coronal magnetic field. largely the case for eruptive prominences. Hα observations of the onset of the activation of a filament (prominence) frequently end very soon due to the 2. DATA USED IN THE ANALYSIS departure of this line from the filter band due to the Studies of statistical connections between coronal Doppler effect, the decrease of the emission due to ejections and other forms of activity have shown that, the decrease in the density of expanding material, among phenomena observed in the lower layers of and the ionization of hydrogen due to heating of the the solar atmosphere, CMEs are most closely as- prominence plasma. The subsequent evolution can sociated with eruptive prominences [6–9]. However, sometimes be observed in the ultraviolet or radio, some analyses have shown the correlation between but the fate of a prominence usually becomes visi- eruptions and coronal mass ejections to be weak (10– ble only when it reaches the lower boundary of the 30%) [10, 11]. Such low correlations are probably due field of view of a spaceborne coronagraph (∼2R) to the fact that the authors also include into eruptive and becomes visible in white light as a coronal mass prominences the thermal disappearance of filaments ejection (CME). A typical CME consists of three [12], sprays and surgers, and the activation of fila- parts: a bright core that is the remnant of an eruptive ments with finite motion [13]. Recent studies distin- prominence; a large, dark, lower-density surrounding guishing between eruptive and activated or between cavity; and an outer, rather diffuseenvelopehaving radially and tangential moving prominences confirm the projected shape of a closed loop with its legs fixed the presence of high correlations between the two on the Sun [1, 2]. However, detailed comparisons phenomena (at the 83–94% level) [14–16]. of the positions of filaments on the disk that have We have proceeded from the assumption that a suddenly disappeared and CMEs reveal certain prob- “genuine” eruption of a prominence, with all or most lems in explaining CMEs as continuations of filament of a filament departing from equilibrium and acceler- eruptions in the upper corona. The heliolatitudes of ating upward to a large height in the corona, always the disappeared filaments and subsequent coronal produces a coronal ejection. We are interested in the ejections sometimes differ by tens of degrees [3–5]. path of a prominence up until the moment when it If the two phenomena are causally connected, this becomes a bright CME core. raises the question of why they are so widely separated Figures 1–3 show the development of an eruptive in latitude (position angle). phenomenon on October 19, 1997, as an example.

1063-7729/04/4802-0145$26.00 c 2004 MAIK “Nauka/Interperiodica” 146 LOZHECHKIN, FILIPPOV

(‡) Oct. 18, 1997 15:33 UT(b) Oct. 19, 1997 15:49 UT

Fig. 1. (a, b) Hα ﬁltergrams of the Sun obtained at Big Bear Observatory and (c, d) Meudon Observatory spectroheliograms in the K3 Ca line.

In the filtergrams of October 18, a large filament is the C2 LASCO coronagraph on board the SOHO visible as a bulging prominence at the eastern limb spacecraft, illustrating the development of a coronal of the Sun (Fig. 1). The eruption of part of the fil- ejection that followed the eruption of a prominence. ament (prominence) was observed in the ultraviolet There were some overlapping streamers at the eastern by the EIT telescope on SOHO and the Nobeyama limb, which filled the region within ±30◦ of the equa- radioheliograph. The dark prominence above the limb tor. Some disturbance crossed the C2 field of view vanished from the EIT images between 1:26 UT and along the streamers from 5 UT to 6 UT but did not 1:55 UT on October 19. By 2:25 UT, some of the dark upset the ray structure. This feature probably had a filament on the disk also disappeared and the rest be- wave nature. The first signs of the development of a came low-contrast. In snapshots taken from 2:46 UT CME appear about 8 UT as an inflation of the feet of until 3:26 UT, a bright node moving eastward above the streamers at the edge of the occulting disk. The the limb is visible. It is obvious that this is material ejection encompasses the entire system of streamers, from the prominence, heated during the eruption and though it does not destroy them completely after its so evolving from absorption of the ultraviolet radiation transit. As a whole, it is arranged symmetrically about to emission [17]. The path of the prominence is shown the equatorial plane and is moving along this plane. in Fig. 2 as a chain of dark and bright nodes above The eruption pattern observed on October 19, the prominence. These nodes trace the positions of 1997, is typical of activity minimum. At such epochs, the eruptive prominence at different times. This image the filaments are mainly concentrated at middle was obtained by combining a snapshot at 1:26 UT latitudes, and it is natural that eruptions are initiated with a series of time-difference images taken from from these regions. Coronal structures are clustered 2:46 UT to 3:26 UT. The path of the prominence ◦ about the equator. Unfortunately, the motions of deviates by more than 30 southward from the radial prominences can only rarely be traced to the boundary direction, shown in Fig. 2 by the white dashed line. of the field of view of the spaceborne coronagraph; After 6 UT, arches of emission appear above the place therefore, we had to compare the coordinates of where the filament was located, which increase in size the beginnings of eruptions (prominences, disap- until 12 UT and then slowly fade. In filtergrams taken peared filaments) and those of CMEs at distances on October 19 (Fig. 1), we can see that the filament of ≥ 2R. We have analyzed a set of SOHO ob- did not decay completely but reduced its length to less servations of CMEs for 1996–1998 together with than half its initial value. There is also no prominence observations of disappearing filaments and eruptive above the limb visible in the Meudon Observatory prominences. The latter were taken from the catalog spectroheliograms (Fig. 1d). of solar events presented by the NOAA at http: Figure 3 shows a sequence of snapshots made by //www.sec.noaa.gov/ftpmenu/indices/events.html.

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We have selected only events that can confidently be associated with CMEs, where the CME can be considered to be a continuation of the eruption of a specific filament in time and space (table). We have compared the distributions of eruptive prominences and CMEs about the solar magnetic equator, since it is likely that only electromagnetic forces can control the motions of plasma structures tangent to the solar surface. We have taken the neutral line on maps of the magnetic field of the source surface calculated and published monthly in the So- lar Geophysical Data as the solar magnetic equator. Unfortunately, although we know a CME’s position angle, we can say nothing about its heliographic longitude. The neutral line is a fairly sinuous curve, and we cannot readily tell at what angular distance from this line the CME appeared. Therefore, we have plotted on the map a segment with the given latitude and ◦ ◦ a scatter in heliographic longitude of 90 (±45 with Oct. 19, 1997 respect to the limb). If CMEs are separated from the ◦ limb by more than 45 , they are likely to be halo-type Fig. 2. Composite image combined from a snapshot at CMEs, which we did not consider. We then found the 1:26 UT in the 195-A˚ ultraviolet band of the SOHO EIT mean distance of the given segment from the neutral telescope and several time-difference images in the same line obtained between 2:46 and 3:26 UT. A chain of dark line after integrating using the method of trapezoids. and bright nodes above the prominence transferred from We adopted this quantity for the distance of the CME the time-difference images traces the path of the eruptive from the magnetic equator. In the table, the letters B prominence. The path deviates more than 30◦ southward and A before a time denote that the event began before from the radial direction, shown with the white dashed line or after this time. A minus sign before the CME’s (we have used data from the joint ESA–NASA project distance from the magnetic equator indicates that the SOHO). prominence and CME were located on opposite sides of the equator. The notation EPL denotes an eruptive a long distance along the solar limb rather than rise prominence observed above the limb, and DFS, the radially. disappearance of a filament on the solar disk. Figures 4c and 4d present data for 1998. Promi- nences disappeared, on average, 33◦ from the mag- ◦ 3. RESULTS netic equator, with a dispersion of 23 , while CMEs appeared, on average, 37◦ from the equator, with ◦ In 1996–1997, prominences disappeared, on av- a dispersion of 25 . The picture for CMEs is very erage, at an angular distance of 25◦ from the mag- “blurred,” and we can see that, in many cases, the netic equator, with a dispersion of 19◦,andCMEs eruption proceeded nearly radially. appeared, on average, 15◦ from the equator, with a dispersion of 15◦. The positions of 81% of CMEs are closer to the magnetic equator than are those 4. DISCUSSION AND COMPARISON of the disappearing filaments. The distributions are WITH THE MODEL shown as histograms in Figs. 4a and 4b. The dis- The distributions we have obtained are consistent ◦ ◦ tribution of prominences peaks at 20 –29 , whereas with the results of other studies [3–5, 16], though ◦ ◦ the maximum of the CME distribution is at 0 –9 . ours are based on a different sample and are plotted as It can be shown that CMEs and disappearances of functions of the angular distance from the magnetic filaments could differ in position angle by as much as equator, not as functions of heliolatitude. The plane ◦ ◦ 70 , with the difference being most often between 16 of the magnetic equator virtually coincides with the ◦ and 33 . Thus, the positional relationship is such that plane of the heliographic equator during solar min- eruptions of prominences take place fairly far from the imum, the two differ during the phase of increasing plane of the magnetic equator, while CMEs tend to activity, and they can be perpendicular near solar concentrate toward this plane. Therefore, if a CME maximum. The situation near solar minimum is suit- indeed represents a further development of an eruptive able for analyses of the development of eruptive pro- prominence, the latter would have to have traveled cesses in space, because it is close to axisymmetric

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06:24 UT 10:27 UT

12:51 UT 15:20 UT

16:40 UT 18:27 UT

19:41 UT 20:42 UT

Fig. 3. Sequence of images from the SOHO C2 LASCO coronagraph illustrating the development of a coronal ejection that followed the eruption of a prominence (we have used data from the joint ESA–NASA project SOHO). with the axis perpendicular to the line of sight, which polarity-inversion lines in the photosphere and a is convenient for observations. We can see that the dipole character for the field in the upper corona paths of plasma structures at the beginning and end (Fig. 5). Thus, the third harmonic (octupole) of the of the eruption can differ widely. Because of this, the expansion of the magnetic-field potential in spherical coordinates of the “source” (filament) and coronal functions dominates at the photospheric surface, ejection can also differ by tens of degrees, not only while the first harmonic (dipole) [18] dominates at in latitude, as we have established here, but also in a distance of about one solar radius. Filaments and longitude. This is very important for estimating the prominences “select” mainly middle-latitude neu- geoeffectiveness of CMEs. tral lines, whereas coronal phenomena (streamers, According to classical concepts, a quiescent CMEs) are clustered toward the neutral line of the prominence is located above a polarity-inversion dipole field, near the equator. If the equilibrium and line of the photospheric magnetic field. The mean motion of a prominence is determined by electromag- theoretical pattern of the global solar magnetic field netic forces, the global magnetic field will determine at activity minimum assumes the presence of three the path of a prominence in the corona. We have

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Angular distance of CMEs and eruptive prominences from the magnetic equator

CME EP Distance Distance from from Position mag- Beginning End Type mag- Date Time angle, netic (UT) of Coordinates netic (UT) (UT) deg equator, event equator, deg deg Oct. 19, 1996 17:39 156 83 B16:08 B17:00 DSF S19E55 42 Dec. 8, 1996 18:30 87 17 B15:48 B16:09 DSF N41E46 60 Dec. 12, 1996 14:30 94 25 A07:35 B07:58 DSF S22E10 3 Dec. 19, 1996 16:30 269 −10 B16:33 B17:13 DSF N02W34 23 Dec. 23, 1996 13:38 284 35 20:11 20:48 EPL S13W67 37 Jan. 10, 1997 10:04 85 −8 B05:15 10:15 DSF N21E15 13 May 31, 1997 18:39 280 −5 B12:02 13:49 DSF N43W90 40 July 30, 1997 19:32 95 −7 B16:35 16:50 DSF N45E21 25 Sept. 29, 1997 15:47 75 −10 B12:54 14:59 DSF N22E90 23 Sept. 29, 1997 18:31 78 −13 A14:47 14:57 EPL N23E90 25 Oct. 13, 1997 11:52 265 −3 B10:00 10:26 EPL S34W29 30 Oct. 21, 1997 01:27 108 30 A23:49 B00:56 DSF S33E59 32 Nov. 10, 1997 12:02 257 22 B07:30 07:58 DSF N30W70 23 Nov. 12, 1997 04:33 270 8 A23:56 03:01 DSF N25W39 25 Dec. 9, 1997 12:27 253 20 11:50 11:54 EPL S27W90 17 Dec. 27, 1997 00:34 279 7 B23:52 01:16 EPL N35W90 18 Feb. 15, 1998 10:27 322 28 B03:50 B07:35 DSF N33E23 62 Feb. 23, 1998 23:27 86 −10 20:58 21:55 DSF N28E11 28 March 1, 1998 20:13 268 −8 18:59 00:00 EPL S26W86 20 March 6, 1998 01:19 256 20 B17:26 A18:34 EPL S52W90 60 March 28, 1998 22:42 326 52 18:26 20:38 DSF N19W26 5 March 31, 1998 13:17 97 38 A11:02 11:29 DSF S27E51 50 Apr. 28, 1998 23:43 63 7 18:51 B19:14 DSF S24E31 12 May 9, 1998 15:18 228 40 15:20 16:08 EPL S22W90 43 May 28, 1998 22:31 140 45 19:06 19:24 EPL N20W90 10 May 29, 1998 22:31 281 2 18:32 18:52 EPL S26W50 27 June 5, 1998 07:03 205 79 A05:48 06:40 DSF S41W18 65 June 15, 1998 14:55 195 30 B11:09 11:24 DSF S71E01 73 June 15, 1998 17:27 106 33 16:55 17:37 EPL S18E90 35 Oct. 26, 1998 21:37 288 32 17:10 17:39 EPL N23W90 47 Nov. 9, 1998 01:26 80 32 23:18 23:30 DSF N24E25 2 Nov. 10, 1998 22:18 106 3 15:20 16:08 EPL S22W90 27

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N 15 1996–1997 N 12 1998 10 EP EP 8

5 4

0 0 0–9 20–29 40– 49 60–69° 0–9 20–29 40–49 60–69° 10–19 30–39 50–59 70–79° 10–19 30–39 50–59 70–79°

16 1998 1996–1997 12 12 CME CME 8 8

4 4

0 0 0–9 20–29 40–49 60–69° 0–9 20–29 40–49 60–69 80–90° 10–19 30–39 50–59 λ 10–19 30–39 50–59 70–79° λ

Fig. 4. Histograms of the numbers N of eruptive prominences (EPs) and CMEs as functions of the angular distance λ from the solar magnetic equator in 1996–1997 and 1998. developed an axisymmetric model for the eruption of a fields for the subphotospheric sources in the range filament represented by a toroidal magnetic-flux rope 1–10 G, a filament in the corona at a height of [19, 20]. In a cylindrical coordinate system (ρ, ϕ, z) about 10–30 Mm can be in equilibrium if the current with its origin at the center of the Sun and its z strengths are 1010–1011 A. For typical densities of axis aligned with the rotational axis, the equations ≈ 105gcm−1, the prominence mass in this model of motion of a segment of a torus with unit length are does not appreciably affect the final equilibrium posi- d2ρ I tion in the lower corona, since the gravitational force m = (B(ex) + B(m) (1) is more than an order of magnitude weaker than the dt2 c z z electromagnetic forces acting on the filament. Two (I) 2 ρ + B ) − mgR , equilibrium points on the curve representing the locus z (ρ2 + z2)3/2 of equilibrium points of the filament correspond to each current. One lies higher and one lower than d2z I z m = − (B(ex) + B(m)) − mgR2 , some critical point; a unique solution of (1)–(2) cor- 2 ρ ρ 2 2 3/2 dt c (ρ + z ) responds to the current Ic at this point. No equilib- (2) rium in the corona is possible for currents greater than (ex) Ic. Only the lower equilibrium position is stable; if where B is the magnetic field created by subpho- the critical current is exceeded, a “catastrophe” (i.e., tospheric sources and currents in the region of the eruption of the filament) occurs. solar wind, B(m) is the field of induction currents in the photosphere that hinder the coronal-current field Having reached the critical point after a slow evolution due to the changing external field or its own from penetrating into the Sun, B(I) is a field created by a current flowing along the axis of a ring, M is current, the filament loses stability and erupts. The the mass per unit length of the filament, and g is the behavior of a filament can be studied by numerically gravitational acceleration at the photosphere level. solving the equations of motion (1)–(2) jointly with the induction equation Solving (1)–(2) with zero left-hand sides for var- LI ious currents I, we can derive the final equilibrium Φ=Φ + = const, (3) position of a filament. For specified coronal magnetic ex c

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determined by the complex magnetic field of the corona can be, at first glance, unexpected. Instead of displaying purely radial (i.e., only vertical at a I given point) and rectilinear motion, the plasma can move a considerable distance along the solar surface. If, in addition, the main phase of acceleration is hidden from the observer, phenomena in the lower and upper corona could appear to be essentially unrelated, whereas they actually represent the development of a Fig. 5. Lines of force of the global solar magnetic field single phenomenon. adopted in the model. The dashed lines show the polarity- inversion lines at the photospheric surface, and the heavy dashed line, the path of a filament during its eruption. ACKNOWLEDGMENTS This work was supported by the Russian Founda- tion for Basic Research (project nos. 03-02-17736, L where is the self-induction of the current ring. 03-02-06044) and the Federal ScientificandTech- Thepathofthefilament during the eruption is nological Program in Astronomy. shown by the heavy dashed line in Fig. 5. The path is initially close to the equilibrium curve, then deviates from it, presumably because inertial forces begin to REFERENCES exert their influence. Due to its inertia, the filament 1. F. Crifo, J. P. Picat, and M. Cailloux, Sol. Phys. 83, crosses the equatorial plane and continues along vir- 143 (1983). tually a straight line at a small angle (≈20◦)tothe 2. D. G. Sime, R. M. MacQueen, and A. J. Hundhausen, equatorial plane. The calculated path of the filament J. Geophys. Res. 89, 2113 (1984). is very similar to the behavior of the prominence and 3. D. F. Webb, in IAU Coll. No. 167: New Perspectives ejection of October 19, 1997, described above. It is on Solar Prominences, Ed. by D. Webb, D. Rust, and B. Schmieder (Astron. Soc. Pacific, San Francisco, also probably typical of all eruptive phenomena at ac- 1998); ASP Conf. Ser. 150, 463 (1998). tivity minimum. The angular distributions of a collec- 4. E. W. Cliver and D. F. Webb, in IAU Coll. No. 167: tion of such phenomena at low heights and distances New Perspectives on Solar Prominences,Ed.by beyond two solar radii will be similar to those shown D. Webb, D. Rust, and B. Schmieder (Astron. Soc. in Figs. 4a and 4b. Pacific, San Francisco, 1998); ASP Conf. Ser. 150, 479 (1998). 5. N. Gopalswamy, Y. Hanaoka, and H. S. Hudson, Adv. 5. CONCLUSIONS Space Res. 25, 1851 (2000). Based on LASCO SOHO observations in 1996– 6. R. H. Munro and A. J. Hundhausen, Sol. Phys. 108, 1998 and SEC NOAA data for the same period, 383 (1987). we have studied the mutual spatial arrangements 7. N. R. Sheeley, Jr., R. A. Howard, M. J. Koomen, and D. J. Michels, Astrophys. J. 272, 349 (1983). of a number of CMEs and nearly contemporaneous 8.D.F.WebbandA.J.Hundhausen,Sol.Phys.108, filament disappearances and prominence eruptions. 383 (1987). All the events were plotted on maps of the magnetic 9. O. C. St. Cyr and D. F. Webb, Sol. Phys. 136, 379 field at the source surface presented in the Solar Geo- (1991). physical Data. In 1996–1997, during solar minimum, 10. H. Wang and P. Goode, in Synoptic Solar Physics, the distributions of the angular distances of CMEs Ed. by K. S. Balasubramanian, J. W. Harvey, and and prominences from the zero line differ drastically: D. M. Rabin (Astron. Soc. Pacific, San Francisco, CMEs appear appreciably closer to the magnetic 1998); ASP Conf. Ser. 140, 497 (1998). equator than the locations where filaments disappear. 11. G. Yang and Y. Wang, in Proc.COSPARColloq.,Ser. If we consider coronal ejections to be continuations 14: Magnetic Activity and Space Environment,Ed. of prominence eruptions, these distributions provide by H. N. Wang and R. L. Xu (Pergamon, Boston, evidence for nonradial and nonrectilinear motions of 2002), p. 113. prominences in the corona. Such behavior of eruptive 12. Z. Mouradian, I. Soru-Escaut, and S. Pojoga, Sol. 158 prominences can be explained in a model with a Phys. , 269 (1995). 13. M. M. Molodenskiı˘ , B. P. Filippov, and N. S. Shylo- magnetic-flux rope in equilibrium with the coronal va, Astron. Zh. 69, 181 (1992) [Sov. Astron. 36,92 magnetic field. The global magnetic field not only (1992)]. determines the sites where the prominences can 14. H. R. Gilbert, T. E. Holzer, J. T. Burkepile, and appear (above large-scale polarity-inversion lines) A. J. Hundhausen, Astrophys. J. 537, 503 (2000). but also controls their motion in the corona during 15. K. Hori and J. L. Culhane, Astron. Astrophys. 382, the eruption. The evolution for a plasma clump 666 (2002).

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16. N. Gopalswamy, M. Shimojo, W. Lu, et al.,Astro- 19. B. P.Filippov, N. Gopalswamy, and A. V.Lozhechkin, phys. J. 586, 562 (2003). Sol. Phys. 203, 119 (2001). 17. B. Filippov and S. Koutchmy, Sol. Phys. 208, 283 20. B. P. Filippov, N. Gopalsvamy, and A. V. Lozhechkin, (2002). Astron. Zh. 79, 462 (2002) [Astron. Rep. 46, 417 18. S. Bravo, G. A. Stewart, and X. Blanco-Cano, Sol. (2002)]. Phys. 179, 223 (1998). Translated by G. Rudnitski ˘ı

ASTRONOMY REPORTS Vol. 48 No. 2 2004 Astronomy Reports, Vol. 48, No. 2, 2004, pp. 153–160. Translated from Astronomicheski˘ı Zhurnal, Vol. 81, No. 2, 2004, pp. 176–183. Original Russian Text Copyright c 2004 by Kitchatinov.

Diﬀerential Rotation and Meridional Circulation near the Boundaries of the Solar Convection Zone L. L. Kitchatinov Institute for Solar–Terrestrial Physics, Siberian Division, Russian Academy of Sciences, P.O. Box 4026, Irkutsk, 664033 Russia Received May 8, 2003; in ﬁnal form, August 8, 2003

Abstract—Although the theory of differential rotation is in satisfactory agreement with helioseismological data for the deep convection zone, there are considerable discrepancies near the solar surface.This disagreement can be eliminated if the anisotropy of turbulent convection is taken into account together with the effects of nonuniformity of the medium, on which the most recent models for differential rotation are based.The model for the di fferential rotation of the convection envelope is supplemented by computations for the transition layer between nonuniform and rigid-body rotation in the upper layers of the solar radiative zone.These are the first computations of differential rotation for the entire volume of the Sun. c 2004 MAIK “Nauka/Interperiodica”.

1.INTRODUCTION The ratio of the contributions of nonuniformity and anisotropy indicated above is favorable for construct- Starting from the pioneering work of Lebedins- ing models of differential rotation.The anisotropy of kiı˘ [1], the nonuniform rotation of the Sun has been the convective turbulence has been studied little, and explained by the interaction between rotation and if it were substantial, this would require an additional convection.As was found in [1], if the convective tur- free parameter in the models.On the contrary, the bulence possesses anisotropy in a specificdirection, nonuniformity is well understood.The strati fication random turbulent motions produce a regular angular- of stellar convection zones is very close to adiabatic. momentum flux due to the Coriolis force, which dis- Therefore, the model [11] based on nonuniformity ac- rupts the uniformity of the rotation.This phenomenon tually does not contain any free parameters.Never- was later called the Λ effect [2]. theless, it is in reasonable agreement with helioseis- The main problem in the theory of differential ro- mological data on the rotation of the convection zone, tation is that the Coriolis number, and the predictions of differential stellar rotation [12, ∗ Ω =2τΩ, (1) 13] are confirmed by observations [14, 15]. On the other hand, a detailed comparison of model determining the intensity of the interaction between computations with available data shows discrepan- convection and rotation is not small for the Sun and cies near the boundaries of the solar convection zone. most similar stars.In (1), τ is the characteristic time Helioseismology predicts an acceleration of the ro- for convective motions, and Ω, the angular velocity. tation with increasing depth near the solar surface Therefore, the corresponding theory must be nonlin- [16, 17], while the theoretical model [11] exhibits the ear in the Coriolis number.Weakly nonlinear e ffects opposite behavior.In addition, the calculated merid- were studied in [3, 4], and full nonlinear expressions ional flow changes its direction near the surface, so for the convective angular-momentum fluxes were that the surface flow is directed toward the equator, found in [5, 6].In addition, it was established that while observations reveal a flow from the equator not only anisotropy but also nonuniformity of the toward the poles in the solar photosphere [18].As we turbulent medium results in the Λ effect [7, 8].The will demonstrate below, these discrepancies can be nonuniformity of the density is most important in removed by taking the anisotropy of the convective this connection [9].Its contribution to the Λ effect motions into account.The reason is that the Coriolis is substantially greater than the contribution from number (1) decreases near the solar surface.For giant anisotropy at large Coriolis numbers, in particular, convection in deep layers of the convection zone, for the value Ω∗ 6 that is typical for the Sun [10] Ω∗ 6, and anisotropy can be neglected (Fig.1). (Fig.1).This seems to be associated with the fact that However, Ω∗ 0.5 in the surface layers of super- rotation can change the anisotropy of the turbulence granules, and the angular-momentum fluxes arising but has virtually no effect on nonuniformity. due to the anisotropy of the convective motions must

1063-7729/04/4802-0153$26.00 c 2004 MAIK “Nauka/Interperiodica” 154 KITCHATINOV

Nonuniformity Anisotropy 0.25 0 I1 0.20 –0.2 0.15 J0 J1 0.10 –0.4 I0 –functions –functions

0.05 Λ

Λ –0.6 0 0.1 1.0 0.1 1.0 Coriolis number Coriolis number

Fig. 1. Comparison of the contributions from nonuniformity of the medium and anisotropy of the turbulence to the angular- momentum fluxes.The e ffect of nonuniformity is dominant for large Coriolis numbers. be taken into account.As will be shown below, this − ρν Ω n U (n ε + n ε ) approach enables us to attain agreement with the T k l i jkl j ikl observed radial nonuniformity of the rotation and the (n · Ω) surface meridional flows. − H (Ω ε +Ω ε ) . Ω2 i jkl j ikl There is a thin transition layer (from differential to rigid-body rotation) beneath the convection envelope Here, N is the effective viscosity tensor; V, the large- [16, 17].This transition seems to be governed by scale velocity; U and H, the dimensionless angular- the magnetic field of the radiative zone [19, 20].In momentum fluxes in the radial direction and along the previous studies, the base of the convection zone was axis of rotation, respectively; and n, the radial unit the lower boundary for calculations of the differential vector.The presence of nondissipative terms in (3) rotation.We will supplement the di fferential-rotation is called the Λ effect, as discussed in the Introduc- model with an analysis of the transition layer and tion.Let us write the nondissipative components of compute the differential rotation in the entire volume the Reynolds stresses that are sources of differential of the Sun. rotation: Λ − − 2 Rrφ = ρνT Ωsinϑ U H cos ϑ , (4) 2.CONVECTIVE ANGULAR-MOMENTUM Λ 2 R = −ρνT ΩH cos ϑ sin ϑ. FLUXES: COMPARING ϑφ THE CONTRIBUTIONS If H>0, the meridional angular-momentum flux is FROM NONUNIFORMITY directed toward the equator, and we expect a decrease AND ANISOTROPY in the angular velocity with latitude.When H<0, − Turbulent motions can redistribute the angular the opposite behavior should be observed.When U 2 momentum between various spatial regions, result- H cos ϑ>0, angular momentum is transported from ing in nonuniformity of the rotation.The angular- the bottom to the surface of the convection zone, resulting in a decrease of the rotational velocity with momentum fluxes in the meridional (−r sinϑRϑφ)and depth. radial (−r sinϑRrφ) directions are proportional to the off-diagonal components of the Reynolds stress ten- The presence of the vector n in (3) indicates that sor: the existence of a special direction is necessary for the Λ effect to appear.This direction can be provided R = −ρu u , ij i j (2) by either the nonuniformity of the medium or the where the angular braces denote averaging over the anisotropy of the turbulence. ensemble of turbulent fluctuations, u is the fluctu- The contribution of nonuniformity was analyzed in ational velocity, and r, ϑ, φ are the usual spherical detail in [8].The following expressions were obtained coordinates.It is important for the theory of di fferen- for the quantities U and H in (4): tial rotation that, along with the contributions from 2 2 turbulent viscosity Rν, the Reynolds stresses for a ∗ ∗ U = J0 (Ω ) ,H= J1 (Ω ) , (5) rotating medium include nondissipative components Hρ Hρ Λ R , which are proportional to the angular velocity Ω where is the correlation length of the turbulent rather than its spatial derivatives: motions and Hρ = −ρ/(dρ/dr) is the density scale ν Λ N ∂Vk height.The correlation length is commonly assumed Rij = Rij + Rij = ρ ijkl (3) ∂rl to be proportional to the pressure scale height: =

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αMLTHp.Since Hρ = γHp in a convection zone with differential-rotation models based on the Λ effect due almost adiabatic nonuniformity (where γ is the adi- to nonuniformity are satisfactory.On the other hand, abatic index), it can easily be shown that /Hρ = there is small-scale convection near the solar surface, where the Coriolis numbers are not so large.Con- αMLT/γ in (5).The dependence on the Coriolis number (1) in (5) is given by the functions sequently, anisotropy must be taken into account to explain the surface features of differential rotation. ∗2 ∗2 ∗ 1 2Ω Ω +9 ∗ J0 (Ω )= 9 − − arctanΩ , 2Ω∗4 1+Ω∗2 Ω∗ (6) 3.THE MODEL ∗2 3.1. Differential Rotation in the Convection Zone ∗ 1 ∗2 4Ω J1 (Ω )= 45 + Ω − 2Ω∗4 1+Ω∗2 We shall consider global flows in the convective ∗4 ∗2 envelope of a star, whose spatial scales are consider- Ω − 12Ω − 45 ∗ + ∗ arctanΩ . ably greater than the correlation length for turbulent Ω motions.These flows will be described by the gas- If the anisotropy of the turbulence is taken into dynamical equations for the mean fields, which can be account, we obtain a contribution to the Λ effect (4) derived by averaging over the ensemble of turbulent that is additive in the nonuniformity.This can be fluctuations.Let us assume that the average charac- written in the form [6] teristics of the star are time independent and axially symmetric with respect to the axis of rotation.Such 2 2 ∗ ∗ a mean axially-symmetric state can be modeled by U = a I0 (Ω ) ,H= a I1 (Ω ) . the joint solution of the three basic equations for the Hρ Hρ (7) angular velocity, meridional circulation, and specific entropy. The ratio /Hρ was introduced here to simplify the The velocity V of the large-scale, axially-symmet- comparison with the contribution from nonunifor- ric flow can be written as a mity.The dimensionless anisotropy parameter is not 1 ∂Ψ −1 ∂Ψ completely free and must satisfy the inequality V = , ,rsin ϑΩ , (10) ρr2 sin ϑ ∂ϑ ρr sin ϑ ∂r γ γ 1 − ≤ a ≤ 2 +1 , (8) α α where Ψ is the flow function.The flow (10) satisfies MLT MLT the Reynolds equation which follows from the condition that the spectral ρ (V ·∇) V = ∇·R −∇P + ρg, (11) functions [21] for the vertical and horizontal components of the fluctuational velocities be positive.In the where R is the Reynolds stress tensor (3) and g is the case αMLT = γ (for which all subsequent computa- free-fall acceleration. ≤ ≤ tions will be done), it follows from (8) that 0 a 4. The model equations are cumbersome, and we The functions of the Coriolis number in (7) have the will not write them in explicit form but will describe form the “recipe” used to derive them fairly completely.In ∗ 3 2 addition, the physical meaning of the model will be I0 (Ω )=− 1+ (9) 4Ω∗4 1+Ω∗2 discussed in detail.This will enable us to clarify the ∗2 reasons for previous discrepancies with observations, Ω − 3 ∗ + arctanΩ , as well as to show the means for their removal. Ω∗ The azimuthal component (11) leads to the equa- ∗2 ∗ 3 2Ω tion of continuity for the angular-momentum fluxes, I1 (Ω )= − 15 + 4Ω∗4 1+Ω∗2 which determines the angular-velocity distribution in the convection envelope: ∗2 3Ω +15 ∗ + arctanΩ . 1 ∂(r3R ) 1 ∂(sin2 ϑR ) Ω∗ rφ + ϑφ (12) r2 ∂r sin2 ϑ ∂ϑ Figure 1 compares the contributions from the non- 2 2 1 ∂(sin ϑΩ) ∂Ψ − 1 ∂(r Ω) ∂Ψ uniformity and anisotropy.At large Coriolis numbers, + 2 2 =0. the contribution of the nonuniformity (namely, the sin ϑ ∂ϑ ∂r r ∂r ∂ϑ corresponding angular-momentum flux H (7) along We can seen that, along with the Reynolds stresses, the axis of rotation) is dominant.Precisely large Cori- meridional circulation can also transport angular mo- olis numbers are typical for the large-scale convec- mentum and, therefore, produce nonuniformity of the tion in the Sun and most cool stars.As a result, rotation.The most convenient form of the equation

ASTRONOMY REPORTS Vol.48 No.2 2004 156 KITCHATINOV for the meridional flow follows from the azimuthal which is Ta ∼ 107 in the Sun.Consequently, the two component of the curl of (11): sources of meridional circulation almost compensate ∂Ω2 g ∂S each other in most of the convection zone.It is ob- D (Ψ) = sin ϑr − , (13) vious that deviations from rotation with constant an- ∂z cpr ∂ϑ gular velocity on cylindrical surfaces [23] are possible only if the thermodynamics of the situation is taken ∂/∂z =cosϑ∂/∂r − r−1 sin ϑ∂/∂ϑ where is the spa- into account in an appropriate way [11, 24].Another tial derivative along the axis of rotation and S = γ important fact is that the mutual compensation of the cvln (P/ρ ) is the specific entropy of a perfect gas two terms on the right-hand side of (13) (the Taylor– (cp and cv are the specific heat capacities at con- Proudman balance) is inconsistent with the stan- stant pressure and density, respectively).The left- dard boundary conditions for zero-surface external hand side of this equation describes the deceleration forces.As a result, boundary layers of thickness ∼ of the meridional circulation due to turbulent viscos- − RT a 1/4 ity: [13] are formed near the boundaries where the meridional circulation is concentrated. ∂ 1 ∂ ∂V D (Ψ) = −ε ρN k . (14) The entropy distribution is determined by the φmi ∂r ρ ∂r ijkl ∂r heat-transfer equation m j l An expression for the effective viscosity tensor N div Fconv + Frad + ρT V · ∇S =0. (15) derived from (3) and (14) taking into account its dependence on the rotational velocity can be found in In the convection zone, we can use the Kramers for- −7/2 24 [22] and will not be presented here. mula for the opacity, κ = cκρT (cκ =2.04 × 10 The right-hand side of (13) contains two sources in cgs units), to determine the radiative heat flux: of meridional flows.The first is the nonpotential part 16σT 3 of the centrifugal force (with the potential component Frad = − ∇T. (16) balanced by the pressure).When the angular velocity 3κρ decreases with distance to the equator, a flow toward Theconvectiveheatflux depends on the latitude be- the pole forms at the surface and a flow toward the cause of the anisotropy of the effective thermal diffu- equator near the bottom of the convection zone.This sivity χ due to rotation: “centrifugal circulation” is very sensitive to the local ∂S direction of the transport of angular momentum.For F conv = −ρT χ , (17) i ij ∂r small Coriolis numbers, i.e., near the surface, the j angular-momentum flux H (5) due to nonuniformity ∗ ∗ ΩiΩj is directed toward the pole (Fig.1).As a result, the χij = χT φ(Ω )δij + φ||(Ω ) . Ω2 meridional circulation in the model of [11] changes its direction near the surface.Our analysis takes into The functions φ and φ|| of the Coriolis number account the contribution of the anisotropy (7).As will be shown below, if the anisotropy parameter a is are designated here using the same notation as in sufficiently large, the flux H remains positive at any [22], where they were written in explicit form.The depth and the meridional flow at the surface is directed quantities νT and χT are the viscosity and thermal toward the poles, in agreement with observations. diffusivity of the so-called initial turbulence, which The vertical fluxes U from (5) and (7) are insignificant. is the turbulence that would occur for a nearly su- Therefore, for simplicity, we shall assume that U =0. peradiabatic distribution without rotation.The corre- The second source of meridional flow is the latitude sponding transfer coefficients can be described by the dependence of the specific entropy, which results in relations for a nonrotating medium so-called baroclinic circulation.If the convective heat τ2g ∂S flux depends on the latitude (for example, increases νT = − ,χT =5νT /4, (18) 15c ∂r with latitude), hotter material at high latitudes will p flow upward toward the surface, while the relatively which close the set of model equations. cool fluid near the equator will flow downward toward To avoid considering a surface layer with a very the bottom, resulting in a meridional stream toward high degree of nonuniformity, the outer boundary of the equator near the surface. the model region was located at re =0.95 R.The Comparing the three terms in (13) in order of standard boundary conditions [11] were applied.The magnitude, we can see that both terms on the right- surface density of the external forces was taken to be hand side are much greater than the left-hand side. zero (Rrφ = Rrϑ =0).We assumed that the merid- For example, the ratio of the centrifugal force to the ional flow did not penetrate into the stably stratified 2 4 viscous force is the Taylor number Ta =4Ω R /νT , radiative zone; i.e., the radial velocity was zero at the

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Frequency of rotation, nHz 460 440 420 400 380 360 340 320 0.5 0.6 0.7 0.8 0.9 Normalized radius

Fig. 2. Computed angular-velocity contours (left panel) and dependences of the angular velocity on depth (right panel) at the equator (solid), latitude 45◦ (dashed), and the poles (dot–dashed) in the absence of anisotropy (a =0).The dotted curve in the left panel shows the base of the convection zone.

Meridional flow, m/s 4

–2

–4

–6

–8 0.75 0.80 0.85 0.90 0.95 Normalized radius

Fig. 3. Stream lines of the meridional flow (left panel) and dependence of the meridional velocity on the normalized radius at latitude 45◦ in the absence of anisotropy (a =0).Negative velocities correspond to flows toward the equator.Solid and dotted stream lines show circulation in the counterclockwise and clockwise directions, respectively. boundaries of the convection envelope.A constant 3.2. Transition Layer 2 heat flux F = L/(4πr ) flows into the convective r i Solving for the differential rotation in the convec- zone through the lower boundary, while the outer tive envelope gives us the boundary conditions at the boundary emits blackbody radiation. surface of the radiative zone required to describe the The presence of the aforementioned boundary lay- transition to rigid-body rotation in deep layers of the ers requires that the numerical methods used have Sun.The physical conditions in this region di ffer con- high resolution.As a result, the commonly used re- siderably from those in the convection zone, and its treatment represents a separate problem.The small laxation method of [11, 25] with an explicit finite- thickness of the transition layer between the zones of difference (in time) numerical scheme is not conve- differential and rigid-body rotation is probably asso- nient.We used the matrix sweep method in radius ciated with the (relict) poloidal field [20, 26]. and expansion in Legendre polynomials in latitude to obtain the stationary solutions.This method is stable The distributions of the angular velocity and due to the presence of viscosity and heat conductivity. toroidal field B satisfy the equations Because of the nonlinearity of the equations under ρν ∂ ∂Ω 1 ∂ ∂Ω sin3 ϑ + r4ρν (19) consideration, an iterative procedure must be used. sin3 ϑ ∂ϑ ∂ϑ r2 ∂r ∂r

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Meridional flow, m/s toward the equator.The amplitude of this acceleration decreases slightly with depth.A thin transition 5 layer between the zones of differential and rigid-body rotation is localized beneath the convection zone.The rotation is almost uniform within most of the radiative 0 zone. However, there are substantial disagreements –5 with the observational data near the outer surface. First of all, we see a change in the direction of the meridional circulation at small depths such that the 0 1 2 3 4 flow at the surface is directed toward the equator. Anisotropy parameter a The observations show a flow toward the poles [18]. In addition, the calculated velocity of the rotation at Fig. 4. Dependenceoftheamplitudeofthemeridional flow at the surface on the anisotropy parameter a (8). low latitudes decreases with depth near the surface, Positive velocities denote flows toward the poles.The whereas helioseismological data show the opposite point corresponding to Figs.5 and 6 is marked. tendency [17]. These disagreements are very likely associated 1 ∂A ∂(sin ϑB) ∂A ∂(rB) with the anisotropy of the turbulence, which was not = r − sin ϑ , included.We can see in Fig.1 that the contribution 2 3 ∂r ∂ϑ ∂ϑ ∂r 4πr sin ϑ of the anisotropy to the development of nonuniform ∂ 1 ∂(sin ϑB) ∂ ∂(rB) rotation is small for large Coriolis numbers.The Cori- η + r η ∂ϑ sin ϑ ∂ϑ ∂r ∂r olis number is substantially greater than unity in ∂Ω ∂A ∂Ω ∂A most of the convection zone, so that anisotropy is not = r − , ∂ϑ ∂r ∂r ∂ϑ important.However, this situation changes near the surface.The Coriolis numbers for supergranules are where ν and η are the microscopic ordinary and mag- Ω∗ 0.5, so that the anisotropy can be important. netic viscosities, respectively, and A is the potential of The problem is that solar convection has not been the poloidal field: studied sufficiently well to be certain of its anisotropy. 1 ∂A −1 ∂A Thus, we must elucidate whether we can choose a value of the anisotropy parameter from the interval (8) B = 2 , ,B(r, ϑ) . (20) r sin ϑ ∂ϑ r sin ϑ ∂r that eliminates the inadequacies of the theoretical We will represent the dipolar poloidal field in the same model. way as in [26]: The dependence of the amplitude of the computed 2 q meridional flow at the surface on the anisotropy pa- r r 2 A = B0 1 − sin ϑ, (21) rameter is shown in Fig.4.The flow changes di- 2 ri rection when a becomes sufficiently large, and the disagreement with the observed circulation disap- where B0 is the amplitude of this field and q is a parameter determining the degre of concentration of the pears.As a increases, the thickness of the surface field toward the stellar center.In our computations, layer with clockwise circulation (Fig.3) decreases, B =0.1 Gandq =5. and this layer disappears completely near a =2.The 0 entire convection zone is encompassed by counter- A joint consideration of the transition layer and clockwise circulation when a>2.In addition, as a differential rotation in the convection envelope en- increases, the deceleration of the rotation with depth ables us to determine the angular-velocity distribu- near the surface is replaced by acceleration, and the tion in the entire volume of the Sun and compare the latitude nonuniformity of the angular velocity de- results obtained with helioseismological data in more creases slightly.The best agreement with the avail- detail. able data is obtained when a 3. The differential rotation and meridional circulation 4.RESULTS AND DISCUSSION computed taking into account the anisotropy of the The characteristics of the global flows when the turbulence are shown in Figs.5 and 6, which were anisotropy of the turbulence is neglected are shown in obtained for a =3.The characteristic radial velocities for the initial turbulence exceed the horizontal Figs.2 and 3.In general, these results are in agree- velocities only slightly for this value of the anisotropy ment with helioseismological data on the rotation of 2 2 2 2 inner layers of the Sun.There is a so-called equatorial parameter: ur / uϑ = ur / uφ =12/11. acceleration in the entire volume of the convection Thus, it is not necessary to revise the theory of zone: the angular velocity increases from the poles global solar circulation to obtain agreement with the

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Frequency of rotation, nHz 460

440

420

400

380

360

340 0.5 0.6 0.7 0.8 0.9 Normalized radius

Fig. 5. Computed angular-velocity contours (left panel) and dependences of the angular velocity on depth for the equator (solid), latitude 45◦ (dashed), and the poles (dot–dashed) for the anisotropy parameter a =3.Thebaseoftheconvectivezone is shown by the dotted curve in the left panel.

Meridional flow, m/s 6

–2

–4

–6 0.75 0.80 0.85 0.90 0.95 Normalized radius

Fig. 6. Stream lines of the meridional flow (left panel) and dependence of the meridional velocity on the normalized radius at latitude 45◦ for the anisotropy parameter a =3.Negative velocities correspond to flows directed toward the equator. observations.It is su fficient to take into account the ACKNOWLEDGMENTS anisotropy of the convective turbulence together with the effects of nonuniformity. This work was supported by the Russian Foun- dation for Basic Research (project nos.02-02-16044 Note that including the effects of anisotropy leaves and 02-02-39027) and INTAS (project no.2001- theoretical predictions for the differential rotation 0550). of stars virtually unchanged [13].In the same way as in the Sun, anisotropy can substantially affect REFERENCES meridional flows near the surface and fine features in the distribution of the angular velocity of giants and 1. A.I.Lebedinski ı˘ ,Astron.Zh.18, 10 (1941). slowly rotating solar-type stars, but it is impossible 2.G.R udiger,¨ Differential Rotation and Stellar Con- vection (Akademie, Berlin, 1989). to accurately estimate such peculiarities from obser- 3. B.R.Durney and H.C.Spruit, Astrophys.J. 234, vations of such stars.In principle, Doppler –Zeeman 1067 (1979). imaging [27] provides high resolution measurements 4.G.R udiger,¨ Geophys.Astrophys.Fluid Dyn. 25, 213 for rapidly rotating stars; however, in this case, the (1983). Coriolis number (1) is large at all depths, so that 5. L.L.Kitchatinov, Geophys.Astrophys.Fluid Dyn. 35, anisotropy is negligible. 93 (1986).

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6. L.L.Kitchatinov, Astron.Zh. 64, 135 (1987) [Sov. 17. J.Schou, H.M.Antia, S.Basu, et al., Astrophys.J. Astron. 31, 68 (1987)]. 505, 390 (1998). 7. L.L.Kitchatinov, Geophys.Astrophys.Fluid Dyn. 38, 18. R.W.Komm, R.F.Howard, and J.W.Harvey, Sol. 273 (1987). Phys. 147, 207 (1993). 8. L.L.Kitchatinov and G.R udiger,¨ Astron.Astrophys. 19. L.L.Kichatinov and G.R udiger,¨ Pis’ma Astron.Zh. 276, 96 (1993). 22, 312 (1996) [Astron.Lett. 22, 279 (1996)]. 9. L.L.Kichatinov, Pis’ma Astron.Zh. 12, 410 (1986) 20. K.B.MacGregor and P.Charbonneau, Astrophys.J. [Sov.Astron.Lett. 12, 172 (1986)]. 519, 911 (1999). 10. B.R.Durney and J.Latour, Geophys.Astrophys. 21. A.Monin and A.M.Yaglom, Statistical Hydrome- Fluid Dyn. 9, 241 (1978). chanics (Nauka, Moscow, 1967), Part 2 [in Russian]. 11. L.L.Kitchatinov and G.R udiger,¨ Astron.Astrophys. 22. L.L.Kitchatinov, V.V.Pipin, and G.R udiger,¨ Astron. 299, 446 (1995). Nachr. 315, 157 (1994). 12. L.L.Kichatinov and G.R udiger,¨ Pis’ma Astron.Zh. 23. P.A.Gilman and J.Miller, Astrophys.J.,Suppl.Ser. 23, 838 (1997) [Astron.Lett. 23, 731 (1997)]. 61, 585 (1986). 13. L.L.Kitchatinov and G.R udiger,¨ Astron.Astrophys. 24. B.R.Durney, Astrophys.J. 338, 509 (1989). 344, 911 (1999). ¨ ¨ 14.J.R.Barnes, A.C.Cameron, D.J.James, and 25.G.R udiger, M.K uker, and K.L.Chan, Astron.As- J.-F.Donati, Mon.Not.R.Astron.Soc. 314, 162 trophys. 399, 743 (2003). (2000). 26.G.R udiger¨ and L.L.Kitchatinov, Astron.Nachr. 318, 15. J.-F.Donati, M.Mengel, B.D.Carter, et al.,Mon. 273 (1997). Not.R.Astron.Soc. 316, 699 (2000). 27. J.-F.Donati and A.C.Cameron, Mon.Not.R.As- 16. P.R.Wilson, D.Burtonclay, and Y.Li, Astrophys.J. tron.Soc. 291, 1 (1997). 489, 395 (1997). Translated by Yu. Dumin

ASTRONOMY REPORTS Vol.48 No.2 2004 Astronomy Reports, Vol. 48, No. 2, 2004, pp. 161–169. Translated from Astronomicheski˘ı Zhurnal, Vol. 81, No. 2, 2004, pp. 184–192. Original Russian Text Copyright c 2004 by Markov, Sinitsyn.

Spectral–Correlational and Kinetic Models for the Motion of the Earth Yu. G. Markov and I. N. Sinitsyn Moscow State Aviation Institute, Moscow, Russia Institute of Informatics Problems, Moscow, Russia Received May 12, 2003; in ﬁnal form, August 8, 2003

Abstract—We consider linear and nonlinear spectral–correlational and kinetic models for ﬂuctuations in the motion of the Earth based on the Fokker–Planck–Kolmogorov and Pugachev equations. Particular attention is paid to Gaussian linear and nonlinear (statistically linearized) models. The results of analytical statistical modeling are presented. c 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION 2. KINETIC EQUATIONS FOR THE NONLINEAR THEORY Spectral methods for analyzing measurements of OF FLUCTUATIONS OF THEMOTION OF THEEARTH the motion of the Earth have been widely used in modern astronomy (see, for example, [1–5]). Lin- Let us first introduce the following notation and ear and nonlinear, analytical, dynamical, stochastic assumptions. models for the motion of the deformable Earth based (1) We denote the state vector by Y =[Y Y Y ]T , on celestial mechanics have been developed in [6– 1 2 3 where Y = p , Y = q , Y = δr , δr = r − r∗ (r∗ is 8]. The general dynamical theory of one- and multi- 1 t 2 t 3 t t t the axial-rotation velocity of the Earth), and pt, qt, dimensional distributions based on the equations of and rt are projections of the instantaneous angular- Fokker–Planck–Kolmogorov (for Gaussian pertur- rotational velocity of the Earth on the associated bations) and Pugachev (for non-Gaussian perturba- axes [5, 11, 12]. tions) is set out in [9, 10]. −1 (2) We will assume that, over a day, T∗ =2πr∗ , The current paper continues the work of [7, 9, 10] the axial and centrifugal moments of the inertia tensor J { } and is dedicated to the construction of (i) spectral– = Jij (i, j = p, q, r) of the deformable Earth correlational and (ii) one- and multi-dimensional A = Jpp, B = Jqq, C = Jrr, Jpq = Jqp, Jqr = Jrq, kinetic models for the motion of the deformable and Jrp = Jpr canbewrittenintheform Earth based both directly on the Fokker–Planck– ∗ Jij = Jij + Jij,1 sin r∗t + Jij,1 cos r∗t Kolmogorov and Pugachev equations and on parametrized versions of these equations. The paper + Jij,2 sin 2r∗t + Jij,2 cos 2r∗t, has seven sections. The Introduction presents a brief with the harmonics 3, 4, ...being negligible. Here, a survey of works in this area and the formulation star denotes constant components of the moments of of the two problems considered. In Section 2, we inertia and single or double primes denote the ampli- derive the kinetic equations for the nonlinear theory tudes of harmonics. of fluctuations of the motion of the Earth. Section 3 is concerned with parametrized, one- and multi- (3) We will call the following dimensionless differences of the axial moments of inertia averaged over a dimensional versions of the kinetic equations, con- −1 centrating on a normal (Gaussian) approximation day, T∗ =2πr∗ ,effective daily humps: ∗−1 for these equations. Section 4 contains linear and u1 = (C − B)A cos ϕ, statistically linearized, spectral–correlational models − ∗−1 for fluctuations of the Earth’s motion. In Section 5, u2 = (C A)B sin ϕ , ∗−1 we present linear and statistically linearized kinetic u3 = (B − A)C sin 2ϕ, models for the fluctuations. Section 6 is concerned ∼ with stochastic and analytical modeling of fluctua- where ... denotes averaging (ϕ = r∗t),withu1 tions of the Earth’s motion and a discussion of the u2; u3 u1,2. results obtained. We present our main conclusions in (4) We will call the following dimensionless mag- the final section. nitudes of the centrifugal moments of inertia averaged

1063-7729/04/4802-0161$26.00 c 2004 MAIK “Nauka/Interperiodica” 162 MARKOV, SINITSYN

−1 3 over a day, T∗ =2πr∗ ,effective daily bulges: 2 R1 = R1(t, pt,qt,δrt, u)= u3ω∗(1 (6) ∗−1 ∗−1 2 u4 = JqrA ,u5 = JqrC sin ϕ, − 2 2 − 3 2 2 ∗−1 ∗−1 b1 cos ω∗t) u5bω∗ cos ω∗t +3u9bω∗ cos ω∗t u6 = JqrA cos 2ϕ,u7 = JqrB sin 2ϕ, 2 ∗−1 ∗−1 2 − 2 2 − u8 = JprB ,u9 = JprC cos ϕ, +3u15ω∗(1 b1 cos ω∗t) D3δrt. ∗−1 ∗−1 ∗ ∗ ∗−1 u10 = JprB cos 2ϕ,u11 = JprA sin 2ϕ, In (1)–(6), N∗ =(C − B )A ω∗ is the Chan- ∗−1 ∗−1 dler frequency; the parameters u and u are u12 = JpqC ,u13 = JpqA sin ϕ, 1...3 4...15 determined by assumptions 3 and 4 above; ω∗ (r∗ = u = J B∗−1 cos ϕ,u= J C∗−1 cos 2ϕ, 14 pq 15 pq 365ω∗) corresponds to the yearly period; b and b1 ≤ ≤ −1 ≈ with u ∼ u , u u ,andu u . (0, 4 b (4/3)π ) with b1 b are known param- 4...7 3 8...11 4...7 12...15 8...11 P Q R (5) We will include only the moments of the grav- eters [12, 13]; and the functions 1, 1,and 1 are itational force of the Sun relative to the associated written with accuracy to squared terms and deriva- T axes [5, 11, 12]. The amplitude of the gravitational tives with respect to u =[u1,...,u15] , pt, qt, δrt, moment of the Moon exceeds that of the Sun by a and the daily averaged rates of the variations in the factor of two to three; however, due to the substantial axial and centrifugal moments of inertia. difference between the eigen and forcing frequencies, Equations (1)–(3) can be compactly written in the the Moon’s influence leads to monthly oscillations vector form with amplitudes that are a factor of 15 to 20 lower Y˙ = a(Y,t)+V,Y(t )=Y , (7) than the amplitude of yearly oscillations. Essentially 0 0 no oscillations of the pole with a monthly period are where T T detected [13]. Y =[ptqtδrt] ,Y0 =[p0q0δr0] , (6) We will take into consideration the external, V =[V V V ]T , linear, fluctuational–dissipative moments of forces 1t 2t 3t M fd = V − D p M fd = V − D q M fd = 1 1t 1 t, 2 2t 2 t,and 3 a = a(Y,t)=[a a a ]T , (8) V − D δr ,whereV , V ,andV are the specific 1 2 3 3t 3 t 1t 2t 3t − − P P external fluctuational moments of the forces, which a1 = D1pt N∗qt + 10 + 11 cos ω∗t 2 2 are uncorrelated Gaussian (normal) white noise with + P12 cos 2ω∗t − u4[(r∗ + δrt) − r∗], the intensities νi = νi(t)(i =1, 2, 3),andD1,2,3 a = −D q + N∗p + Q + Q cos ω∗t are the specificcoefficients of the moments of the 2 2 t t 10 11 2 2 dissipative forces. + Q12 cos 2ω∗t + u8[(r∗ + δrt) − r∗], Adopting assumptions (1)–(6), the equations of a3 = −D3δrt + R10 + R11 cos ω∗t + R12 cos 2ω∗t; motion of the Earth acquire the form [5, 9, 10] 3 3 p˙ + N∗q = P + V ,p= p , (1) P − 2 − 2 t t 1 1t t0 0 10 = u4ω∗ 1 b1 (9) − Q 2 2 q˙t N∗pt = 1 + V2t,qt0 = q0, (2) 3 b 3 b2 R − ∗ 2 1 2 1 2 δr˙t = 1 + V3t,δrt0 = r0 r , (3) − u ω∗ 1 − − u ω∗ 1 − − u r∗, 2 6 2 2 11 2 4 where 2 2 3 2 b1 3 2 3b1 P = P (t, p ,q ,δr , u)=3u bω cos ω∗t (4) Q10 = u7ω∗ 1 − + u8ω∗ 1 − 1 1 t t t 1 ∗ 2 2 2 2 3 3 − 2 − 2 2 − 2 2 u4ω∗(1 3b1 cos ω∗t) u6ω∗(1 3 2 b1 2 2 2 − u ω∗ 1 − + u r∗, 2 10 2 8 − 2 2 − 3 2 − 2 2 b1 cos ω∗t) u11ω∗(1 b1 cos ω∗t) 2 2 2 3 2 b1 2 b1 R10 = u3ω∗ 1 − +3u15ω∗ 1 − , 3 2 2 2 2 2 + u ω∗b cos ω∗t − u (r∗ + δr ) − D p , 2 13 4 t 1 t 2 3 2 2 P11 =3u1bω∗ + u13bω∗, (10) Q1 = Q1(t, pt,qt,δrt, u)=−3u2bω∗ cos ω∗t (5) 2 Q − 2 − 2 3 2 − 2 2 3 2 11 = 3u2bω∗ 3u14bω∗, + u7ω∗(1 b1 cos ω∗t)+ u8ω∗(1 2 2 3 2 2 R11 = − u5bω∗ +3u9bω∗, 2 2 3 2 2 2 2 − 3b cos ω∗t) − u ω∗(1 − b cos ω∗t) 1 2 10 1 2 2 9 2 2 3 2 2 3 2 2 − 3u bω∗ cos ω∗t + u (r∗ + δr ) − D q , P = u b ω∗ + u b ω∗ + u b ω∗, (11) 14 8 t 2 t 12 4 4 1 4 6 1 4 11 1

ASTRONOMY REPORTS Vol. 48 No. 2 2004 MODELS FOR THEMOTION OF THEEARTH 163 3 9 3 Q − 2 2 − 2 2 2 2 ∂gn T ∂ − 1 T 12 = u7b1ω∗ u8b1ω∗ + u10b1ω∗, = iλn a ,tn λn νλn gn, (16) 4 4 4 ∂tn ∂iλn 2 3 2 2 3 2 2 R = − u b ω∗ − u b ω∗. gn(λ1,...,λn; t1,...,tn−1,tn−1)=gn−1 12 4 3 1 2 15 1 × (λ1,...,λn−2,λn−1 + λn; t1,...,tn−1), Equations (7) are linear in pt and qt and quadratic T T in δrt. Sources of perturbations include, first, additive, where λ0 =[λp0 λq0 λδr0 ] and λn =[λpn λqn λδrn ] . regular components with the coefficients P10, Q10, The notation ((∂/∂iλ),t) indicates that the variable R10, P11, Q11, R11, P12, Q12,andR12 and, second, y in the polynomial is to be replaced by the differenti- the additive irregular (fluctuational) components V1t, ation operator ∂/∂iλ. V2t,andV3t. The parameters u appear linearly and, in contrast to [6, 18], are taken to be known and not The kinetic equations (12) and (13) lie at the basis random. of the nonlinear theory for fluctuations of the Earth’s motion. This is a system of related partial-differential Let us turn to the derivation of kinetic equa- equations with their corresponding initial conditions. tions for the nonlinear theory of the fluctuations Currently, direct methods for the numerical analysis applied to (7). We denote f = f (y; t) and f = 1 1 n of these equations can be realized only on supercom- f (y ,...,y ; t ,...,t ) n n 1 n 1 n to be the one- and -dim- puters. ensional densities, (n =2, 3,...),andg1 = g1(λ; t) and gn = gn(λ1,...,λn; t1,...,tn) to be the corresponding characteristic functions. As applied to 3. PARAMETRIZATION OF THE KINETIC the nonlinear, stochastic differential equation (7) EQUATIONS FOR FLUCTUATIONS for Gaussian density perturbations, f1 and fn are OF THEEARTH’S MOTION determined by partial-differential equations called the Fokker–Planck–Kolmogorov equations. These Via parametrization of the one- and n-dimen- equations together with their initial conditions have sional distributions using the consistent orthogonal the form [14, 15] expansions of [14, 15], we can represent the one- dimensional (n =1)and multi-dimensional (n>1) ∂f ∂T 1 ∂ ∂T 1 = − (af )+ tr (νf ) , (12) models in the form ∂t ∂y 1 2 ∂y ∂y 1   n∗ f1(y; t0)=f0(y);   f1(y; t) ≈ w1y (y) 1+ cµtpµ(y) , (17) l=3 | | ∂f ∂ 1 ∂ ∂T µ =l = − (af)+ tr (νf) , (13) ∂t ∂y 2 ∂y ∂y fn(y1,...,yn; t1,...,tn) ≈ wny (y1,...,yn) (18) | − f(y; τ η; τ)=δ(y η); n∗ × 1+ cµ1,...,µn fn(y1,...,yn; t1,...,tn)=f1(y1; t1) (14) l=3 | | ··· | | µ1 + + µn =l × f(y2; t2|y1; t1) ×···×f(yn; tn|yn−1; tn−1),

× pµ ,...,µ (y1,...,yn) . fn(y1,...,yn−1,yn; t1,...,tn−1,tn−1) 1 n

= fn−1(y1,...,yn−1; t1,...,tn−1) Here, wnst = wny (y1,...,yn) is the known standard × δ(y − y − ), n n 1 consistent density series for (n ≥ 1) having the same T T y0 =[p0q0δr0] ,yn =[pt qt δrt ] first- and second-order moments as fn, as a conse- n n n (n =2, 3,...). quence of which the wny depend on the following parameters: the mathematical expectation mt,covaria- | Here, f = f(y; t η; τ) is called the transition density. tion matrix Kt, and the covariation function K(t ,t ) Due to the polynomial character of the nonlinear of the vector Y for t ,t = t1,...,tn. {pµ(y),qµ(y)} functions a = a(Y,t), the Pugachev equations for the are the known system of orthogonal polynomials, g g { } characteristic functions 1 and n can be written as and pµ1,...,µn (y1,...,yn), qµ1,...,µn (y1,...,yn) are a system of partial-differential equations in operator the known consistent series of orthogonal polynomi- form [14, 15]: als. We have for the parameters mt, Kt, cµt, K(t ,t ), ∂g ∂ 1 and cµ µ ,...... ,cµ ,...,µ in (17) and (18) a system 1 = iλT a ,t − λT νλ g , (15) 1 2 1 n ∂t ∂iλ 2 1 of ordinary differential equations, which, in general, are nonlinear in m , K ,andthecoefficients of the 2 − t t g1(λ; t0)=g0(λ)(i = 1); consistent orthogonal expansions.

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δr 0 In particular, in a Gaussian approximation (the +2(r∗ + mt )δrt , normal approximation of [14, 15]), the two-dimensional, nonlinear, kinetic model is described by the Eqs. (7) take the following form for the mathematical p,q,δr equations expectations mt : 3| | −1/2 p − p − q − δr 2 f1 = f1(y,mt,Kt; t)=[(2π) Kt ] (19) m˙ t = D1mt N∗mt u4[(mt ) (24) × exp{−(y − m )T K−1(y − m )}, δr δr P P t t t + Dt +2r∗mt ]+ 10 + 11 cos ω∗t p p T 1 T + P12 cos 2ω∗t, m = m , g1(λ; t)=exp iλ mt − λ Ktλ ; t0 0 2 q − q p δr 2 δr m˙ t = D2mt + N∗mt + u8[(mt ) + Dt m˙ t = MN a(Y,t),mt = m0, (20) δr Q Q Q 0 +2r∗mt ]+ 10 + 11 cos ω∗t + 12 cos 2ω∗t, ˙ T − T q q Kt = MN [a(Y,t)(Y mt ) (21) mt0 = m0, T +(Y − mt)a(Y,t) ]+ν, Kt = K0; δr − δr R R 0 m˙ t = D3mt + 10 + 11 cos ω∗t δr δr f = f (22) R ∗ 2 2 + 12 cos 2ω t, mt0 = m0 ; × (y1,y2,mt1 ,mt2 ,Kt1 ,Kt2 ,K(t1,t2); t1,t2) 0 0 0 and for the centered components pt , qt ,andδrt , 2| ¯ | −1/2 =[(2π) K2 ] 0 0 0 p˙ = −D1p − N∗q (25) 1 t t t × − T − T ¯ −1 − δr 0 0 0 exp (¯y2 m¯ 2 )K2 (¯y2 m¯ 2) , − 2u (r∗ + m )δr + V ,p= p 2 4 t t 1t t0 0 0 0 0 q˙ = −D q + N∗p 1 t 2 t t ¯ ¯T − ¯T ¯ ¯ δr 0 0 0 gt1,t2 (λ)=exp iλ m¯ 2 λ K2λ ; +2u (r∗ + m )δr + V ,q= q 2 8 t t 2t t0 0 δr˙0 = −D δr0 + V ,δr0 = δr0. T T T T T T t 3 t 3t t0 0 y¯2 =[yt1 yt2 ] , m¯ 2 =[mt1 mt2 ] , ¯ T T T p,q,δr λ =[λ1 λ2 ] , Equations (24) are nonlinear in mt , while   0 0 0 Eqs. (25) are linear in pt , qt ,andδrt . Equations (24) K(t1,t1) K(t1,t2) and (25) form a joint system of stochastic diﬀer- K¯2 = , ential equations determining the mathematical ex- K(t2,t1) K(t2,t2) p,q,δr pectation of the processes mt and the centered ∂K(t ,t ) 0 0 0 1 2 − T ﬂuctuational components pt , qt ,andδrt . Using the = MN [(Yt1 mt1 )a(Yt2 ,t2) ], (23) ∂t2 numerical methods of statistical modeling [15], we K(t1,t1)=K(t1), obtain estimates of the mathematical expectations, the spectral–correlational characteristics, and the where a subscript N by the sign of the mathematical 0 0 one- and multi-dimensional distributions pt , qt ,and expectation indicates that it is calculated for the 0 equivalent normal (Gaussian) distribution (19) or δrt (see Section 6). (22) with the unknown parameters mt, Kt,and Due to the linearity of Eqs. (25), we have the K(t1,t2). Knowing the one- and two-dimensional following equations for the covariation matrix Kt and normal (Gaussian) distributions, the old distributions the covariation-function matrix Kt1t2 [14, 15]: f and g (n>2) are calculated in the normal n n K˙ = a K + K aT + ν, K = K , approximation. t 1 t t 1 t0 0 (26)

We will construct spectral–correlational models ∂Kt1t2 T = Kt1t2 a1(t2) ,Kt1t2 = Kt1 (27) for the motion of the Earth based on equations (19)– ∂t2 (23). at t1

ASTRONOMY REPORTS Vol. 48 No. 2 2004 MODELS FOR THEMOTION OF THEEARTH 165   ∂Kδrq ν (t)0 0 t1t2 δrq δrp  1  = −D K + N∗K   ∂t 2 t1t2 t1t2 ν = ν(t)=  . 2  0 ν2(t)0 δr δrδr ∗ +2u8(r + mt2 )Kt1t2 ; 00ν3(t) for the initial conditions Using (26) and (27), we arrive at the following ex- Kii = Di ,Kij = Kij(i, j = p, q, δr). p,q,δr t1t2 t1 t1t2 t1 pressions for the dispersion Dt and the covaria- pq,qδr,pδr tions Kt : For covariationally stationary processes involving 0 0 0 ˙ p − p pq pt , qt ,andδrt (i.e., processes for which the covari- Dt = 2(D1Dt + N∗Kt ) (28) 0 0 0 ation functions pt , qt , δrt depend only on the time δr pr +2u4(r∗ + mt )Kt + ν1, diﬀerence t2 − t1 = τ), we can use the following ex- q q pq pressions for the covariation functions and the spec- D˙ =2(−D D + N∗K ) t 2 t t tral and mutual-spectral densities: δr qr +2u8(r∗ + mt )Kt + ν2, ∞ D˙ δr = −2D Dδr + ν , p,q,δr p,q,δr p,q,δr iωτ t 3 t 3 k (τ)=Kt,t+τ = s (ω)e dω; (31) −∞ ˙ pq − pq p − q Kt = (D1 + D2)Kt + N∗(Dt Dt ) (29) ∞ δr qδr δr pδr p,q,δr 1 p,q,δr −iωτ − 2u4(r∗ + m )K +2u8(r∗ + m )K , s (ω)= k (τ)e dτ. (32) t t t t 2π ˙ pδr − pδr − qδr −∞ Kt = (D1 + D3)Kt N∗Kt − δr δr 2u4(r∗ + mt )Dt , In the linear theory of the ﬂuctuations [8], the 2 2 ˙ qδr pδr − qδr terms u4,8[(r∗ + δrt) − r∗] are neglected in (7). Kt = N∗Kt (D2 + D3)Kt δr δr Equation (7) then breaks up into two independent lin- +2u8(r∗ + mt )Dt , ear systems of stochastic equations for the motion of ¯ T and with t

ASTRONOMY REPORTS Vol. 48 No. 2 2004 166 MARKOV, SINITSYN − − min(t1,t2) wqq(t − τ)=e 1/2(D1+D2)(t τ) δr − δr − + w˜ (t1 τ)˜w (t2 τ)ν3dτ; D2 − D1 × − sin N∗(t − τ)+cosN∗(t − τ) , t0 2N∗   wpq(t − τ)=wqp(t − τ) (45) pp pq s (ω) s (ω) ∗ −1/2(D +D )(t−τ) s¯(ω)=  =Φ(iω)¯νΦ(iω) , (37) = −e 1 2 sin N∗(t − τ); sqp(ω) sqq(ω) where the elements w(t − τ) have been written with δr δr 2 accuracy to within squared terms and derivatives of s (ω)=|Φ (iω)| ν3. (38) −1 the D1,2N∗ , 1(t − τ) is a unit step function, and Here, a star denotes complex conjugation, and − −     wδr(t − τ)=e D3(t τ), (46) p p δr −D3(t−τ) mt  m0 w˜ (t − τ)=e 1(t − τ). m¯ t = , m¯ 0 = , (39) q q mt m −1    0  With D1,2N∗ 1 and constant ν1, ν2,andν3, p pq formulas (31), (32), (37), and (38) take the form ¯  D0 K0  ν1 0  K0 = , ν¯ = , 2D ν − (D − D )ν pq q p 1 2 2 1 1 K D 0 ν2 D = , (47) 0 0 2D1(D1 + D2)   2D2ν1 +(D2 − D1)ν2 P P P Dq = , (48)  10 + 11 cos ω∗t + 12 cos 2ω∗t  2D (D + D ) α0 = , (40) 1 1 2 Q + Q cos ω∗t + Q cos 2ω∗t ν1D2 − ν2D1 10 11 12 Kpq = , (49) R R R N∗(D1 + D2) α0(t)= 10 + 11 cos ω∗t + 12 cos 2ω∗t. (41) ν In addition, the quantities Φ(iω), w˜(t − τ), w(t − Dδr = 3 , (50) τ) and Φδr(iω), w˜δr(t − τ), wδr(t − τ) in (33)–(39) 2D3 denote the transfer functions, weighting functions, pp −2 2 2 2 and fundamental solutions corresponding to the lin- s (ω)=∆1 (ω)[ν1(D2 + ω )+ν2N∗ ]; (51) qq −2 2 2 2 ear equations (7). According to [14, 15], these are s (ω)=∆1 (ω)[ν1N∗ + ν2(D1 + ω )]; (52) determined by the formulas pq −2 s (ω)=∆ (ω)N∗[−ν1(D2 + iω) (53) − − −1 1 Φ(iω)= (α iωI2) (42) qp   + ν2(D1 − iω)] = s (ω); 1 −(D2 + iω) −N∗ = −   , δr −2 2 2 2 ∆(ω) s (ω)=∆2 (ω)ν3, ∆2(ω)=D3 + ω , (54) N∗ −(D1 + iω)     where − − ∗ 2 2 − 2 2 2 2  D1 N  10 ∆1(ω)=(N∗ + D1D2 ω ) + ω (D1 + D2) α = ,I2 = ; 2 2 2 2 2 N∗ D2 01 ≈ (N∗ − ω ) + ω (D1 + D2) .

− Equations (24) and (28)–(32) lie at the basis of the Φδr(iω)=(D + iω) 1; (43) 3 algorithm for analytical modeling and analysis of the spectral–correlational characteristics of fluctuations ∆(ω)=ω2 − ω2 +2εiω, c of the Earth’s motion (see Section 6). 2 2 ≈ 2 ωc = N∗ + D1D2 N∗ , 2ε = D1 + D2;   5. LINEAR AND LINEARIZED KINETIC wpp(t − τ) wpq(t − τ) MODELS FOR THE FLUCTUATIONS w(t − τ)=  , (44) wpq(t − τ) wqq(t − τ) We now turn to more subtle probability characteristics: the one- and multi-dimensional distributions. w˜(t − τ)=w(t − τ)1(t − τ), Such characteristics are important for estimating the − − significance of outliers and level intersections. wpp(t − τ)=e 1/2(D1+D2)(t τ) Let us first consider the kinetic equations (12) in D2 − D1 application to (7) in our approximate (linear) the- × sin N∗(t − τ)+cosN∗(t − τ) , 2N∗ ory, neglecting terms containing u4,8 in (7). Then,

ASTRONOMY REPORTS Vol. 48 No. 2 2004 MODELS FOR THEMOTION OF THEEARTH 167 ¯T T Eq. (12) for the one-dimensional characteristic func- + iλ2 [pt2 qt2 ] , T tions of the variables [ptqt] and δrt T T λ¯1 =[λ11λ12] , λ¯2 =[λ21λ22] , g1(λ1,λ2; t)=E exp{iλ1pt + iλ2qt}, (55) δr { } δr g1 (λ ; t)=E exp iλ δrt (56) { } gt1t2 (λ1,λ2)=E exp iλ1δrt1 + iλ2δrt2 (64) takes the form satisfy equations of the form (57) and (58). According ∂g1 T ∂g1 T − 1 T to [14, 15], we have the following implicit formulas for = λ α + iλ α0(t) λ ν¯(t)λ g1, Gaussian initial distributions and t

(58) × w(tl − t0)¯m0 + w(tl − τ)α0dτ T where λ =[λ1λ2] ; α0(t), α0(t),andα are defined in t0 (40) and (41); and b = I2. 2 − 1 ¯T − ¯ − T We assume that the initial distributions in p0, q0, λl w(tl t0)K0w(th t0) δr 2 and 0 are Gaussian: l,h=1 T 1 T min(tl,th) g0(λ)=exp iλ m¯ 0 − λ K¯0λ , (59) 2 − − T + w(tl τ)¯ν(τ)w(th τ) dτ λh ; 1 2 gδr(λ )=exp iλ mδr − λ Dδr . (60) t0 0 0 2 0 2 Then, using (57) and (58), in accordance with [14, gδr (λ ,λ )=exp i λ T (66) 15], we have the following implicit formulas for the t1t2 1 2 l l=1 Gaussian one-dimensional characteristic functions: tl − − − − T × δr D3(tl t0) D3(tl τ) g1(λ; t)=exp iλ w(t − t0)¯m0 (61) m0 e + e α0(τ)dτ t 0 t 2 1 1 − − + w(t − τ)α (τ)dτ − λT − λ Dre D3(tl+th 2t0) 0 2 2 l 0 t l,h=1 0 min(tl,th) T − − × w(t − t0)K¯0w(t − t0) D3(tl+th 2τ) + ν3(τ)e dτ λh. t t 0 + w(t − τ)¯ν(τ)w(t − τ)T dτ λ ; Expressions for the multi-dimensional characteristic functions for n>2 can be written analogously. t0 The partial-differential equations (57) and (58) lie at the basis of the differential, linear, one-dimensional, δr δr −D (t−t ) kinetic models with Gaussian white noise. Equa- g (λ ; t)=exp iλ m e 3 0 (62) 1 0 tions (60), (61), (65), and (66) underlie the finite, linear, one- and two-dimensional Gaussian kinetic t models for the case of Gaussian white noise and −D (t−τ) 1 2 δr −2D (t−t ) + e 3 α (τ)dτ − λ D e 3 0 Gaussian initial distributions of pt , qt ,andδrt . 0 2 0 0 0 0 t0 We now turn to the nonlinear theory. Linearized t kinetic models for nonlinear fluctuations can be obtained by statistically linearizing the nonlinear func- −2D3(t−τ) + e ν3(τ)dτ . tions in (7) and applying the methods of the theory

t0 of linear stochastic systems [14, 15]. This leads to expressions (24) and (28)–(30) for the mathemati- The two-dimensional characteristic functions cal expectations and correlational characteristics ap- ¯ ¯ ¯T T gt1t2 (λ1, λ2)=E exp iλ1 [pt1 qt1 ] (63) pearing in (19) and (22).

ASTRONOMY REPORTS Vol. 48 No. 2 2004 168 MARKOV, SINITSYN

δ p,q,δr s r(ω) densities s (ω) and dispersions and covariations p,q,δr δr ν D∞ obtained for the linear theory [formulas (33)– δ D∞ = 3/2D3 |Φ r(0)| |Φδr ω | (53)] with white-noise intensities ν = ν = ν , ν > 1 2 (i *) 1 2 0 3 δr s1 ν0 and specificcoefficients for the moments of the |Φδr | 3 (iN*) dissipative forces D1 = D2 = D0, D3 >D0.Theco- variation functions have the form |Φδr ω | 4 (2i *) p,q p,q −D |τ| δr 0 ∗ s2 k (τ)=D∞ e cos N τ, (67) δr s3 δr δr δr −D3|τ| s4 k (τ)=D∞e . (68) According to [4, 5], various estimates of the relaxation ω ω ω − 0 * N* 2 * 1 time D0 yield values between 10 and 100 years. The (365 days) (433 days) p,q δr ≈ ν 2 δr ≈ ν ω2 δr ≈ ν 2 δr ≈ ν 4ω2 level of the mean square deviations D∞ will be s1 3/D3 , s2 3/ * , s3 3/N* , s4 3/ * (2–4) × 10−2 . Fig. 1. Taking into account nonlinear terms with the co- −8 efficients u4,8 ∼ 10 at the frequency N∗ via the ana-

% |Φp, q | lytical modeling yields corrections of 2 (in the math- 3 (iN*) p, q ν D∞ = 0/2D0 ematical expectation), 6% (in the second moments), p, q ω p, q s ( ) p, q K∞ = 0 and 10% (in the spectral density). These estimates s3 were obtained via statistical modeling. In addition to analyses of the accuracy of the theory, (33)–(53) can be applied to measurements |Φp, q ω | |Φp, q ω | to identify both the basis parameters of the input 2 (i *) 4 (2i *) model (1)–(3) (N∗, D ) and the parameters u, |Φp, q(0)| 1,2,3 1 which are important for estimating trends. p, q s2 p, q The results of our statistical analyses indicate s4 p, q D0/2 that spectral–correlational models based on the s3 linear theory can provide suﬃcient accuracy on time ω ω ω 0 * N* 2 * intervals of three to ﬁve years. The kinetic models in (365 days) (433 days) the Gaussian approximation are fully based on the ν ()ω2 2 ν ()ω2 2 spectral–correlational characteristics. pq, 2 pq, 0 * + N* pq, 2 pq, 0 4 * + N* s ≈ ν ⁄ N ,,,s ≈ ------s ≈ ν ⁄ 2D s ≈ ------1 0 * 2 2 3 0 0 4 2 ()2 ω2 ()2 ω2 N* Ð * N* Ð * 7. CONCLUSIONS

Fig. 2. Dynamical, Gaussian, linear and nonlinear, spectral–correlational models for fluctuations in the motion of the Earth can be applied to identify basis Thus, we use the linearized, Gaussian, stochastic dynamical parameters using real measurements and differential equations (24) and (25) for the statisti- analyze their accuracy [1–5]. The models not only cal modeling. For analytical modeling in the non- preserve the qualitative properties of the fluctuations stationary case, we use the deterministic equa- of the Earth’s motion but also describe them with tions (24) and (28)–(30), also applying formulas (31) sufficient accuracy on intervals of three to five years. and (32) in the stationary case (see Section 6). The Gaussian, spectral–correlational and two- dimensional, kinetic models lie at the basis of stan- 6. ANALYSIS, MODELING, dard methods for reducing measurements collected AND DISCUSSION OF RESULTS for studies of the statistical dynamics of the Earth’s rotation. Dynamical spectral–correlational and kinetic Kinetic models for second-order fluctuations of models for fluctuations of the Earth’s motion (Sec- the Earth’s motion enable the refinement of spectral tions 2–5) provide basic information about the fun- characteristics and the characteristics of level inter- damental problem of the statistical dynamics of the sections and trends. High-order kinetic models are Earth’s rotation [16]. required to study higher-order extrema. In addition, Figures 1 and 2 show the stationary amplitudes the kinetic characteristics can be used to estimate of the mathematical expectations |Φp,q,δr(iω)| at fre- the errors in the spectral-analysis methods applied quencies ω∗, N∗,and2ω∗, together with the spectral in [1–3]. Such models form the basis of nonstandard

ASTRONOMY REPORTS Vol. 48 No. 2 2004 MODELS FOR THEMOTION OF THEEARTH 169 methods for reducing measurements in studies of sta- 5. N. S. Sidorenkov, Physics of Earth Rotation Insta- tistical dynamics of the Earth’s rotation. bilities(Nauka, Moscow, 2002), p. 384 [in Russian]. Software developed at the Institute of Informatics 6. Yu. G. Markov and I. N. Sinitsyn, Dokl. Akad. Nauk Problems of the Russian Academy of Sciences in 385, 189 (2002) [Dokl. Phys. 47, 531 (2002)]. 2000–2003 enables the realization on a PC of analyt- 7. Yu. G. Markov and I. N. Sinitsyn, Dokl. Akad. Nauk ical, statistical, and combined modeling of regular and 387, 482 (2002) [Dokl. Phys. 47, 867 (2002)]. ﬂuctuational motions of the Earth in both stationary 8. Yu. G. Markov and I. N. Sinitsyn, Astron. Zh. 80, 186 and nonstationary regimes. (2003) [Astron. Rep. 47, 161 (2003)]. 9. Yu. G. Markov and I. N. Sinitsyn, Dokl. Akad. Nauk 390, 343 (2003). ACKNOWLEDGMENTS 10. Yu. G. Markov and I. N. Sinitsyn, Dokl. Akad. Nauk 391, 194 (2003). The authors thanks V.I. Sinitsyn, Yu.V. Chumin, ` 11. W. Munk and G. MacDonald, The Rotation of the V.V. Belousov, and E.R. Korepanov for their consid- Earth (Cambridge, 1960; Mir, Moscow, 1964), p. 384. erable work in creating the programs used for this 12. L. D. Akunenko, S. A. Kumakshev, and analysis. This work was supported by the Russian Yu. G. Markov, Dokl. Akad. Nauk 382, 199 (2002) Foundation for Basic Research (project nos. 01-02- [Dokl. Phys. 47, 78 (2002)]. 17250 and 01-01-00758). 13. L. D. Akunenko, S. A. Kumakshev, Yu. G. Markov, andL.V.Rykhlova,Astron.Zh.79, 952 (2002) [As- tron. Rep. 46, 858 (2002)]. REFERENCES 14. V. S. Pugachev and I. N. Sinitsyn, Stochastic Dif- 1.L.V.Rykhlova,G.S.Kurbasova,andT.A.Taidako- ferential Systems. Analysis andFiltration ,2nded. va, Astron. Zh. 67, 151 (1990) [Sov. Astron. 34,79 (Nauka, Moscow, 1990), p. 632 [in Russian]. (1990)]. 15. V. S. Pugachev and I. N. Sinitsyn, Theory of 2. L. V.Rykhlova, G. S. Kurbasova, and M. N. Rybalova, Stochastic Systems (Logos, Moscow, 2000) [in Rus- Nauchn. Inform. Inst. Astron. Akad. Nauk SSSR 69, sian]. 3 (1991). 16.I.N.Sinitsyn,Yu.G.Markov,V.I.Sinitsyn,et al., 3. L. V. Rykhlova and G. S. Kurbasova, Astron. Zh. 75, in Proceedings of Internation School–Seminar on 632 (1998) [Astron. Rep. 42, 558 (1998)]. Automatization andComputerization in Science 4. M. Arato, Linear Stochastic Systems with Con- stant Coeﬃcients (Springer, Berlin, 1982; Nauka, andTechnology, ACS’2002 , http://www.elicsnet.ru. Moscow, 1989). Translatedby D. Gabuzda

ASTRONOMY REPORTS Vol. 48 No. 2 2004 Astronomy Reports, Vol. 48, No. 2, 2004, pp. 81–88. Translated from Astronomicheski˘ıZhurnal, Vol. 81, No. 2, 2004, pp. 99–107. Original Russian Text Copyright c 2004 by Uryson.

Seyfert Nuclei as Sources of Ultrahigh-Energy Cosmic Rays A. V. Uryson Lebedev Physical Institute, Leninski ˘ı pr. 53, Moscow, Russia Received September 28, 2002; in ﬁnal form, August 8, 2003

Abstract—Particles can be accelerated to ultrahigh energies E ≈ 1021 eV in moderate Seyfert nuclei.This acceleration occurs in shock fronts in relativistic jets.The maximum energy and chemical composition of the accelerated particles depend on the magnetic field in the jet, which is not well known; fields in the range ∼5–1000 G are considered in the model.The highest energies of E ≈ 1021 eV are acquired by Fe nuclei when the field in the jet is B ≈ 16 G.When B ∼ (5–40) G, nuclei with Z<10 are accelerated to E ≤ 1020 eV, while nuclei with Z ≥ 10 acquire energies E ≥ 2 × 1020 eV.Only particles with Z ≥ 23 acquire energies E ≥ 1020 eV when B ∼ 1000 G.Protons are accelerated to E<4 × 1019 eV, and do not fall into the range of energies of interest for any magnetic field B.The particles lose a negligible amount of their energy in interactions with infrared photons in the accretion disk; losses in the thick gas–dust torus are also negligible if the luminosity of the galaxy is L ≤ 1046 erg/s and the angle between the normal to the galactic plane and the line of sight is sufficiently small, i.e., if the axial ratio of the galactic disk is comparatively high.The particles do not lose energy to curvature radiation if their deviations from the jet axis do not exceed 0.03–0.04 pc at distances from the center of R ≈ 40–50 pc.Synchrotron losses are small, since the magnetic field frozen in the galactic wind at R ≤ 40–50 pc is directed (as in the jet) primarily in the direction of motion.If the model considered is valid, the detected cosmic-ray protons could be either fragments of Seyfert nuclei or be accelerated in other sources.The jet magnetic fields can be estimated both from direct astronomical observations and from the energy spectrum and chemical composition of cosmic rays. c 2004 MAIK “Nauka/Interperiodica”.

1.INTRODUCTION We considered X-ray (as the most powerful) pul- The hypothesis that active galactic nuclei (AGNs) sars from the list [16], radio galaxies from the cat- are sources of cosmic rays is not new and was widely alogs [17, 18], and AGNs from the catalog [19] as discussed in the 1980s [1].We have returned to the possible sources.For each of these types of objects, idea that cosmic rays are accelerated in the nuclei of we calculated the probability that their coordinates active galaxies after the direct identification of AGNs were in the 3-σ region for a shower by chance.The as possible sources of cosmic rays with energies E> probabilities for chance coincidence proved to be fairly ≤ 4 × 1019 eV [2–6], which was carried out as follows. low for moderate Seyfert nuclei with redshifts z Various objects located within 100 Mpc of our Galaxy 0.0092 (P ≈ 3.20σ) and for BL Lac objects (P ≈ and lying within the 1-σ,2-σ,and3-σ error regions 3.15σ; σ is the parameter of the Gaussian probability for the arrival direction of the particles were ana- distribution).According to probability theory, val- lyzed.This means of selecting potential sources was ues P>3σ indicate that the corresponding objects dictated by two factors.First, particles arriving from are not located in the search field by chance.The very large distances would unavoidably lose energy probabilities of chance coincidence for the remaining in photon–pion reactions with background radiation objects considered were P<3σ, testifying that they [7–9].Second, the actual coordinates of the particle most likely lie in the search field by chance.The value arrival direction are located within the 3-σ region with of P depends strongly on the choice of the search a probability of 99.8%, but within the 1-σ region with area: probabilities of chance coincidence for the 2-σ a probability of only 66.8% [10]. The shower statistics region surrounding the shower are several orders of included showers detected at the AGASA, Fly’s Eye, magnitude lower than those for the 3-σ region. Havera Park, and Yakutsk installations [11–14], for Specific models for the acceleration of cosmic rays which errors in the arrival direction are published.The in active galaxies were proposed in [20, 21].In the errors in the Yakutsk showers were calculated in our model of [20], the magnetic field in an accretion disk paper [3], and the errors for the Havera Park showers surrounding a supermassive black hole evolves as a were presented in [15].Of these showers, 11 have consequence of differential rotation of plasma with energies E>1020 eV. frozen-in magnetic field.As a result, there is an ex-

1063-7729/04/4802-0081$26.00 c 2004 MAIK “Nauka/Interperiodica” 82 URYSON plosive (more rapid than exponential) growth in the In both models, the particle acceleration occurs in electric field in some sections of the disk.Particles the presence of extremely strong electric fields.The can be ejected from the flow of plasma in regions of maximum particle energies ∼1021 eV coincide with low plasma density at the surface of the disk and can the maximum energies detected for cosmic rays [12]. be accelerated by this strong electric field.The accel- The model of [20] predicts sporadic flares of ra- eration continues until the particles leave the region diation associated with the ejection of accelerated of explosive electric-field growth or until the growth particles, while the model of [21] predicts a flux of in the field begins to slow.Numerical computations accelerated particles from the galactic nucleus.The of the trajectories of individual particles show that moderate Seyfert galaxies that have been identified as particles in a disk with particle density ∼1016 cm−3 possible sources of cosmic rays emit low radio and 7 around a black hole with mass ∼10 M can be accel- X-ray fluxes and do not display the characteristic fea- erated to energies of ∼1021 eV.The accelerated parti- tures predicted by these models.In addition, it is not cles lose some of their energy to synchrotron radiation known whether there exist in these galaxies the con- and the formation of pairs in reactions with photons. ditions required to accelerate particles in the proposed Synchrotron losses will be negligible if the particle scenarios.We have proposed another model for the velocities are aligned with the magnetic field.(Ac- acceleration of cosmic rays that supposes that various celeration mechanisms that can operate in the case conditions can exist in the cosmic-ray sources, with of other orientations of the particle velocities include various acceleration mechanisms being realized.Our acceleration in harmonic fields: if a particle is trapped model does not predict any specific features of the between the crests of a wave and the magnetic field source radiation apart from the acceleration of par- ≈ 21 has a component perpendicular to the particle veloc- ticles to energies of E 10 eV. ity, the particle will be accelerated [22, 23].) Losses In sources of cosmic rays, particles are not only to pair formation depend on the density distribution accelerated to ultrahigh energies, but also leave of the photons.In addition, in the case of collisions the acceleration region without appreciable energy between particles and photons, these losses depend losses.Sections 2 –4 present our analysis of data for on the angle between the directions of motion of the AGNs and demonstrate that the conditions required particle and photon.It is therefore possible that some for the acceleration and free escape of particles exist of the accelerated particles could leave the source in moderate Seyfert galaxies.Section 5 presents galaxy virtually without any loss in their energy. estimates of the energy of galactic sources based on the ultrahigh energy cosmic-ray spectrum observed In the model of [21], the particle acceleration oc- at the Earth.In the Conclusions, we discuss other curs in the electric field induced near a supermas- models for the origin of cosmic rays and the charac- 9 sive black hole with mass ∼10 M in periods of teristics of cosmic rays predicted by them. low activity of the black hole, when the accretion is diminished.The model is based on two main as- 2.CONDITIONS IN ACTIVE GALACTIC sumptions.The first is that the magnetic field of NUCLEI the black hole can be as high as ∼2 × 1010 G, in The nuclear activity of AGNs is most likely due contrast to the value ∼104 G derived in [24] and 2 to the existence of a supermassive black hole with the limit derived in [25], B ∼ 8πρ,whereρ is the 9 a mass of M ≈ 10 M and an associated accretion density of matter in the accretion disk.Second, it disk at the center of the galaxy.The accreting ma- is assumed that there is a region with a very low terial is provided by gas from stellar winds and su- plasma density in quiescent states of the black-hole pernova explosions surrounding the nucleus, stellar activity, in which a very strong induced electric field remnants that are disrupted by tidal forces near the cannot be compensated by the volume charge of the black hole, and entire stars that are captured by the plasma.Regions in which such very strong electric hole.The thickness of the accretion disk appears to fields can exist are located near the magnetic poles be ∼1 pc [26].Models in which the disk thickness is and rotational axis of the black hole.Particles can ∼ 16 18 10 cm have also been discussed in the literature be accelerated to energies 10 Z GeV in these fields. [27].The accretion disks in AGNs can apparently be The accelerated particles lose energy in interactions either optically thin or optically thick [26, 28].Two with photons in the disk and via curvature radiation. jets of material flowing out from the inner accretion In quiescent phases, losses to direct pair production disk emerge along the rotational axis.Jets of both and losses in photon reactions are negligible.The ordinary and electron–positron plasma and fluxes of particle energy can be decreased to 1012Z GeV due to electromagnetic radiation have all been considered in curvature radiation, and the maximum proton energy the literature; the jet material can be ejected either is accordingly 1021 eV. in the form of individual “blobs” ofplasmaorasa

ASTRONOMY REPORTS Vol.48 No.2 2004 SEYFERT NUCLEI AS SOURCES OF UHE COSMIC RAYS 83 continuous flow [28].The material ejected from the radio loudness is correlated with the spin of the cen- inner regions of the disk flows through the disk along tral black hole: in radio-loud objects, the spin is close two funnel-like channels formed along the rotational to its maximum value, while the spin is low in radio- axis. quiet objects [44].Falcke et al.[33] propose a uni fied The ejected particles interact with material in the model for AGNs in which the different properties of funnel walls and with their radiation.As a result of pp radio-quiet and radio-loud objects, and also of FRI and pγ interactions, electron–positron pairs or pions and FRII galaxies, are associated with different ge- are born, which decay into positrons and photons, so ometries for a torus surrounding the central black that an electromagnetic cascade is generated inside hole: the opening angle of the funnel is apprecia- the torus, giving rise to collimated beams of gamma bly broader for radio-quiet objects and Seyferts than radiation [27–30], providing an explanation for the it is for radio-loud objects.As a consequence, the gamma-ray emission of AGNs. jets in radio-quiet objects and Seyfert galaxies are The conditions under which plasma from the disk more poorly collimated and weaker and interact less and funnel walls does not fall into the channel of the with the torus (therefore, the production of electron– funnel are considered in [31]: this is hindered by a positron pairs in the funnel is negligible, so that these magnetic field of ∼104 G oriented parallel to the rota- nuclei emit weakly in gamma rays).It is proposed tional axis of the black hole.(As a result, the electric in [45–47] that, in the vicinity of the central black field inside the funnel cannot be short-circuited by the holes of radio-quiet objects, there are stellar cores volume charge of the plasma, analogous to the model with fairly large masses and colliding hot gas flows, in ∼ ∼ of [21], and particles are accelerated inside the funnel which the jets are disrupted (by 90%) within 1– by this field.) 3 pc of the nucleus. The central region of the galaxy is surrounded by Our model supposes that moderate Seyfert galax- a geometrically and optically thick dusty torus that ies have relativistic jets. radiates in the infrared; the thickness of this torus can A cocoon of perturbed jet material forms around reach ∼100 pc [32].Models in which the inner walls the jet as it propagates [48].Shock waves are excited of the torus are constantly bombarded by radiation, in the jet and cocoon as a consequence of the non- leading to the liberation of torus material that then linear development of instabilities at the jet surface, falls into the jet, are considered in the literature (see collisions with dense clouds, and velocity fluctuations [33] and references therein). [28, 49, 50].Relativistic particles can be accelerated As is clear from this brief summary of various to ultrahigh energies in the fronts of shocks with models, disk material can either be contained in the regular magnetic field (see the reviews [51, 52] and jet from the beginning or fall into the jet at some point. references therein). We will assume that the jets contain plasma from the We suppose that cosmic rays are accelerated in accretion disk. moderate Seyfert nuclei in the fronts of shock waves Different types of AGNs differ in various ways, in- in the jet.The magnetic- field strength in the jets is cluding the power of their radiation in various bands, not known.Only various estimates are available: the the ratio of the radio luminosity to the luminosity of field in the magnetosphere of a black hole is expected the accretion disk (radio “loudness” or radio “quiet- to be ∼104 G [24, 25, 42], though the estimates of [21] ness”), and the strength of the jets.Radio images suggest it could reach ∼1011 G; the fields at distances show that radio galaxies have jets extending to tens 1 pc from the black hole could be ∼103 G [53]; jet or hundreds of kiloparsecs [34, 35], while the lengths fields of ∼0.01 Gand∼100 G are considered in [54]; of the jets in Seyfert nuclei are ∼1–10 pc [36–40]. the field in the funnel along which the jets propagate Seyfert galaxies are usually considered to be radio- is estimated to be 700 G [27].In our model, the quiet objects (although it is possible that this is not field strength in the jet is an unknown.We obtain the case for all types of Seyferts [41]). an estimate of the field based on the conditions for How can we explain the moderate jets and radio- maximum acceleration of cosmic rays in shock fronts quiet nature of Seyfert galaxies? Unified schemes for in the jet. AGNs are discussed in the literature, in which ob- What chemical compositions are possible for the served differences in properties are due, for example, accelerated cosmic rays? Since we assume that the to differences in the accretion conditions (the mass jet contains material from the accretion disk, the of accreted gas and the accretion rate) [42], the ori- composition of the cosmic rays should reflect that entation of the AGN relative to the observer [26], the of the disk.The chemical composition of AGNs was mass of the central black hole [43], etc.In addition, investigated in [55] and a number of works referenced various authors have explained the radio-loud/radio- in [55].We will assume that both protons and nuclei quiet dichotomy in various ways.It is possible that are present in the jet.

ASTRONOMY REPORTS Vol.48 No.2 2004 84 URYSON 3.ACCELERATION OF COSMIC RAYS strength for which the particle is accelerated to the IN SEYFERT NUCLEI maximum energy, × 2 2/3 −1/3 The main assumptions of our model are the fol- BCR =(3.5 10 ) Z , (5) lowing.In most Seyfert galaxies, relativistic jets form and the values of this maximum energy for nuclei in the vicinity of a massive central black hole.(It is (A/Z ≈ 2): possible that they are disrupted at distances of 1– ≈ × 20 1/2 3 pc inside a massive stellar core.) The transverse Emax A 6.6 10 (Z/B) eV, (6) cross section in the central region of the jet is S = and for protons: 3 × 1031 cm2, and the jet Lorentz factor is γ =10 ≈ × 20 −1/2 [45].Both protons and nuclei are present in the jets. Emax p 1.65 10 B eV. (7) The jet magnetic field is treated as an unknown, and (These same formulas were obtained in a somewhat we will obtain an estimate of the field based on the different way in our paper [58].) If the field is B ∼ conditions for maximum acceleration of cosmic rays (5–40) G, nuclei with Z ≥ 10 acquire energies E ≥ in shock fronts in the jet. 2 × 1020 eV, while lighter nuclei are accelerated only 20 The jet field is parallel to the jet axis, the shocks are to energies E ≤ 10 eV: for protons, Bp ≈ 19.6 G also parallel, and particles are accelerated to energies 19 and Emax p ≈ 3.7 × 10 eV; for He nuclei, BHe ≈ ≈ 20 Ej ZeβjBRj erg (1) 39.5 GandEmax He ≈ 1.5 × 10 eV; and for Fe nu- 20 clei, BFe ≈ 16 GandEmax Fe ≈ 8 × 10 eV.In a field when they scatter off inhomogeneities in the magnetic of B ∼ 100 G, particles with Z>2 are accelerated field due to turbulence, where Ze is the charge of the E ≥ 1020 B ∼ 1000 particle, β is the ratio of the jet velocity to the velocity to energies eV.In a field of G, j only heavy particles with Z ≥ 23 acquire energies of light, B is the magnetic fieldinahotspotinthejet, E ≥ 1020 eV. and Rj is its transverse size [56].For the jet parameters presented above, the velocity and transverse size The particles acquire such energies over distances 15 of R ≈ Tac from the base of the jet.Taking the B of the jet will be βj ≈ 0.99 and Rj ≈ 3 × 10 cm and fields from above, we find that this distance is RHe ≈ the maximum particle energy will be 15 6 × 10 cm≈ 20Rg for He nuclei and RFe ≈ 3 × 18 15 Ej ≈ 1.9 × 10 ZB eV. (2) 10 cm ≈ 10Rg for Fe nuclei, where Rg is the gravi- 9 In the hot-spot magnetic field, the particle simul- tational radius of a black hole with a mass of 10 M. taneously is accelerated and loses energy to syn- In a field of B ≈ 1000 G, particles with Z =23are chrotron radiation.We will assume that, under these accelerated to E =1020 eV over a distance of R ≈ conditions, the particle can accumulate the maximum 0.1Rg. energy if it loses less than half its acquired energy Formulas (5)–(7) and these numerical estimates to synchrotron radiation over the acceleration time 15 were obtained for a jet radius of Rj ≈ 3 × 10 cm.In Ta.In other words, we assume that the acceleration a jet with radius kRj, the maximum particle energy time Ta does not exceed the time Ts over which the 1/3 particle energy decreases by a factor of two: Ta ≤ Ts. changesbyonlyafactorofk .Therefore, these The acceleration time is equal to [52] estimates are valid for jets with cross sections in the range ∼5 × 1029–1033 cm2. T ≈ l/(β2c), (3) a s Thus, protons are accelerated to E<4 × 1019 eV, where l is the diffusion mean free path (in the vicinity and do not fall into the energy range of interest for of a shock, this will be equal to the Larmor radius of any values of B.Therefore, if this model is correct, × 19 the particle, l ≈ rL)andβs is the ratio of the velocity cosmic-ray protons with energies E>4 10 eV of the shock to the velocity of light.The Larmor radius were not accelerated in Seyfert nuclei.They are frag- ofaparticleis[1]rL ≈ E/(300ZB),whererL is in ments of nuclei, or were accelerated in other sources cm, the energy E is in eV, and the magnetic field B is (such as BL Lac objects).Further, the magnetic fields in Gauss.The time Ts is equal to [57] in jets can be estimated not only based on astronomical observations but also based on the energy 18 2 3 2 Ts ≈ 3.2 × 10 /B⊥(A/Z) 1/Z (Mc /E), (4) spectrum and chemical composition of cosmic rays. where B⊥ is the field component perpendicular to the particle’s velocity, A is the atomic number of 4.ESCAPE OF THE PARTICLES FROM THEIR SOURCES the particle, M = Amp is the mass of the particle, and mp is the proton mass.Assuming Ta ≈ Ts and Accelerated particles that have left the hot spot do E = Emax ≈ 1/2Ej , we obtain from (2)–(4) the field not interact with the leading wave excited in the flow

ASTRONOMY REPORTS Vol.48 No.2 2004 SEYFERT NUCLEI AS SOURCES OF UHE COSMIC RAYS 85 of hot gas by the jet, since this wave propagates more develop in the jets due to their interaction with the slowly than the jet due to the fact that the density of walls of the gas–dust torus.This enhances the ra- the gas is lower than that of the jet [50].The particles diation density in the jet, leading to additional en- lose energy in photon–pion reactions with infrared ergy losses by the accelerated particles.However, photons and to synchrotron and curvature radiation. in Seyfert galaxies, the interaction of the jets with Let us first consider photonic losses. the torus walls is insignificant, so that no cascades develop.For this reason, these AGNs radiate only weakly in gamma rays [33]. 4.1. Energy Losses of the Particles in Interactions An accelerated particle will not intersect the dust with Photons torus if it travels at an angle i to the normal to the During interactions of the particles with pho- galactic plane such that tan it. (11) In addition, a particle can lose energy in interac- We find the distance Rline as follows.The accelerated tions with radiation emitted in the jet.The emission particle freely leaves the galaxy, having reached re- of such radiation is considered in [33].Cascades can gions where the field has decreased enough that the

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Larmor radius of the particle becomes roughly rL ≥ equal to 5 kpc [63].(Here, we have taken the characteristic ∞ dimensions of the spiral galaxies that are the host L = F (E)EdE, (14) galaxies of most Seyferts to be the same as those of UHECR g our Galaxy.) For ultrarelativistic particles, the Larmor E ≈ −γ radius is equal to [1] rL E/(300ZB) (the energy where Fg(E)=KE is the differential particle- E is measured in eV, the field B in Gauss, and rL generation spectrum in a Seyfert galaxy.If the in cm), and, for particles with various Z values and spectrum of high-energy cosmic rays is only weakly energy E = Emax, the condition rL ≥ 5 kpc is fulfilled − distorted by interactions in intergalactic space (this is when the field is B ≤ 10 5 G.Taking the magnetic possible if the sources are located within R ∼ 50 Mpc field to decrease with distance as B ∼ R−3 [64], and of the Earth), the particle-generation spectrum in the the field at R ∼ 1 pc to be B ∼ 1 G [53], we obtain source Fg(E) and the observed cosmic-ray spectrum ≈ 19 Rline 46 pc.The curvature radius of the field lines in I(E) for E>5 × 10 eV will have the same form, 2 −γ adipolemagneticfield is ρc =4R /3a,whereR and and Fg(E)=KE with γ ≥ 3.1.The intensity of a are the distances from the center and from the axis cosmic rays with energy E in the Universe is [1] of the dipole [21].Based on this information and (9) – (11), we can estimate the maximum deviation from I(E)=(c/4π)Fg (E)nSyTCR, (15) − − the jet axis of a particle with energy E = Emax that where n =2× 10 77 cm 3 is the density of Seyfert ≈ Sy has traveled a distance R 46 pc with small curva- galaxies and T is the age of the UHE cosmic ≈ CR ture losses: for protons, ap 0.01 pc; for He nuclei, rays.Since the distances from which the cosmic rays ≈ ≈ aHe 0.03 pc; and for Fe nuclei, aFe 0.04 pc. can arrive without interacting with the background ∼ ≈ Let us determine the fraction of accelerated parti- radiation are R 50 Mpc, the ages are TCR = R/c cles that leave the source without significant curva- 1.7 × 108 yr.According to measurements made using ture losses.For such particles, the deviation from the various installations [66], the spectrum I(E) at E> jet axis will be 5 × 1019 eV is I(E) ≈ 10−39–10−40 (cm2 ssreV)−1. ≤ × −4 We obtain from (14) and (15) the observed luminos- θ a/Rline =6.5 10 . (12) ity of the galactic sources in cosmic rays with E> × 19 ≈ 41 42 ≈ In the wave system, the particles are scattered 5 10 eV, LUHECR 10 –10 erg/s (γ 3.1). isotropically.Therefore, the fraction of particles in The real luminosity is higher than the observed value 43 which we are interested can be found as follows.The by a factor of 1/η and is LUHECR,eff ≈ 3 × 10 – ∗ angle θ between the velocity vector and jet axis in the 3 × 1044 erg/s.(Note that, if the mass of the black 9 2 wave system is related to the angle θ by the expression hole is M ≈ 10 M and all its energy Mc is spent [65] on accelerating UHE cosmic rays at the power found − 2 1/2 ∗ −1 ∗ above, the energy of the black hole is depleted over tan θ =(1 β ) (β +cosθ ) sin θ (13) 13 14 ∗ ∗ 10 –10 yr, much longer than the age of the Uni- =0.14 sin θ (0.99 + cos θ ) 10 verse, TMg ≈ 1.3 × 10 yr.) with β ≈ 0.99.For angles θ<0.02, sin θ∗ ≈ θ∗, ∗ ∗ ∗ 6.CONCLUSIONS cos θ ≈ 1 and θ ≈ 0.07θ .Hence, θ ≈ 0.01,and the fraction of particles deviating from the axis by We have shown that particles can be accelerated angles (12) is η =0.01/π ≈ 3 × 10−3; i.e., roughly to ultrahigh energies E ≈ 1021 eV in the nuclei of one in 300 of the accelerated particles leave the source moderate Seyfert galaxies.In our model, the acceler- without curvature losses. ation takes place in shock fronts in relativistic jets in these objects.Jets in Seyfert nuclei have been studied We considered the propagation of the particles in in references listed in Section 2.We adopted the jet the magnetic fields of intergalactic space and in our parameters presented in [45] for our estimates.We Galaxy in [58]. have assumed that the jets contain material from the accretion disk, so that both protons and nuclei are 5.ESTIMATES OF THE ENERGY present in the jets.The maximum energy and chemi- OF GALACTIC COSMIC-RAY SOURCES cal composition of the accelerated particles depend on the magnetic field in the jet, which is not well known, Thus, based on the flux of cosmic rays measured at and so was treated as an unknown parameter in the the Earth, we can estimate the observed luminosities model.We considered fields in the range ∼5–1000 G. in ultrahigh energy (UHE) cosmic rays for galac- The highest energy E ≈ 8 × 1020 eV is acquired by tic cosmic-ray sources.This luminosity LUHECR is Fe nuclei when the field in the jet is B ≈ 16 G.When

ASTRONOMY REPORTS Vol.48 No.2 2004 SEYFERT NUCLEI AS SOURCES OF UHE COSMIC RAYS 87 the field is B ∼ (5–40) G, nuclei with Z ≥ 10 ac- rays should be nuclei (nuclear fragments), with no quire energies E ≥ 2 × 1020 eV, while lighter nuclei excess cosmic rays from the Galactic center. are accelerated to energies E ≤ 1020 eV.In a field If our model is correct, detected protons with en- of B ∼ 1000 G, only particles with Z ≥ 23 acquire ergies E>4 × 1019 eV are either fragments of nuclei energies E ≥ 1020 eV.Protons are accelerated only or were accelerated in other sources (such as BL Lac to E<4 × 1019 eV and do not fall into the energy objects).The jet magnetic fields can be estimated range of interest for any value of B.The estimates not only based on astronomical observations but also we have obtained are valid for relativistic jets with based on the energy spectrum and chemical compo- cross sections of ∼5 × 1029–1033 cm2.The particle sition of the cosmic rays. acceleration occurs over distances ∼(0.1–10)Rg. Our model can be verified using measurements The accelerated particles leave the source galaxy of the spectrum and composition of UHE cosmic without significant energy losses under the following rays.Such measurements will be carried out us- conditions.First, the particles do not lose energy ing AGASA, as well as using future giant installa- in photon–pion reactions in the accretion disk if its tions such as HiRes, Pierre Auger, Telescope Array, optical depth is τ ≤ 1 [21, 31, 56, 59].Second, losses satellite instruments (see references in the reviews in the thick gas–dust torus are insignificant if the [71, 72]) and the ShAL-1000 installation [73]. galaxy luminosity is L<1046 erg/s [56] and the angle between the normal to the galactic plane and the line ACKNOWLEDGMENTS of sight is sufficiently small, i.e., if the axial ratio of the I am grateful to E.Ya. Vil’koviskiı˘, V.A. Dogel’, galactic disk is comparatively high.Third, particles do A.V. Zasov, Ya.N. Istomin, and O.K. 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ASTRONOMY REPORTS Vol.48 No.2 2004 Astronomy Reports, Vol. 48, No. 2, 2004, pp. 89–102. Translated from Astronomicheski˘ı Zhurnal, Vol. 81, No. 2, 2004, pp. 108–123. Original Russian Text Copyright c 2004 by Abubekerov, Antokhina, Cherepashchuk.

Masses of X-ray Pulsars in Binary Systems with OB Supergiants

M. K. Abubekerov1, E.´ A. Antokhina2,andA.M.Cherepashchuk1,2 1Moscow State University, Vorob’evy gory, Moscow, 119992 Russia 2Sternberg Astronomical Institute, Universitetski ˘ı pr.13, Moscow, 119992 Russia Received June 15, 2003; in ﬁnal form, August 8, 2003

Abstract—We describe the results of a statistical approach to analyzing the combined radial-velocity curves of X-raybinaries with OB supergiants in a Roche model, both with and without allowance for the anisotropyof the stellar wind. We present new mass estimates for the X-raypulsars in the close binarysystems Cen X-3, LMC X-4, SMC X-1, 4U 1538-52, and Vela X-1. c 2004 MAIK “Nau- ka/Interperiodica”.

1. INTRODUCTION rich observational data that are now available and cannot be used to correctlyestimate the masses of Neutron stars are a natural product of the evolu- the X-raypulsars in binarysystems.This is especially tion of stars with masses in excess of 10 M.They true for systems with OB supergiants. were predicted theoreticallybyLandau [1] in the early 1930s, and the first radio pulsar was found in the cen- In X-raybinaries with OB supergiants, the op- ter of the Crab nebula in the early1960s. At the be- tical star nearly fills its critical Roche lobe, which ginning of the 1970s—an epoch of active programs in determines the limiting equilibrium figure in a ro- near-Earth space exploration—the “Uhuru” satellite tating reference frame fixed to the binary. Therefore, discovered objects with periodic X-rayvariability[2] the shape of the optical component deviates from a (X-raypulsars), later identi fied with neutron stars in sphere, and the velocity field at the stellar surface binarysystems. becomes stronglynonuniform. The surface of the star Thousands of compact X-raysources have now that is turned toward the X-raypulsar is heated bythe been discovered in our Galaxyand other nearby pulsar’s radiation, and the nonuniform gravitational galaxies. Most of these are close binarysystems, acceleration on the companion leads to anisotropyof in which the optical component supplies matter the stellar wind. All these factors distort the radial- velocitycurve of the optical star. The distortions of the to the neutron star. Accretion with subrelativistic velocities onto the compact component’s surface radial-velocitycurve due to the e ffects of outflow and releases immense amounts of X-rayenergywith the stellar wind in close binaries with OB supergiants are comparable to the orbital velocity. These facts 1036 1039 luminosities of the order of – erg/s [3–7]. show the necessityof proper fitting of close-binary From the point of view of the equation of state of the radial-velocitycurves in a Roche model taking into superdense matter, the keycharacteristic of an X-ray account the anisotropyof the stellar wind [14]. pulsar is its mass. The theorypredicts two stable modes for neutron stars, based on a “soft” equation of state, with Mmax 1.4 M,anda“hard” equation of 2. DESCRIPTION OF THE ROCHE MODEL state, with Mmax 1.8 M [8–12]. The question of whether or not we should adopt the hard equation of The radial-velocitycurves of X-raybinaries are state for neutron stars remains unanswered. Among usuallyanalyzed using models with two point masses, known X-raypulsars, onlyVela X-1 has a mass though it is not appropriate to approximate the stars close to 1.8 M [13]. Studies of X-raybinaries have in most close binaries as point masses moving in been carried out for more than 30 years. Extensive Keplerian orbits for the following reasons. spectroscopic observations have been accumulated 1. Due to tidal deformation, the optical compo- which not onlyenable but also demand for their nent is ellipsoidal or pear-shaped due to the gravi- interpretation a more accurate allowance for effects tational action of the relativistic companion, leading resulting from the closeness of the components. A to stronglynonuniform surface temperatures (gravi- point-mass model approximating the optical com- tational darkening). For mass ratios q = mx/mv < 1 ponent as a source of electromagnetic radiation with (with mx and mv being the masses of the optical infinitesimal size is not adequate for analysis of the and X-raycomponents), the binarycenters of mass

1063-7729/04/4802-0089$26.00 c 2004 MAIK “Nauka/Interperiodica” 90 ABUBEKEROV et al. are inside the bodyof the optical star, leading to account the Doppler effect to obtain the total profile significant distortion of the radial-velocitycurve. for the star at the corresponding phase of the orbital 2. X-rayheating of the side of the optical star fac- period. The theoretical total profile is subsequently ing the relativistic component results in a nonuniform used to determine the star’s radial velocityand derive temperature distribution over the stellar surface. the radial-velocitycurve over the course of the orbital period (for more details, see [15]). 3. It is important for early-type stars in close binaries that the cores of strong absorption lines formed in The input parameters of the algorithm used to the outermost atmospheric layers, at the base of the compute the radial-velocitycurve of an X-raybinary stellar wind, experience radial Doppler shifts. These in the Roche model are collected in Table 1. shifts are due to the regular outflow of the stellar-wind plasma with velocities of the order of the sound speed, 3. COMPARISON OF THE THEORETICAL 1/2 Vs ∼ T . The temperature and local gravitational RADIAL-VELOCITY CURVES acceleration varystronglyat the surface of the tidally IN THE ROCHE AND POINT-MASS deformed star, leading to appreciable distortion of the MODELS optical component’s radial-velocitycurve. For early- Our main goal was to refine available estimates of type stars, the velocity of the stellar wind at its base the masses of X-raypulsars in close binaries. Before becomes comparable to the orbital velocityof the analyzing the observations, we carried out test com- optical component, of the order of 10–20 km/s. putations to compare the theoretical radial-velocity For these reasons, the radial-velocitycurves of curves in the Roche and point-mass models. We X-raybinaries with OB supergiants should be an- computed the radial-velocitycurves using the H γ line alyzed using Roche models. We applied our algo- for a hypothetical X-ray system with parameters (Ta- rithm for synthesizing light curves, radial-velocity ble 1) close to those of the studied systems: Cen X-3, curves, and absorption-line profiles, described in de- LMC X-4, SMC X-1, Vela X-1, and 4U 1538–52 tail in [15, 16], to analyze observations of massive (Table 2). close X-raybinaries. We describe the optical star To identifythe in fluence of reflection effects on using a Roche model, treating its companion as a the radial-velocitycurves, we computed them in the point X-raysource. We brie flysummarize the main Roche model for low and high incoming X-ray fluxes, principles of our algorithm below. kx =0.05 and kx =1.4. The computations of the The optical star and point relativistic object with radial-velocitycurves from the H γ lines demonstrate masses of mv and mx move in elliptical orbits about that the semiamplitude of the curves decreases in the their common center of mass. The inclination of the presence of stronger X-rayheating; a similar result binaryorbit to the plane of the skyis i. In general, the was obtained earlier [15]. We can explain this bythe star’s axial rotation is not assumed to be synchronous fact that, in the case of a stronger X-ray flux incident with the orbital motion. The star’s shape coincides on the star and stronger heating of the surface facing with an equipotential surface of the Roche model [16, the companion, the contribution of the light from the 17]. The size of the star is determined bythe filling co- star’s “nose” to the combined luminosityincreases. efficient for the critical Roche lobe, µ = R/R∗,where The areas on the nose contribute more stronglyto the R and R∗ are the polar radii for partial and complete star’s mean radial velocity. The velocities of such ar- filling of the critical Roche lobe at the orbit periastron. eas are lower than the star’s center-of-mass velocity, The star’s tidallydistorted surface is subdivided into and so the mean velocityof the star decreases. thousands of mass elements, with the emergent radi- The difference of the star’s radial-velocitycurve ation computed for each of them, after which the con- from the curve for its center of mass increases from tributions of the various areas are added taking into phase 0.0 to phase 0.5 as the star moves along its account the visibilityof each area element at various orbit and its heated surface turns toward the observer. phases of the orbital period. The computed radiation At phase 0.0, the star is in front of the X-raysource. fluxes from each area element include the effects of The difference between the star’s radial-velocity gravitational darkening, heating of the stellar surface curves in the Roche and point-mass models should bythe radiation from the companion (the “reflection” increase with decreasing q = mx/mv. effect), and limb darkening. The absorption line profile Figure 1a compares the theoretical radial-velocity and line equivalent width for each visible area with its curves of a system (for its parameters, see Table 1) temperature T and local gravitational acceleration g in the Roche and point-mass models. Curves are were calculated using the tables of Kurucz [18] for the plotted for the Hγ line and for the case of low heating, Balmer lines and an interpolation procedure. We add kx =0.05. The semiamplitude of the curve is lower the local profiles over the stellar surface after first nor- in the Roche model, and the semiamplitudes of the malizing to the continuum for each area, taking into radial-velocitycurves can be the same onlywhen the

ASTRONOMY REPORTS Vol. 48 No. 2 2004 MASSES OF X-RAY PULSARS 91

Table 1. Numerical parameters used to synthesize an X-raybinaryradial-velocitycurve in the Roche model

Relativistic component’s mass mx, M 1.35

Optical component’s mass mv, M 20 Period P ,days 4 Eccentricity e 0 Longitude of periastron of the optical component ω, degrees 0

Radial velocityof the system’s center of mass Vγ ,km/s 0 Orbital inclination i, degrees 80 Roche-lobe filling coefficient for the optical component at periastron µ 0.99 Rotational asynchronism coefficient f 0.95

Eﬀective temperature of the optical component Teff ,K 30000 Gravitational darkening coeﬃcient β 0.25 Ratio of the relativistic component’s X-rayluminosity

to the optical component’s bolometric luminosity kx = Lx/Lv 0.05 и 1.4 Coeﬃcient for reprocessing of the X-rayradiation A 0.5 Limb-darkening coeﬃcient u 0.3

Table 2. Observed characteristics of X-raybinaries with OB supergiants (from the literature [13, 19, 20])

Name Spectral type Teff ,K P ,days e i, degrees µ f Kx,km/s

+3 +0.005 +0.27 ± Cen X-3 O(6-9) II-III 35000 2.0871390 <0.0008 83−3 0.995−0.005 0.95−0.25 414.1 0.9 +3 ∼ +0.23 LMC X-4 O7 III-V 35000 1.40839 <0.01 63−3 1.0 0.65−0.19 400.6 +5 +0.03 +0.34 ± SMC X-1 B0.5Ia 25000 3.89229118 <0.00004 65−5 0.97−0.03 0.95−0.27 301.5 4 +3 +0.04 +0.09 ± Vela X-1 B0.5Ibeq 25000 8.964368 0.0898 73−3 0.95−0.04 0.69−0.08 278.1 0.3 +5 +0.04 +0.32 ± 4U 1538–52 B0Iabe 25000 3.72844 0.08 60−5 0.95−0.04 0.94−0.25 309.0 11

The orbital inclination i was derived from the duration of the X-rayeclipse (see, e.g., [20]). The values of e and Kx were derived from the X-raypulsar’s radial-velocitycurve. mass of the relativistic component is increased from enable us to determine whether the masses of the 1.35 M to 1.45 M. To obtain equal semiamplitudes neutron stars in X-raybinaries with OB supergiants for phase 0.25, the mass of the X-raypulsar must exceed the masses of radio pulsars, which are equal, ± be increased to 1.55 M. For stronger heating, kx = on average, to 1.35 0.04 M [21]. 1.4, with the remaining parameters unchanged, equal semiamplitudes in the Roche and point-mass models require that the X-raypulsar’s mass be increased to 4. OBSERVATIONAL MATERIAL 1.65 M. Five eclipsing X-raybinaries with OB supergiants Thus, all other conditions being the same, the use have been discovered to date (Table 2). During the of the Roche model leads to an increase in the mass last 30 years, extensive observational material has of the relativistic component by ∼7–20% compared been accumulated for these systems, making it pos- to the mass obtained in the point-mass model. This sible to test hypotheses using statistical methods. It suggests that using the more realistic close-binary is not possible to correctlytake into consideration the model to analyze the radial-velocity curves should components’ mutual inﬂuence with the widelyused

ASTRONOMY REPORTS Vol. 48 No. 2 2004 92 ABUBEKEROV et al.

Vr, km/s Vr, km/s 30 (a) 60 (a) 20 40 10 20 0 0 –10 –20 –20 –40 –30 –60 (b) 20 (b) 40 10 20 0 0 –10 –20 –20 –40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 φ φ

Fig. 2. (a) Master observed radial-velocitycurve for the Fig. 1. (a) Theoretical radial-velocitycurves of the op- close X-raybinaryCen X-3 (triangles show spectro- tical star in a hypothetical X-rayclose binary(for its scopic data for 1976 from [22]; circles, data for 1978 parameters, see Table 1). The curves are derived from from [23]; crosses, data for 1990 from [24]; and squares, the Hγ line. The dashed curve corresponds to the optical data for 1997 from [24]). For comparison, the theoretical star in a point-mass model with mx =1.35 M,and radial-velocitycurves are shown for the Roche model the solid curve, to a Roche model with mx =1.35 M; (solid curve) and point-mass model (dashed curve), with the dotted curve corresponds to mx =1.45 M,and mx =1.22 M corresponding to the minimum residual the dot–dash curve, to mx =1.55 M. (b) Comparison for the Roche model computed using method 2. The of the theoretical radial-velocitycurves of a star in the parameters are collected in Table 3. (b) Radial velocities Roche model based on the Hγ line (solid curve) and averaged within phase intervals (ﬁlled circles are the mean the HeI 4471 A˚ line (dot–dash curve). The parameters radial velocities for the phase bins, and open circles, the are close to those of Vela X-1 (mv =23 M, mx = radial velocities corrected for the normalized anisotropy ◦ 1.8 M, Porb =8.96 days, e =0.0898, ω = 332.59 , i = function of the optical star’s wind). The theoretical radial- ◦ 73 , µ =0.99, f =0.69, T = 25 000 K, β =0.25, kx = velocitycurves for the Roche model (solid curve) and 0.05, A =0.5, u =0.38). The dashed curve is the radial- point-mass model (dashed curve) with mx =1.22 M velocitycurve for the optical star’s center of mass. are shown for comparison.

Monte Carlo method in the framework of the point- observed radial velocities derived from the shifts of mass model. For the reasons indicated above, earlier the HeI 4471 A,˚ HeI 5875 A˚ absorption lines and estimates of the masses of X-raypulsars must be the hydrogen Balmer lines. The line shifts in [22, reﬁned and revised using the Roche model. 24] were derived relative to the spectrum averaged We constructed a master radial-velocitycurve for over all phases, so that the γ velocitywas taken each of the systems. We describe the spectroscopic into account in the method itself. The mean radial data used for each radial-velocitycurve below. The velocities presented in [22] were derived from the center of the X-rayeclipse was taken as the zero HeI 4471 A,˚ HeII 4541 A˚ absorption lines and phase for all the systems. the hydrogen Balmer lines. Of the radial velocities Cen X-3. The system consists of an O giant from [24], we used onlythose measured for the H γ (O(6–9)II–III), with mass 17–18 M and the X-ray line. A total of 79 data points were available to us, pulsar, and has an orbital period of 2.0871390 days. distributed approximatelyuniformlyin phase. We The orbital eccentricityderived from timing of the used these data to construct a master radial-velocity X-raypulses is verylow ( e<0.0008 [13]), and we curve for the Cen X-3 system from the Hβ,Hγ,Hδ, have assumed a circular orbit. HeI 4471 A,˚ and HeI 5875 A˚ absorption lines. Our We used data from [22–24] acquired in 1976– test computations demonstrated that the theoretical 1997 to construct the master radial-velocitycurve radial-velocitycurves for the Roche model based on (Fig. 2a). The data of [23] were corrected onlyfor Hγ and other lines (Hβ,Hδ, HeI 4471 A)˚ coincided, the close binary’s γ velocity. The corrections for all other parameters being the same (Fig. 1b; see also the velocityof the stellar wind were alreadyintro- the text below). Thus, we used the theoretical radial- duced bythe authors when combining the mean velocitycurves for the H γ line when ﬁtting the master

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Vr, km/s Vr, km/s 80 (a) 80 (a) 60 60 40 40 20 20 0 0 –20 –20 –40 –40 –60 –60 –80 –80 50 (b) 40 (b) 30 20 10 –10 0 –30 –20 –50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 –40 φ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 φ Fig. 3. (a) Master observed radial-velocitycurve for the close X-raybinaryLMC X-4 ( filled triangles show spec- Fig. 4. (a) Master observed radial-velocitycurve for the troscopic data for 1975 from [25]; circles and filled squares close X-raybinarySMC X-1 ( filled squares show spec- show spectroscopic data for 1982 from [26] acquired with troscopic data for 1973 from [27]; open circles, spec- 2.5-m and 4-m telescopes). For comparison, theoretical troscopic data for 1976 from [28]; and open triangles, radial-velocitycurves are shown for the Roche model spectroscopic data for 1989 from [29]). For compari- (solid curve) and point-mass model (dashed curve), with son, theoretical radial-velocitycurves are shown for the mx =1.63 M corresponding to the minimum residual Roche model (solid curve) and point-mass model (dashed for the Roche model computed using method 2. The curve), with mx =1.48 M corresponding to the min- parameters are collected in Table 3. (b) Radial velocities imum residual for the Roche model computed using averaged within phase intervals (filled circles are the mean method 2. The parameters are collected in Table 3. (b) Ra- radial velocities for the phase bins, and open circles, the dial velocities averaged within phase intervals (filled cir- radial velocities corrected for the normalized anisotropy cles are the mean radial velocities for the phase bins, and function of the optical star’s wind). For comparison, the- open circles, the radial velocities corrected for the normal- oretical radial-velocitycurves are shown for the Roche ized anisotropyfunction of the optical star’s wind). For model (solid curve) and point-mass model (dashed curve) comparison, theoretical radial-velocitycurves are shown with mx =1.63 M. for the Roche model (solid curve) and point-mass model (dashed curve) with mx =1.48 M. radial-velocitycurves for Cen X-3, as well as for the other close binaries. The system’s γ velocityderived 17–18 M and the X-raypulsar. Its orbital period is using the spectroscopic data of [23] is 39 km/s. 3.89229118 days. LMC X-4. The system consists of the X-ray pulsar The mean radial-velocitycurve (Fig. 4a) was con- and an optical star of spectral type O7III–Vwith structed using the data of [27–29]. The mean radial a mass of 14–15 M. The eccentricityis verylow velocities measured from the HeI and SiIV absorption (e<0.01), and we assumed the orbit to be circular. lines and the hydrogen Balmer lines are presented The orbital period derived from studies of the X-ray in [27]. The radial velocities determined byHutchings pulsar is 1.40839 days. et al. [28] were based on the mean of the shifts of We used the data of [25, 26] acquired in 1975– the hydrogen Balmer lines and the HeI, SiIV, and ˚ 1982 to plot the master light curve (Fig. 3a). The ear- NIII absorption lines in the blue (3900–4000 A). The lier observational data presented in [25] were already radial velocities in [29] were computed using a cross- used in [26]. Thus, we used the newest spectroscopic correlation method based on the entire spectrum from data of [26]. The radial velocities were derived from 4000 to 4395 A˚ relative to the spectrum of the B3V the hydrogen absorption lines. Prior to being included standard star HR 1174, which has a similar spectral in the master radial-velocitycurves, the data were type. corrected for the γ velocityin [26]. The system’s γ Before plotting the mean curve, we corrected the velocityderived using the spectroscopic data of [26] spectroscopic data of [27–29] for the γ velocity. We is 275 km/s. computed the γ velocitybyminimizing the sum of SMC X-1. The system contains an optical com- squared deviations between the radial velocities that ponent of spectral type B0.5Ia with a mass of about were observed and synthesized in the Roche model

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Vr, km/s Vr, km/s 60 60 (a) (‡) 40 40 20 20 0 0 –20 –20 –40 –40 –60 –60 40 60 (b) (b) 40 20 20 0 0 –20 –20 –40 –60 –40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 φ φ

Fig. 5. (a) Master observed radial-velocitycurve for the Fig. 6. (a): Master observed radial-velocitycurve for close X-raybinary4U 1538 –52 (filled circles are the the close X-raybinaryVela X-1 ( filled circles are the spectroscopic data for 1978 of [31], and open circles, spectroscopic data for 1973 of [33], and open circles, the the spectroscopic data for 1991 of [32]). For compari- spectroscopic data for 1995–1996 of [34]). For compar- son, the theoretical radial-velocitycurves are shown for ison, the theoretical radial-velocitycurves are shown for the Roche model (solid curve) and point-mass model the Roche model (solid curve) and point-mass model (dashed curve), with mx =1.18 M corresponding to the (dashed curve), with mx =1.93 M corresponding to the minimum residual in the Roche model computed using minimum residual in the Roche model computed using method 2. The parameters are collected in Table 3. (b) The method 2. The parameters are collected in Table 3. (b) The radial velocities averaged within phase intervals (filled radial velocities averaged within phase intervals (filled circles are the mean radial velocities for the phase bins, circles are the mean radial velocities for the phase bins, and open circles, the radial velocities corrected for the and open circles, the radial velocities corrected for the normalized anisotropyfunction of the optical star’s wind). normalized anisotropyfunction of the optical star’s wind). For comparison, the theoretical radial velocitycurves are For comparison, the theoretical radial-velocitycurves are shown for the Roche model (solid curve) and point-mass shown for the Roche model (solid curve) and point-mass model (dashed curve) with mx =1.18 M. model (dashed curve) with mx =1.93 M. for the same phase. The γ velocityenters the algo- γ velocity. We derived the γ velocities −168.5 and rithm used to synthesize the radial-velocity curves −146.7 km/s using the spectroscopic data of [31, 32], additively, so that the condition that the areas of the respectively. In this case, the large negative value of radial-velocitycurve above and below the γ velocity the γ velocitymayre flect radial expansion of the line- be equal was fulfilled automatically. The γ velocities formation region near the base of the stellar wind. derived from the spectroscopic data of [27–29] were Vela X-1. This system contains a B0.5Ibeq su- 173.0, 181.0, and 172.7 km/s. pergiant with a mass of 24–25 M and the X-ray 4U 1538-52. This system consists of the X-ray pulsar. The components are in an elliptical orbit with pulsar and a B0Iabe supergiant with a mass of 17– e 0.0898, and we adopted this eccentricity. The 18 M. The system’s orbital period is 3.72844 days. system’s orbital period is 8.964368 days. The eccentricityderived from timing of the X-ray The observational material for this system is very pulses is 0.07 ± 0.09 [30]. Due to the large uncertainty rich. We used onlythe radial velocities of [33, 34] for of this value, we assumed zero eccentricity. the hydrogen Balmer lines to construct the master The master radial-velocitycurve (Fig. 5a) is based radial-velocitycurve (Fig. 6a). As a result, data for on the spectroscopic data of [31, 32]. Radial veloc- 1973 and 1995–1996 were included in the master ˚ curve; these demonstrated good agreement despite ities derived from the Hβ,Hγ, HeI 4921 A, and the long time interval separating them. ˚ HeI 4471 A absorption lines are presented in [31]. The The Balmer absorption lines from [33] derived from radial velocities in [32] were computed via the cross- the absolute shifts of the cores of the H2–H10 ab- correlation method using the spectroscopic standard sorption lines relative to their laboratorywavelengths HD 133955 and HeI absorption lines in the range indicated increasing stellar-wind velocities with de- 6290–6710 A˚ . These data are in good agreement. We creasing line number in the series—a Balmer pro- corrected the data from [31, 32] for the close binary’s gression [35, 36]. This effect is due to the negative

ASTRONOMY REPORTS Vol. 48 No. 2 2004 MASSES OF X-RAY PULSARS 95 temperature gradient in the star’s photosphere, re- Vr, km/s sulting in a dependence of the observed stellar-wind 20 (a) velocityon the excitation potential of the absorption line used to measure this velocity. Thus, the Hβ ab- 10 sorption line is formed in higher and, consequently 0 cooler, layers of the stellar photosphere, so that the stellar-wind velocityderived from this line is higher –10 than the stellar-wind velocitymeasured from Balmer –20 absorption lines with higher numbers. 10 (b) According to the data of [37], the γ velocityderived from the Hβ line and HeI, NIII, and SiIV ion lines is 5 −3.5 ± 0.8 km/s. The shifts of the lines’ cores relative 0 to their laboratorywavelength are due not onlyto the γ velocitybut also to the velocityof the stellar wind –5 at its base. If the system’s γ velocityis much lower –10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 than this stellar-wind velocity, it is not possible to φ determine it from the shifts of the H2–H10 lines due to the strong stellar wind from the OB supergiant. Fig. 7. (a) Anisotropyfunction of the stellar wind. For example, according to the spectroscopyof [33], (b) Same after normalization. the radial velocities derived from the H2,H3,H5,and H6 lines are −29.0, −15.5, −14.0,and−8.5 km/s, respectively. It is evident from these results that the binaries studied here, we deemed it to be admissible velocityassociated with the wind out flow is much to synthesize the theoretical radial-velocity curves higher than the systematic velocity of the center of using our earlier algorithm, which requires far fewer mass, the latter being −3.5 ± 0.8 km/s [37]. For this computer resources. The theoretical radial-velocity reason, we do not present a γ velocityfor Vela X-1. curves for the Hγ and HeI 4471 A˚ lines shown in The radial velocities derived using the H2–H10 lines Fig. 1b demonstrate good agreement, and, as was from [33] were corrected for the systematic radial alreadymentioned, we used the theoretical radial- velocity(the velocities of the center-of-mass motion velocitycurves based on the H γ line for our analysis and of the stellar wind at its base) individuallyfor of the master set of observational data for the Balmer each line before including them in the master radial- lines and HeI absorption lines. velocitycurve. The spectral cross-correlation method used in [34] 5. THE EFFECT OF ANISOTROPY to compute the radial velocities made it unnecessary OF THE STELLAR WIND for us to introduce anyfurther corrections for either IN THE ATMOSPHERE OF THE OB STAR the velocityof the stellar wind at its base or the γ velocity. Early-type stars posses high mass-loss rates, up −7 −5 ˙ We can see from the above discussion that the to 10 –10 M/year. The outflow velocityin the radial velocities we used to fit the observational data zone in which the absorption lines are formed can were derived from both the hydrogen and HeI ab- be as high as ∼10–20 km/s, and, thus, becomes sorption lines. Therefore, as was mentioned above, we comparable to the orbital velocityof the OB giant in carried out test computations to check the agreement the system. between the theoretical radial-velocitycurves based The isotropyof the stellar wind in a close bi- on the Hγ and HeI 4471 A˚ lines (Fig. 1b). The radial naryis disrupted bythe gravitational action of the velocities for the HeI 4471 A˚ line were computed compact object. The optical star is alreadyin a using a new version of our algorithm for the synthe- nonuniform gravitational field. The wind velocitynear sis of theoretical absorption-line profiles and of the the Lagrangian point L1 increases; this is observed radial-velocitycurves for close X-raybinaries [38]. In (Figs. 2a, 3a, 4a, 5a, 6a) as an excess negative radial contrast to the earlier algorithm, which is described velocity(toward the observer) at phase 0.5, when the in the cited paper and uses the Balmer profiles from X-raysource is in front of the OB star. the tables of Kurucz [18] for the local profiles, the This stronglyhinders correct interpretation of the new algorithm computes the local profiles for area radial-velocitycurve for close binaries with early- elements on the stellar surface after constructing the type optical components. The anisotropy of the stel- model atmosphere irradiated bythe external X-rays. lar wind is a source of errors when searching for Since the X-rayheating is not strong for most of the a system’s spectroscopic orbital elements [14], and

ASTRONOMY REPORTS Vol. 48 No. 2 2004 96 ABUBEKEROV et al. this anisotropymust likewise be taken into account Roche model, taking into account the anisotropyof when fitting the radial-velocitycurve of the optical the stellar wind from the OB star. component of a close binary. Bearing in mind the difficulties of constructing theoretical models for an We applied a simple method with an exhaustive anisotropic wind from a distorted star, we approached parameter search and found multiple solutions vary- this problem empirically. ing the desired parameter. In our case, the unknown m Before fitting the radial-velocitycurves of the sys- was x and the mass of the compact object in the tems, we calculated the phase dependences of the close X-raybinarywas varied during the solutions, with all other parameters of the close binarybeing differences Vobs − Vteor for each system (with the ex- ception of 4U 1538–52, due to the low number of data fixed. The values of these fixed parameters were de- points and their nonuniform distribution in orbital rived from analyses of the photometric light curves, X-rayeclipse data, and timing of the X-raypulses. phase); Vobs is the observed radial velocityfor a given The close-binaryparameters used to synthesize the orbital phase and Vteor is the theoretical value in the Roche model without taking into account the wind. radial-velocitycurves are presented in Table 3. Jumping somewhat ahead, we note that we used the It is often verydi fficult to measure the true radial mass of the relativistic object computed using the velocityof the optical star. Complex processes on the second method (i.e., not taking into account the radial surface of the supergiant and in gaseous structures in velocities observed at phases from 0.4 to 0.6) when the system can lead to both systematic and random computing Vteor. Such mass estimates for X-raypulsars are most reliable. deviations. We were faced with the question of how to best take into account the systematic errors. In The resulting radial-velocitydeviations for all four addition to the effect of anisotropyin the optical star’s systems were combined and averaged within phase wind, tidal gravitational waves on the stellar surface bins with a width of 0.05. Since the structure of the spectroscopic data for Vela X-1 was such that could be a source of systematic errors if the orbit is elliptical. The recent study[40] demonstrated that there radial velocities derived from the H2–H10 absorption lines fell at the same phase, we first averaged these was no correlation between the orbital phase and the within each orbital phase. We adopted the mean ve- phase of the gravitational tidal waves. Consequently, tidal waves on the optical star’s surface are a source locityfrom all the absorption lines for Vobs (i.e., from H2–H10) at a given epoch. Unaveraged data were of random errors and can be reduced byaveraging the used for the other systems. data from manynights of observations. The mean curve of the radial-velocitydeviations, In addition to gravitational–tidal perturbations − Vobs Vteor, for the four systems is presented in of the OB supergiant’s surface, additional absorp- Fig. 7a. We will call this relation the stellar-wind tion and photoionization effects [41] in the complex anisotropyfunction. The relation in Fig. 7a clearly gaseous structures in the close binarycan be sources demonstrates an excess negative radial velocity(to- of random errors in the observed radial velocities. ward the observer) at phases from 0.4 to 0.6, when the nose of the star is turned toward the observer. This To reduce the influence of random errors due result is in qualitative agreement with the physics of to various effects, we averaged the radial velocities anisotropic stellar-wind outflows. within phase intervals. Our subsequent analysis The use of the stellar-wind anisotropyfunction to of the radial-velocitycurves was based on these correct the observed radial velocities is justified by mean observed radial velocities. The averaged radial- the fact that it is essentiallyan empirical relation. We velocitycurves for the program binaries are presented normalized the relation for this purpose. The Vobs − in Figs. 2b, 3b, 4b, 5b, and 6b. Vteor residuals were averaged for the phase intervals 0–0.4 and 0.6–1.0. The resulting mean values, 1.79 We tested various hypotheses using the Fisher ra- and 4.32 km/s, were adopted as the zero points for tio test, which takes into account the most complete correcting the observed radial velocities in the phase information on the averaged data. We describe our intervals 0.0–0.5 and 0.5–1.0, respectively. The nor- adaptation of the Fisher ratio test for the analysis obs malized empirical stellar-wind anisotropyfunction is of the radial-velocitycurves below. Let V¯j be the shown in Fig. 7b. observed mean radial velocityfor the phase interval teor centered at phase φ¯j; Vj , the theoretical radial 6. INTERPRETATION velocityvalue at this phase; and σj, the rms deviation obs OF THE RADIAL-VELOCITY CURVES of V¯j from the radial velocities observed in the given ¯ Our aim was to determine the mass of the X-ray phase interval centered at φj.LetM be the number pulsars from the master radial-velocitycurves in the of the phase intervals, and nj, the number of radial

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Table 3. Numerical parameters used to synthesize the radial-velocitycurves of the close X-raybinaries in the Roche model

Parameters Cen X-3 LMC X-4 SMC X-1 4U 1538–52 Vela X-1 P ,days 2.0871390 1.40839 3.89229118 3.72844 8.964368 e 0.0 0.0 0.0 0.0 0.0898 ω,deg∗ 0.0 0.0 0.0 0.0 332.59 i,deg 82 65 65 65 73 µ∗∗ 0.99 0.99 0.97 0.95 0.95 и 0.99 f 0.95 0.65 0.95 0.94 0.69

Teff ,K 35000 35000 25000 25000 25000 β 0.25 0.25 0.25 0.25 0.25

kx 0.05 1.4 0.25 0.0025 0.003 A 0.5 0.5 0.5 0.5 0.5 u∗∗∗ 0.30 0.30 0.38 0.38 0.38

Kx,km/s 414.1 400.6 301.5 309.0 278.1 ∗ The longitude of the periastron of the optical component is given. ∗∗ The value for Vela X-1 corresponds to the orbit periastron. ∗∗∗ Data of [39].

velocities averaged in a given phase interval. We can Method 3. Using the mean radial velocities at calculate the residual as follows: phases 0.4–0.6 corrected for the stellar-wind aniso- M M tropyfunction. − teor − ¯ obs 2 (nj 1) nj(Vj Vj ) Let us consider the details of our procedures for j=1 j=1 ∆(m )= . (1) determining the compact object’s mass using the first x M M − 2 system, Cen X-3, as an example. nj(nj 1)σj j=1 Cen X-3. The mean radial-velocitycurve for Cen X-3 had 79 data points. Despite the fact that the The variable ∆(mx) will be distributed according points were uniformlydistributed in orbital phase to a Fisher law, F M [42]. Let us adopt (Fig. 2a), we can identifyclose groups of points with M, (nj −1),α j=1 similar orbital phases. In the case of Cen X-3, we the significance level α.Theconfidence set for the isolated 11 such groups, within which we averaged desired parameter, mx, will then consist of the values the observed radial velocities. The radial velocities satisfying the condition averaged within the phase bins are shown in Fig. 2b. The mean radial velocities show the strongest de- ∆(mx) ≤ F M . M, (nj −1),α viations from the standard curve at phases 0.4–0.6. j=1 The supergiant’s excess negative radial velocityat these phases is due to the anisotropyof the stellar We solved the inverse problem using the Roche wind. and point-mass models. The latter was used onlyto Recall that we determined the unknown m by reveal disagreements between the results for the two x carrying out an exhaustive search among its possible models. We analyzed the mean radial-velocity curves values. In the process, we varied the mass of the for each system using three methods. relativistic component so that the semiamplitude of Method 1. Using all the mean observed radial its radial-velocitycurve remained unchanged, Kx = velocities, with no correction for anisotropyof the 414.1 km/s [43]. For more detail on the method used stellar wind. to compute the optical component’s mass byrunning Method 2. Rejecting mean observed radial veloci- through possible masses for the relativistic compo- ties at phases 0.4–0.6 that are most stronglydistorted nent, see the Appendix. The remaining binaryparam- bythe e ffects of the stellar-wind anisotropy. eters (Table 3) were fixed.

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∆, km/s (‡) ∆, km/s (b) 5 4 2 2 2 1 4 1 3 1 1 3 3 3 2 2 3 3

2 1

1 0 0.6 0.8 1.0 1.2 1.4 1.6 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 mx, M mx, M

Fig. 8. Deviations of the observed mean radial-velocitycurves for (a) Cen X-3 and (b) Vela X-1 from the syntheticcurves in the Roche model (solid curve) and point-mass model (dashed curve) obtained using methods 1, 2, and 3. The method used to derive the residuals is indicated bya number near the curve. The horizontal lines correspond to the critical levels in the Fisher ratio test, ∆8,47 =2.14 (a) and ∆18,782 =1.605 (b), for a signiﬁcance level of 5%.

The residual yielded by analyzing the observed ra- bodyof the optical star ( q<1), so that some frac- dial velocities using method 1 is shown in Fig. 8a. The tion of the optical star’s emitting surface is moving significance level adopted for the computations was in the same direction as the relativistic component. 5%. We can see in this figure that the model without This leads to a lower amplitude of the synthesized allowance for the wind anisotropymust be rejected at radial-velocitycurve in the Roche model, all the other this significance level. The mx value corresponding to parameters being the same, and, hence, to a higher the minimum residual is presented in Table 4. mass for the relativistic object. Interpretation of the radial-velocitycurve is hin- In the computations with method 3, we corrected dered bythe strong deviation of the data points from the mean radial velocities at phases 0.4–0.6 for the their computed positions near phase 0.5. This does mean normalized wind anisotropyfunction. This not fit into the Roche model with an isotropic stellar correction makes the deviation between the synthetic wind. We therefore excluded in the next stage ob- and observed radial velocityvalues much smaller served radial velocities at phases 0.4–0.6 during the (Fig. 2b). The differences between the theoretical computations, and our analysis of the fitted curves did and corrected observed radial-velocitycurves are not take them into account (method 2). The behavior presented in Fig. 8a. We can see that, in this case, of the residuals is shown in Fig. 8a. We can see that the Fisher ratio test indicates both the Roche and the point-mass model must be rejected according to point-mass models to be acceptable. the Fisher test, whereas the Roche model, which is The resulting masses, along with the masses of the more adequate for describing the physics of processes X-raypulsars in the other systems,are collected in in close binaries, is acceptable. We accordinglyadopt Table 4. the Roche model, not because it is necessarilycorrect, LMC X-4. The master radial-velocitycurve con- but because there is no reason to reject it (see the tains about 70 data points (Fig. 3a); the averaged review [44] for details). master radial-velocitycurve is presented in Fig. 3b. The mass of the X-raypulsar corresponding to the The mean velocities in the phase bins were formed minimum residual in the Roche model is 1.63 M. using from three to ten points. These mean observed The minimum residual for the point-mass model is velocities are in good agreement with the theoretical reached for 1.47 M. This large a discrepancycan be values. explained bythe di fference in the shapes of the syn- The residuals obtained for the Roche and point- thetic curves for the Roche and point-mass models mass models using methods 1 and 2 indicate that (Fig. 1a). The barycenter of the system is inside the both models are acceptable according to the Fisher

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Table 4. Masses of X-raypulsars in the close binaries with supergiants derived from fitting the mean observed radial- velocitycurves in the Roche and point-mass models (the con fidence interval is given for the 95% confidence level)

Roche model Point-mass model Name Method 1 Method 2 Method 3 Method 1 Method 2 Method 3 +0.15 +0.05 +0.18 Cen X-3 1.14 1.22−0.14 1.10−0.05 1.12 1.14 1.09−0.18 +0.29 +0.42 +0.30 +0.47 +0.08 LMC X-4 1.55−0.27 1.63−0.47 1.60 1.40−0.29 1.47−0.43 1.43−0.05 +0.33 +0.47 +0.49 +0.33 +0.41 +0.49 SMC X-1 1.40−0.29 1.48−0.42 1.40−0.45 1.30−0.31 1.36−0.39 1.30−0.45 +0.28 +0.29 +0.11 4U 1538–52 1.21−0.26 1.18−0.27 1.26 1.16 1.13−0.11 1.21 ∗ +0.25 +0.08 +0.30 +0.21 Vela X-1 (µ =0.95) 2.02 1.93−0.24 1.94−0.10 2.02 1.92−0.28 1.94−0.18 ∗ +0.19 +0.30 +0.21 Vela X-1 (µ =0.99) 2.02 1.93−0.21 1.94 2.02 1.92−0.28 1.94−0.18 ∗ The confidence interval is given for the error of the mean artificiallyincreased twofold. test. Because of the initial good agreement of the Fisher ratio test, whereas the point-mass model is re- mean observed and theoretical radial velocities, the jected. With the radial velocityat phase 0.45 excluded residual increases when the observed radial-velocity (model 2), both the Roche and point-mass models curve is analyzed using method 3. The increase in pass the Fisher test. Since an excess positive radial the difference between the theoretical and observed velocityis observed at phase 0.45, the correction for radial velocities is quite apparent in Fig. 3b. When the the normalized stellar-wind anisotropyfunction only observed mean radial-velocitycurve is analyzed using increases the discrepancybetween the mean observed method 3, the point-mass model is rejected bythe and theoretical radial velocities. Thus, the increased Fisher test, whereas the Roche model is acceptable. residual yielded by method 3 results in both mod- The resulting masses and confidence intervals are els being rejected bythe Fisher test. The resulting presented in Table 4. masses and confidence intervals are presented in Ta- SMC X-1. The mean curve contained 70 data ble 4. points (Fig. 3a). The averaged radial velocities are presented in Fig. 4b; the observed radial velocities at Vela X-1. The master radial-velocitycurve in- phases 0.40 and 0.51 deviate fairlystronglyfrom the cluded 782 data points subdivided into 18 groups theoretical values. (Fig. 6a). The values averaged within each group The analysis of the mean observed radial-velocity are shown in Fig. 6b. Each phase bin used for the curves using anyof the three methods indicates that averaging contains many(28 to 82) data points, lead- both the Roche and point-mass models are accept- ing to low standard deviations. Thus, the residuals able according to the Fisher ratio test. Correction of obtained using method 1, method 2, and method 3 are the mean observed radial velocities at phases 0.40 and 8.59, 6.00, and 6.59, respectively, with the quantiles 0.51 reduces the disagreement between the synthetic being 1.69, 1.605, and 1.69. The low uncertainties in and observed radial velocities (Fig. 4b); i.e., the resid- the averaged data impose stringent requirements on uals obtained using method 3 are lower than those for the models, so that even the Roche model with the method 1. This is reflected in the broader confidence stellar-wind anisotropytaken into account is rejected interval for the X-raypulsar masses satisfyingthe bythe Fisher test, testifying to the complexityof the Fisher test at the 95% confidence level for method 3 physics of the absorption-line-formation processes. compared to method 1 (Table 4). 4U 1538–52. The master radial-velocitycurve Artificiallyincreasing the standard deviations, σj, contains 36 data points (Fig. 5a); the values aver- bya factor of 1.5 reduced the residuals for method 2 aged within each group are presented in Fig. 5b. to 1.92, whereas the critical level for the Fisher ratio Because of the sparseness of the spectroscopic data, test with a 5% significance level is 1.605. The differ- the averaging in some phase intervals was based on ence between the critical level for the Fisher test with radial velocities measured on a single night, making the 5% significance level and the residuals derived it impossible to reduce the influence of random errors. using the other methods using 1.5σj are even higher. This could be the origin of the excess positive velocity When the standard deviations, σj, in (1) were artifi- at phase 0.45. ciallydoubled, the models for methods 2 and 3 were When analyzing the mean observed radial veloc- deemed acceptable bythe Fisher test. The model for ities using method 1, the Roche model passes the method 1 is rejected even after doubling the standard

ASTRONOMY REPORTS Vol. 48 No. 2 2004 100 ABUBEKEROV et al. deviations and analyzing the full, uncorrected radial- Let us now consider the X-raypulsar mass es- velocitycurve. timates for the Roche model. Theyare systemati- The above findings indicate that the Roche model callyhigher than the masses derived in the point- is not adequate to the high-accuracyradial-velocity mass model, byon average ∼ 10% (Table 4). Our test curve of the optical star in the Vela X-1 system, computations for the SMC X-1 system demonstrated making the mass of the X-raypulsar, mx, derived for that the systematic excess of the X-ray pulsar masses this system relatively uncertain. obtained for the Roche model did not depend on the orbital inclination i (see the Appendix). Thus, we Because of the large uncertaintyin the Roche-lobe conclude that the X-raypulsar masses were system- filling coefficient, µ, we carried out computations for aticallytoo low (by5 –10%) in all earlier studies based µ =0.95 and 0.99, keeping the remaining conditions on point-mass models. the same (this corresponds to two lines in Table 4). The deviation of the mean observed radial-velocity The masses of the X-raypulsars in LMC X-4 +0.42 +0.47 curve from the synthetic curve for the Roche model and SMC X-1, 1.63−0.47 M and 1.48−0.42 M, is much lower for µ =0.95 than for µ =0.99, which respectively, are somewhat higher than the standard is reflected in a wider confidence interval for the better mass for a neutron star, 1.35 ± 0.04 M [21], though fit (Table 4). However, the mass of the X-raypulsar theycan be reconciled with a mass of 1.35M within derived from the residual minimum remains the same our confidence intervals. It is difficult to explain the for the two cases. We conclude that the radial-velocity excess of the X-raypulsar masses over the standard curve is insensitive to relativelysmall variations of the value as being due to the accumulation of accreted Roche-lobe filling coefficient. matter from an accretion disk onto the neutron star’s Figure 8b shows the behavior of the residuals (for surface. A mass of about 0.01 M could settle onto µ =0.95) found using each of the methods. Table 4 the pulsar’s surface over the optical component’s life presents the X-raypulsar masses corresponding to time during the stage when the Roche lobe is close the minimum residuals for models rejected bythe to being filled, ∼ 105 years, if the mass-loss rate of −7 −6 Fisher ratio test. It is not possible to derive confidence the optical star is 10 –10 M˙ /year, taking into intervals in these cases. account the size of the close binaryand the relativistic star’s gravitational capture radius. The X-raypulsar masses in the Cen X-3, LMC 7. CONCLUSIONS X-4, SMC X-1, and 4U 1538–52 systems derived This paper has presented estimates of the masses using the most reliable method, in the Roche model of X-raypulsars that are most adequate to the entire excluding the mean radial velocities near phase 0.5, +0.15 +0.42 +0.47 set of observational data for X-raybinaries with OB are 1.22−0.14 M, 1.63−0.47 M, 1.48−0.42 M, supergiants. In contrast to earlier studies, we have +0.29 and 1.18 M, respectively. The mean of the considered not onlythe semiamplitude of the radial- −0.27 four X-raypulsar masses is 1.37 ± 0.15 M at the velocitycurve but also its overall shape. 95% confidence level (the mass of the pulsar in the Our investigation demonstrates that a pure Roche Vela X-1 system was not used for the average due to model is insufficient for analyzing the radial-velocity its anomalouslyhigh value). This mean X-raypulsar curves of X-ray close binaries with early-type optical mass agrees with the standard radio pulsar mass, stars. This model is often rejected bythe Fisher ratio 1.35 ± 0.04 M, within the errors [21]. test due to the presence of an excess negative radial Our test computations in the Roche model for velocitynear phase 0.5. In a number of cases, only various orbital inclinations, i, demonstrated the valid- taking into account the wind anisotropymakes it − ityof the relation m ∼ sin 3i (see the Appendix for possible to accept the model based on the Fisher test, x more details). Thus, the X-raypulsar masses derived therebyenabling us to obtain reliable estimates of the in the Roche model can easilybe recalculated if their neutron-star masses in the X-raybinaries and their orbital inclinations are refined. confidence intervals. An important result of this studyis the mass es- This studyhas yieldedmasses of the X-raypulsars timate for the compact object in the Vela X-1 X-ray derived in the Roche model, with the stellar-wind +0.32 anisotropytaken into account empirically.This model binary. The earlier estimate was 1.86−0.32 M at the is much more realistic than earlier models based on 95% significance level [34], considerablyin excess point masses in Keplerian orbits. Note that the mass of the standard value, 1.35 ± 0.04 M. Our analy- estimates for X-raypulsars in binaries with super- sis of the radial-velocitycurve in the Roche model giants obtained in the point-mass models of [13] are taking into account the wind anisotropyslightlyin- in good agreement with our own determinations us- creased the mass of the X-raypulsar in this system, +0.19 ing the same model with two point masses. to 1.93−0.21 M at the 95% confidence level. Note

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Table 5. Masses of the SMC X-1 pulsar derived using the Roche and point-mass model for various orbital inclinations (conﬁdence intervals are given for the 95% conﬁdence level)

Roche model Point-mass model Orbital inclination Method 1 Method 2 Method 3 Method 1 Method 2 Method 3 ◦ +0.33 +0.33 +0.49 +0.33 +0.41 +0.49 i =65 1.40−0.29 1.48−0.29 1.40−0.45 1.30−0.31 1.36−0.39 1.30−0.45 ◦ +0.27 +0.37 +0.40 +0.27 +0.35 +0.41 i =75 (case A) 1.15−0.24 1.22−0.36 1.15−0.37 1.08−0.25 1.12−0.31 1.43−0.38 i =75◦ (case B) 1.156 1.223 1.156 1.074 1.123 1.074 ◦ +0.24 +0.34 +0.37 +0.24 +0.32 +0.38 i =85 (case A) 1.05−0.22 1.11−0.32 1.05−0.33 0.98−0.24 1.02−0.28 0.98−0.33 i =85◦ (case B) 1.054 1.114 1.054 0.978 1.024 0.978

again that our Roche model for the Vela X-1 system from (2). Substituting the true Kv derived from (2) was rejected bythe Fisher ratio test, so that the X-ray into (3), we find the mass of the optical star, mv,for pulsar’s mass, 1.93 M, cannot be considered very the specified value of i. In this manner, relating the trustworthy. A “hard” equation of state must be used mass of the optical component to that of the rela- to explain the existence of neutron stars with masses tivistic component, we can keep the semiamplitude of this high. Studies in this direction are verypromising. the relativistic component’s radial-velocitycurve, Kx, constant. Note also that (3) contains onlya quadratic dependence on K , whereas the dependence on K ACKNOWLEDGMENTS v x is cubic. This is reflected in the fact that varying mx This studywas supported bythe Russian Founda- bya factor of 2.7 leads to a change in the optical tion for Basic Research (project code 02-02-17524) component’s mass byonlya factor of 1.14. and a grant from the program “Leading Scientific In the model with two point masses, the masses Schools of Russia.” We thank V.V. Shimanskiı˘ for of the optical star, mv, and of the X-raypulsar, mx, helpful advice. −3 depend on the orbital inclination, i,asmv ∼ sin i −3 and mx ∼ sin i. We should check if the relation APPENDIX −3 mx ∼ sin i is valid when the X-raypulsar’s mass is determined using the Roche model for the optical star When solving the inverse problem, we varied the from the minimum difference between the observed mass of the X-raypulsar, mx, so that the semi- mean radial-velocitycurve and the curve synthesized amplitude of its radial-velocitycurve, Kx, remained in the Roche model. unchanged, since this parameter is known to high ac- For this purpose, we also analyzed the mean curacyfrom X-raytiming data (Table 2). To keep Kx radial-velocitycurve of the SMC X-1 systemin the constant as the mass of the relativistic component, Roche model using the three methods for orbital mx, varied, we had to varythe mass of the optical star, inclinations of 75◦ and 85◦. The result is presented in mv, as well. We used the classical formulas to relate Table 5, where it is called “case A.” Having the mass the mass of the optical component, mv, to each value obtained earlier in the Roche model for an orbital of the X-raypulsar’s mass, mx: inclination of 65◦ (Table 4), we recalculated this mass ◦ ◦ 3 −7 2 3/2 75 85 mx sin i =1.038 × 10 P (1 − e ) (2) for the orbital inclinations and according to the formulas × 2 Kv(Kx + Kv) , 3 ◦ ◦ ◦ sin 65 mx(75 )=mx(65 ) , (4) 3 −7 2 3/2 sin3 75◦ mv sin i =1.038 × 10 P (1 − e ) (3) 2 3 ◦ × Kx(Kv + Kx) . ◦ ◦ sin 65 mx(85 )=mx(65 ) 3 ◦ , (5) The point-object model is quite applicable to the sin 85 ◦ ◦ ◦ X-raypulsar, justifying the use of (2). The value of where mx(65 ), mx(75 ),andmx(85 ) are the mass- ◦ Kv it gives for fixed mx and i characterizes the true es of the X-raypulsar for orbital inclinations of 65 , velocityof the optical star’s center of mass. Due to the 75◦,and85◦. The results are presented in Table 5, various reasons mentioned above (the pear-like shape where theyare called “case B.” We can see that the of the stars, anisotropic winds, heating effects, etc.), masses m derived from the minimum residual in the x ◦ ◦ the observed Kv can differ from the value of Kv found Roche model for the orbital inclinations 75 and 85

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