Transient Heliosheath Modulation
aa r X i v : . [ a s t r o - ph . S R ] S e p Transient Heliosheath Modulation
J J Quenby and W R Webber Blackett Laboratory, Imperial College, London, SW7 2BZ, UK. Department of Astronomy, New Mexico State University, Las Cruces, USA
ABSTRACT
Voyager 1 has explored the solar wind-interstellar medium interaction regionbetween the Terminal Shock and Heliopause, following the intensity distributionof galactic cosmic ray protons above 200 MeV energy. Before this componentreached the expcted galactic flux level at 121.7 AU from the sun, four episodes ofrapid intensity change occured with a behaviour similar to that found in ForbushDecreases in the inner solar system, rather than that expected from a mechanismrelated to models for the Long Term Modulation found closer to the sun. Becausethe mean solar wind flow is both expected and observed to be perpendicular tothe radial direction close to the Heliopause, an explanation is suggested in termsof transient radial flows related to possible heliopause boundary flapping. Itis necessary that the radial flows are at the sound speed found for conditionsdownstream of the Terminal Shock and that the relevant cosmic ray diffusionperpendicular to the mean field is controlled by ’slab’ fluctuations accounting for20 % or less of the total power in the field variance. However, additional radialdrift motion related to possible north to south gradients in the magnetic fieldmay allow the inclusion of some diffusion according to the predictions of a theorybased upon the presence of 2-D turbulence The required field gradients may arisedue to field variation in the field carried by solar plasma flow deflected away fromthe solar equatorial plane. Modulation amounting to a total 30 % drop in galacticintensity requires explanation by a combination of transient effects.
1. INTRODUCTION
Since the provision of early models of heliosheath modulation (Potgieter and le Roux1989, Quenby et al. 1990), it has been assumed that this region between the terminal shockand the heliopause is a location where a substantial fraction of the solar modulation of thegalactic cosmic ray intensity occurs (see review by Potgieter (2008)). A very recent descrip-tion of heliosheath modulation in a spherically symmetric approximation is due to Webber 2 –et al. (2013b) while (Potgieter (2013) povides a recent general review. The heliopause repre-sents the boundary beyond which the interstellar cosmic ray intensity would be encountered.Strauss et al. (2013) have recently questioned this assumption and mention various possi-bilities of increased particle scattering beyond the heliopause as the interstellar field wrapsaround the Heliopause. Models and observations related to the interaction of the interstellarmedium (ISM) with the heliosphere allow a solar wind terminal shock and a heliosheathlying between this shock and the heliopause. These models suggest that the heliosheathcomprises a low latitude region where the magnetic structure is determined by reconnectionof the sector structure fields and a high latitude region where field lines connect back to thesolar wind (Opher el al. 2012). The field is carried by the solar wind as it is diverted to highlatitudes and back downwind of the interstellar flow. No interstellar bow shock is expected(McComas et al. 2012) but the external field pressure causes asymmetry in the terminalshock (Opher et al. 2006) and may also explain intermitent observation of shock acceleratedcosmic rays (Jokipii et al. 2004, Stone et al. 2005) prior to the termination shock crossing.The dramatic Voyager 1 observation of two sudden increases of the greater than 200 MeVproton galactic cosmic ray (GCR) component near the heliopause reached at 121.7 AU (Webber and McDonald, 2013a, Stone et al., 2013) presents a challenge to current ideas ofmodulation within the heliosheath. It is the purpose of this paper to provide a simple classof models for these sudden increases in terms analagous to the cause of Forbush Decreasesin the inner solar system. There should exist transient, enhanced radial plasma flows whichmay or not be accompanied by changes in magnetic field gradients suitable to yield enhacedparticle drift speeds. A companion paper (Webber and Quenby, 2014) discusses the observa-tions of the two extra-ordinary increases in GCR intensity in relation to magnetic field dataand the force field modulation model.
2. VOYAGER DATA
The starting point of this work lies in data obtained by Webber and McDonald (2013a)from the Voyager 1 CRS instrument (Stone et al. 1977). We concentrate on the period ofthe final increase in the >
200 MeV proton intensity to attain the expected galactic fluxvalue, witnessed by the steady intensity distribution beyond 121.7 AU. The increase around2012 day 240 is represented by a change of scaled count rate from 4.08 to 4.58 in 3.7 days(see figure 1, Webber and McDonald, 2013a). In figure 2 of Webber and Quenby (2014) itis seen that these data correspond to the final, step like GCR intensity increase before agalactic value is reached and that it is preceeded by two fluctuations of similar magnitudeover the previous month. These two preceeding flutuations do not reach the galactic GCR 3 –value. They could be regarded as two Forbush Decreases exhibiting rapid recovery of theflux to pre-decrease levels or above. If this structure in the GCR intensity is convected pastVoyager 1 with velocity V in units of 100 km/sec, the spatial gradient is typically 0 . /V per AU. Since the spacecraft is moving at only 0.07 AU per week, it is very unlikely thatthe intensity structure is stationary in time. The regions of observed high spatial gradientextend back to about 120.5 AU to include the second region of high spatial gradient discussedby Webber and Quenby (2014). A significant observation by these last authors lies in therelative modulation of the 200 MeV protons and of 10 MeV galactic electrons. Both speciesexhibit the step intensity increases but the relative modulation is higher by about a factortwo for the electron channelIn order to estimate the likely mean solar wind velocity close to the heliopause, we appeal tothe models provided by Opher et al.(2012) who provide alternatives, based upon whether ornot reconnection in the sector region is included. Some verification is provided by using theLow Energy Charged Particle (LECP) Voyager 1 results to deduce the radial and tangentialflow components in the absence of a working plasma probe. Unfortunately the numericalresults need scaling from a modelled heliopause at 162 AU. The Opher et al. (2012) flowvectors are given, both at 120 AU which is within 2 AU of the heliopause and at 110 AUfor the radial, tangential and normal components in Table 1 for alternative models, eitherignoring the low-latitude sector structure or taking into account reconnection associated withthe sector structure.The LECP keV data is given at 110 AU for two components. It appears that the ex-periment seems to agree better with the sector model for V R and the non-sector modelfor V T . However, the data exhibits very large fluctuations throughout the heliosheath.Negligible LEPC data is available for the normal, V N component which clearly becomesthe dominant steady flow at these distances. We estimate the measured V N (110) as theTable 1: Opher et al. (2012) Models and Voyager 1 Flow data in the Heliosheath in km/secSource V R (120) V T (120) V N (120)No-Sector 44 -30 68Sector 4 -5 30Data V R (110) V T (110) V N (110)No-Sector 52 -71 50Sector 12 - 5 15Data 22 -36 (29) 4 –mean ot the other two components. Extrapolation of the experimental results suggests that V R (120) , V T (120) ≤ ±
20 km/sec.Magnetometer data obtained by Burlaga et al. (2013) during the heliopause crossing enableus to estimate the power in the fluctuations of the magnitude of B . These authors observea period of outward pointing polarity in 2012 from DOY 150 to DOY 171, inward polarityfrom DOY 176 to DOY 202 and outward polarity from DOY 204 DOY 238, the last periodincluding two intervals where post-heliopause conditions are apparently encountered. Usingdata from figure 2 of Burlaga et al. (2013), we give in Table 2 the mean and standarddeviations of the fields in these 3 periods, neglecting the post-heliopause data. The quotedexperimental errors are ± . × − Hz is found in the Voyager 1 data at 110 AU. If the field is convected past thespacecraft at a relative speed of 46 km/sec, as suggested by the data, the correlation lengthis 1.7 AU, taking into account the Voyager 1 velocityWebber and Quenby (2014) show large changes in the magnetic field energy density wellcorrelated with the large fluctuations in GCR intensity during the period day 208 to day240. This observation provides indirect evidence of significant change in plasma velocity.Quenby and Webber (2013) have suggested that transient radial plasma flow velocities nearthe boundary, triggered by large changes in plasma pressure, could reach the sound speedobtained by Borovikov et al. (2011) in simulations of post terminal shock conditions. Theirvalue is 228 km/sec. The attainment of this speed out to near the heliopause depends onthe plasma temperature not decaying significantly. If the Alfven speed becomes the domi-nant fast mode speed, a simple estimate can be made based on the continuity of mass flow, nV r = constant where n is plasma number density at radial distance r. Using the observedfield and the post-shock and non-sector velocities of previously mentioned models, we findan Alfven speed of 31 km/sec at 120 AU.Table 2: Voyager 1 Field Magnitude and Standard Deviation, sdPeriod DOY 150-171 DOY 176-202 DOY 204-238B, nT 0.182 0.339 0.225sd, nT 0.0690 0.0596 0.0735 5 –
3. DIFFUSION MEAN FREE PATHS
To carry out our calculation, we need to estimate possible values of the diffusion coef-ficients parallel and perpendicular to the mean field. We will use a theoretical formulationwhich has achieved reasonable agreement with experimental results in the inner Heliosphere,but employing field data obtained far out in the Heliosheath. The waves in the field modelare composed 80% of a 2-dimensional component with fluctuation vectors perpendicular toboth the mean field and wave propagation direction and 20% of a slab component with fluc-tuations perpendicular to the mean field but with wave propagation along the mean field.As derived by Le Roux et al. (1999), this composite field model yields a parallel diffusionmean free path λ || = 2 . B b x,sl ( PcB ) / λ / sl × F (1)where P is particle rigidity, B is the mean field, b x,sl is the x component of the slab fieldfluctuations, λ sl is the correlation length of the fluctuations, which are assumed to be validfor all components and F is function very close to unity in the present application. Pei etal. (2010) employ a very similar result to model the parallel diffusion coefficient throughoutthe Heliosphere.For the perpendicular diffusion coefficient, we follow the Le Roux et al. (1999) non-perturbativeapproach where D sl = 12 λ sl A sl ; D D = λ D A D (2) D sl describes the magnetic field wandering due to slab turbulence with A = b x,sl /B witha corresponding correlation length λ sl . D D denotes the magnetic field line wandering for2-D fluctuations. A D is the amplitude for this turbulence and λ D is the correspondingcorrelation length perpendicular to background field.We note that if the contribution of the 2-D fluctuations is ignored, the perendicular diffusioncoefficient becomes the expression given in Jokipii (1971) K ⊥ = v P xx (0) B (3)where P xx (0) is the power in one slab component at zero frequency. If however, 2D turbulencedominates, we find from le Roux et al. (1999) the modified value K ⊥ , D = 12 vλ D A D (4)Le Roux et al. (1999) also consider the quasi-linear model of Chuvilgin and Ptuskin(1993) whih also takes into account resonant and nonresonant interactions. These last au-thors derive K ⊥ = 0 . A K || (5) 6 –where A is the total fractional deviation in the field. We neglect a relatively unimportantadjustment to the numerical value given in this last equation to obtain agreement with somenumerical simulations performed at 1 AU.In the following section it will become apparent that the simple, Jokipii, (1971) expres-sion is more likely to satisfy the proposed modulation model. In Table 3, values of paralleldiffusion coefficients and mean free paths are calculated from the data of Table 2, using theLe Roux et al. (1999) expression, equation (1), while perpendicular diffusion coefficientsand mean free paths are obtained from the same data using the Jokipii (1971) expression.To estimate the power in the slab component, ie that due to wave propagation along themean field direction, it is assumed that the dominant fluctuation power is transverse. Tofirst order, sdB = 12 δB ⊥ B (6)and if only 20% of the standard deviation results from the slab-like fluctuations, then P xx (0) = 0 . λ sl π δB ⊥ λ sl = 1 . λ ⊥ ∼ − AUfor 100 MeV protons far out in the heliosphere based on models for the development ofturbulence.Using the 2D turbulence model as expressed by Le Roux et al. (1999), we obtain K ⊥ , D =4 . × − AU /s or λ ⊥ , D = 1 . K ⊥ , D = 1 . × − AU /s or λ ⊥ , D = 3 .
4. A TRANSIENT 3-D MODULATION APPROXIMATION
In the first model of the approximation of this section the aim is to attempt to explain theobserved large radial gradients near the heliopause in terms of transient radial velocities andTable 3: Parallel and Perpendicular Diffusion Coefficients and Mean Free Paths derived fromTable 2 Data for 200 MeV protonsPeriod DOY 150-171 DOY 176-202 DOY 204-238 K || AU s − . × − . × − . × − λ || AU 19.6 34.3 21.2 K ⊥ AU s − . × − . × − . × − λ ⊥ AU . × − . × − , × − x in the radial direction y in the azimuthal direction and z to complete the right handed set, the Fokker-Planck equation in the Sun’s reference framefor the differential number density U ( r , t ) is (eg Quenby (1984), ∂U∂t + ∇ S + ∂∂T ( dTdt ) U = 0 (8)where the streaming is S = C V U − K. ∇ U (9)and the Sun frame kinetic energy, T, loss rate is dTdt = V . ∇ ( αT U α = ( T + 2 m ◦ ) / ( T + m ◦ ). m ◦ is rest mass and the Compton Getting factor C is C = (1 − U ∂∂T αT U ) (11)The slope of the energy spectrum and α are assumed to be constant over the limited regionof application of the above Fokker Planck equation, thus both C and the pressure term in theexpression for kinetic energy T are independent of position. The spectral slope at 200 MeVis obtained from Stone et al. (2013). Resolving components, we notice that perpendiculardiffusion applies in the x and z directions while parallel diffusion applies in the y direction.We also allow the transverse diffusion coefficient to yield a drift velocity V d,z in the directionperpendicular to the plane of the spacecrafts orbit due to the large scale Parker field structureand a possible drift V d,x due to a field gradient in the z direction arising from lack of symmetryabout the equatorial plane. The steady state Fokker-Planck becomes K ⊥ ∂ U∂x − (1 . V x + V d,x ) ∂U∂x + K || ∂ U∂y − . V y ∂U∂y + K ⊥ ∂ U∂z − (1 . V z + V d,z ) ∂U∂z = 0 (12) 8 –Neglecting spatial variation in V and K and specifying U ◦ as the differential number densityat the origin of coordinates, that is at the bottom of the region of sudden intensity increase,a seperable, trial solution results in U = U ◦ exp ( (1 . V x + V d,x ) xK ⊥ + 1 . V y yK || + (1 . V z + V d,z ) zK ⊥ ) (13)Hence the solution tracks the initial, outward rise in intensity. U ◦ is determined by theboundary conditions on the heliopause and the relative ease of entry of GCR along the threespatial directions. However this work is not attempting a complete 3D solution, especiallyas the plasma parameters along the three directions are poorly known. Rather we pointout the expected result that a local radial, x , directed gradient requires a matching radial,outward directed plasma flow. In the absence of a suitable drift velocity, such a flow doesnot exist, close to the heliopause in the time independent Heliosphere model of Opher et al.(2012) Therefor, radial gradients close to this boundary may require transient radial flows.We first explore a ’drift free’ model. To obtain modulation below the GCR level near theheliopause, sudden bursts of radial plasma velocity need to appear sufficiently frequently sothere is always one event within about 1 AU of the boundary if the series of steep depressionsshown in Figure 1 of Webber and McDonald, (2013a) are to be explained. It is within thisregion that the intensity switches from near the GCR level to a modulated level. Anylesser occurence frequency would mean a further encroachment of the GCR level within theheliosheath.As source of the transient radial plasma flows, we postulate that the radial plasma flowcorresponds to the flows required by Quenby and Webber (2013) who suggested that theheliopause flapped with the speed of the fast mode, here identified as the sound speed inthe hot solar wind plasma. The flapping could be caused by pressure imbalance with theexternal medium as the solar wind plasma pressure varies.In order to see if the diffusion and flow parameters discussed in sections 2 and 3 can satisfythe observed gradient employing the transient modulation model, we find the radial cosmicray gradient from the adopted solution as given by K ⊥ ∂U∂x = 1 . V x + V d,x (14)On DOY 240, 2012, ∂U∂t s = 0 .
032 per day is the rate of intensity change in the reference frameof V1. In the adopted model DOY 240 is regarded as corresponding to the end of a fieldstructure convected past V1 as the heliopause expands. The gradient then refers to the entryof particles into the region of reduced intensity by diffusion in competition with the windoutward sweeping. Since the measurement of the actual spatial gradient is determined bythe adopted radial wind speed, the above equation can be written as K ⊥ = 1 . × − V x, + 1 . × − V d,x, .V x, AU s − (15) 9 –where V x, and V d,x, are respectively the radial and drift velocities in units of 100 km/sec.Neglecting the addition drift term in V ,d,x on the right hand side of the above equation,if the radial flow is ∼
100 km/sec, the estimated values of K ⊥ found in Table 3 seem tosatisfy the measured gradient. Suitable flow values are the sound speed estimated frompost terminal shock conditions or the Opher et al (2012) model which ignores the sectorstructure. However, this second, model estimate does not provide transient effects. Modelsfor K ⊥ which add 2-D turbulence onto the slab turbulence field line wandering so as to yieldthe dominant effect are less likely to satisfy the cosmic ray data.The average drift motion at the epoch of observation is northward for positive chargedparticles in the northern solar hemisphere because the average solar field is inward at thetime The drift magnitde V d from guiding centre theory is V d = 1 ωR v = 2 vP c BR (16)where ω is the cyclotron frequency, P the rigidity and R estimates both the field line radiusof curvature and the fractional field gradient. Using the mean measured field at 120 AUand assuming R=120 AU, the northward drift speed V d,z = 54 km/sec. However, transientchanges show both a field line polarity switch and a field magnitude gradient which is positivein the outward direction (Burlaga et al., 2013), thus reinforcing the northward proton drift.Hence modulation can be enhanced in the northward direction, as compared with the solarwind sweeping.For charge e, the field gradient and curvature drift are aligned in the e Bx ∇ B direction.Hence for protons, an episode of outward directed north hemisphere B together with alimited region of ∇ B directed in the − z direction produces an outward, x directed drift.According to the Opher et al. (2012) model with the sector structure included, gradients( ∇ B ) /B ≈ . − z direction on the edge of the sectorregion. This would lead to a radial drift of 115 km/sec, outward for outward north hemispherefield epochs. Neglecting for the moment the evidence provided by the electron modulation,one may develop the idea of a drift dominated flow. Burlaga et al. (2013) observed anunexpected field increase close to the heliopause. Averaging over a time period equivalent toa distance scale of 6 cyclotron radii for a 200 MeV proton, a field gradient ( δB ) /B ≈ . − z direction, perhaps as a transient effect, a radialdrift of 4900 km/sec would be possible. The effect of adding these various values of outwardradial drifts on the required value of K ⊥ can be seen by employing equation (15). Asbefore, it is assumed that the end of the passage of a field structure on DOY 240 is beingconsidered. Additionaly a typical plasma motion ∼
100 km/s is used in order to estimatethe spatial gradient. We find K ⊥ = 6 . × − AU s − is allowed if the field gradient is0.78/AU directed southward. However, from the equation for the drift speed, it is foundthat electrons of 10 MV rigidity drift two orders of magnitude slower than 200 MeV protons 10 –and are therefore relatively insensitive to field gradient effects. Since the electron modulationis greater than that of protons, it is very unlikely that drift can dominate the modulationnear the heliopause. This is because the electron scattering mean free path at 10 MV isunlikely to show a large reduction compared with that of a 200 MeV proton. In any case,there still seems to be shortfall by a factor near 10 as compared with the estimated 2Dturbulence model for perpendicular diffusion. It is perhaps significant that two periods oflarge radial gradient seen by Webber and McDonald (2013a) centred on DOY 128 and 209are in periods of outward pointing field.Turning to the GCR sudden increase starting on day 128, the lack of detailed correlationwith change in magnetic field amplitude led Webber and Quenby (2014) to suggest the causelies in structure ∼ − z directed field gradientto allow enough outward GCR motion to counter inward diffusion. The second requirementis that the scale size of the changes to the plasma parameters close or at the heliopauseprovide a depletion region of GCR intensity where particle leakage from the y , z directionsis slower than the current due to the outward convection or drift velocities. If our model isreasonable, the ocurrence of the day 128 increase is both evidence for dramatic movementat or within 1 AU of the heliopause and of a scale size of the movement in directions parallelto the heliopause of several AU.
5. MEAN FREE PATH RIGIDITY DEPENDENCE
Webber and Quenby (2014) discuss the rigidity dependence of the modulation duringthe period of the two increases, days 128 to 238. A problem arising from the data analysisreported by these authors which the work presented here has not explained is the lack ofapparent charge splitting in the relative modulation suffered by protons and alpha particles.This splitting was first predicted by the force field approximation of Gleeson and Axford(1968). However, unlike the previous authors, we do not depend either on assuming no netstreaming in a particular direction or on integrating the solution between the spacecraft andthe boundary of modulation. Instead, we simply consider the local values of the intensity 11 –gradients during the increases. This however limits the discussion to the relative changes inelectron and proton intensity, these being the species for which we have detailed information.From equation (13), the ratio electron to proton gradients in the x direction if V x >> V d,x is1 U ◦ ( P e ) ∂U ( P e ) ∂x / U ◦ ( P p ) ∂U ( P p ) ∂x = K ⊥ ,p ( P p ) K ⊥ ,e ( P e ) = β p λ ⊥ ,p ( P p ) β e λ ⊥ ,e ( P e ) (17)where P e , P p and β e , β p are the magnetic rigidities and velocities of the electron and protonsobserved. From Fig 2 Webber and Quenby, (2014), the ratio of the fractional electron toproton intensity change measured at the step change around day 210, 2012 is 2.6. Takingthe electron mean energy as 10 MeV and the protons as corresponding to 200 MeV, a meanfree path dependence λ ⊥ ∝ P . is required to fit the diffusion of both species if the day210 change represents the ratio of electron to proton intensity gradients. For the day 128step, the gradient ratio is 1.6 and the exponent of the rigidity dependence is 0.29. . This fitfavours the rigidity dependence of equation (5) if equation (1) is used to define K || . However,the absolute value of the perpendicular diffusion coefficient is still more than two orders ofmagnitude too large to balance the expected transient convective radial flow.A drift dominated model seems excluded by the relative size of the electron to proton stepchanges and therefore cannot be used to discuss the mean free path rigidity dependence. Asa conclusion to this section (5) it seems that none of the expressions for the perpendiculardiffusion coefficient discussed simultaneously satisfy both the absolute magnitude and rigiditydepndence implied on either of the two transient modulation models suggested. A smalladdition to the Jokipii (1971), equation (3) expression, dependent on a λ || ∝ P / due tononresonant interactions would seem to provide the closest overall fit.
6. CONCLUSIONS
In the attempt to explain the unexpectedly large radial cosmic ray gradients observedaround 120 AU an appeal has been made to the possible existence of transient outwardplasma flows, approaching the sound speed in the Heliosheath. Interpreting the availablemagnetometer data to yield a suitable value of the perpendicular diffusion coefficient, it wasnecessary to employ a model of scattering dependent only on 20% of the measured fluctua-tion power. This power would constitute the percentage present in waves propagating alongthe magnetic field direction. The resulting modulation model is similar to that expectedfrom a succession of Forbush Decreases in the inner Heliosphere. Field gradient particle drifteffects can also play a role in modulation near the heliopasuse. If these are sufficiently large,additional diffusion based upon 2-D turbulant fields may explain the rigidity dependenceof the modulation. Transient changes in radial plasma speed or field gradient necessary to 12 –cause the observed sudden GCR intensity changes do not necessarilly need to be locatedat the spacecraft but may alternatively occur between the spacecraft and the heliopause.However, although we believe it worthwhile discussing the two transient modulation modelswe provide, problems arise in satisfying the observed rigidity dependence of the modulationin terms of available models and data relating to diffusion in the Heliosheath. Further infor-mation on the relevent plasma parameters, especially transient effects, at 120 AU is requiredto determine whether either of the proposed models will remain successful. Also furtherellucidation of the relation of magnetic field turbulence to the actual cosmic ray diffusionperpendicular to the mean field seems required.
7. ACKNOWLEDGMENTS
W R Webber thanks his Voyager 1 colleagues from the CRS instrument, especially thelate Frank McDonald and Project PI Ed Stone. Magnetometer data were obtained by LenBurlaga and Norman Ness.
8. REFERENCES8. REFERENCES