RADIAL VARIATION OF THE SOLAR WIND PROTON TEMPERATURE: HEAT FLOW OR ADDITION?

Transcription

1 The Astrohysical Journal, 809:16 (1), 015 August The American Astronomical Society. All rights reserved. doi: / x/809//16 RADIAL VARIATION OF TE SOLAR WIND PROTON TEMPERATURE: EAT FLOW OR ADDITION? J. D. Deartment of Physics and Astronomy, University of Iowa, IA, USA; jack-scudder@uiowa.edu Received 015 March 6; acceted 015 June 8; ublished 015 August 14 ABSTRACT The roton temerature rofile of the asymtotic solar wind lasma can be modified by four different hysical effects: PdV work, external heat deosition, D Q, divergence of heat, q, and collisional energy exchange with other secies in the lasma. Suggestions in the literature that DQ heating is required to exlain the roton rofile have often been deduced while neglecting q and energy exchange. Desite the adiabatic aroximation q = 0 having no rigorous justification in low-density lasmas, the simultaneous neglect of energy exchange unnaturally forces the need for DQ to balance adiabatic cooling caused by the wind s exansion. In this aer, the asymtotic wind roton heat flux is determined which balances the inner elioshere s steady state entroy equation for the rotons, ignoring heat addition and energy exchange. The solutions of the energy equation recover both the ower-law trend and amlitude of the 5 year averaged elios temerature rofiles that were segregated by seed. The dimensionless skewness of the heat flow is emirically shown to scale for all wind states below 600 km s 1 as if it were equal to the Knudsen number determined by Coulomb collisions, a relation that is rigorously demonstrated for an infinitesimal Knudsen number, bridging the unusual adiabatic rotons for U 50 km s 1 and the higher seed states of the winds that remain hotter over a wider radial domain. igher seed states (U > 650 km s 1 ) may require additional scattering beyond what Coulomb effects can rovide, although the averaged rofiles for these seeds are not as accurate as those below 600 km s 1. Key words: conduction equation of state hydrodynamics lasmas solar wind waves 1. INTRODUCTION From the earliest measurements, it has been suggested that the roton temeratures in the solar wind were not those redicted by a sherically symmetric thermodynamically adiabatic exansion, which has a rofile 43 T () r µ r -. In the initial interretive Phase 1, sherically symmetric two fluid models with Sitzer transort could not redict the ion temeratures observed at Earth for tyical wind seeds (artle & Barnes 1970). istorically such observations were interreted as evidence that the rotons were being heated by some external agent(s) not in the modeled equations, with wave dissiation suggested. The elios radial rofiles oened Phase of the interretations; they re-enforced the disagreement with the adiabatic redictions (Schwenn et al. 1981; Marsch et al. 198) and also noted the different radial behaviors of the wind as a function of the size of the solar wind seed, U. The longlived elios mission roduced over 5 years of data involving 34 searate traverses of 0.3 r 1 AU. In Phase, the inner helioshere s roton radial variation versus seed state was characterized by Marsch et al. (1983), Schwartz & Marsch (1983), Loez & Freeman (1986, 1987), Freeman & Loez (1985), ellinger et al. (011, 013), among others. Loez & Freeman characterized best-fit radial ower laws versus U using all 5 years of elios data as summarized in Figure 1. The most heavily oversamled regime was the km s 1 states. The reorted inverse radial ower-law exonent, g º- dlnt dlnr, ranged from the adiabatic value of g = 43 (indicated by the red dashed line) at the lowest seeds seen, to generally much shallower radial variations as U increased. The urose of this aer is to exlore the ossible range of conclusions that come from such inner helioshere observations.. PASES OF INTERPRETATION OF T P (R) Theoretically, the variation of T j (r) in the asymtotic wind regime (U U ) is shaed by 4 factors: (i) work done on its volume element P j dv (±); (ii) energy deosition/loss DQ j ; (iii) divergence of heat flux qj ; (iv) energy exchange n ij with other lasma comonents ±, including thermal forces. At the overview level, T (r) is less than 1% of the solar wind budget in a Mach 10 flow. Exlaining and recovering the radial variation of this small and decreasing fraction requires an accurate and comlete hysical icture that is likely to involve all of the factors listed above. At times, the roton (j = ) budget has been looked at alone, as art of the overall turbulent fluid, or as one of two fluid constituents of solar wind. Confusingly, at different times and in different aers, various subsets of these four factors for rotons (and sometimes for the electrons subset) have been exlored when interreting and modeling the solar wind; there has also been some evolution with time in the assessments of the imortance of the usually ignored terms. At other times, there has been no clear statement that the aer s conclusions were contingent on the omissions of these other factors, leaving the casual reader with the imression that the aer s stated conclusions have no other reasonable interretation(s). In setting the context for this aer, a brief overview of the tacit assumtions of various examles of aroaches to exlaining the roton s observed radial behavior will be resented. This discussion is not meant to review all aroaches, but to illustrate the current state of the analysis..1. Phases 1 3 Some Phase 1 interretations roceeded from the observation that exansion cooling (effect (i)) did not exlain the observed T (r), and so heat addition (effect (ii)) is required. Other interretations in this hase included two fluid modeling 1

2 The Astrohysical Journal, 809:16 (1), 015 August 0 Figure 1. Summary of least squares fits to radial ower laws, T µ r - g, for elios ( ) as a function solar wind seed state, U (Loez & Freeman 1986). Error bars and latest udated values are used (Loez & Freeman 1987). Intervals of ±50 km about the labeled seed were used for this analysis. Uncertainties indicated reflect the seed intervals and the reorted errors for the exonents (Loez & Freeman 1987). including effects (i), (iii), and (iv), using Sitzer closure for rotons and electrons, but neglecting comressive PdV effects to reach a similar conclusion. In hindsight, such conclusions are arametric (in the first instance) to being able to exclude the occurrence of effects (iii) and (iv), or(in the second instance) to authenticating the correctness of the Sitzer transort used for both electrons and ions, or (in either aroach) to excluding the role of ositive PdV work in the fluid arcels being modeled. Phase analysis incororated the fitted radial ower-law variation fits to the trend of the roton heat fluxes determined from elios data, q () r, and six other arameters in the internal energy equation, but ignored the dv>0 effects of stream dynamics in (i) and did not consider thermal force forms of energy exchange, enroute to concluding that (ii) the data still require external heating. Conclusions of this tye are arametric in the assumtion that the...model indeendent... characterizations of the elios roton heat flux were sufficiently recise to leverage their conclusions. We return to this assumtion below. Phase 3 of the exlanation of T (r) has centered on the ossible role of turbulent dissiation in the roton heat budget. Three branches of investigation have made contributions: (1) one branch has focussed on modeling the observed features of the roton temerature anisotroy, attributing it to various daming effects D Q (e.g., Isenberg & Vasquez 007; Chandran et al. 011) of an imosed wave source; () another branch has estimated the energy migration via turbulent cascades to shorter scales with a lumed arameter descrition with a significant list of free arameters that assumes the rotons are the ultimate recetacle of the energy cascade generated internally by the solar winds (e.g., Zank et al. 1996; Smith et al. 001). The dominant sources of turbulence are modeled from estimates of the lumed arameter effects of wind shears and ick-u ions. (3) A third branch has used emirical electron heat flux closures to estimate how the resumed DQ energy introduced to the exlain the T (r) rofile would need to be aortioned between electrons and ions (Breech et al. 009; Cranmer et al. 009). All but one of the cited aers in the Phase 3 discussion have ignored (a) the role of the roton heat flux or argued that the roton heat flux was ignorable after first modeling the skew of the VDF as being caused by a delta function moving at the Alfvén seed along bˆ; the one aer that formally retained the roton heat flux did so with a local closure aroximation much in a local form of Grad, while also retaining the Sitzer formulation in the lowest corona for the electron heat flux well outside its domain of validity (see & Karimabadi 013). The aers of the Phase 3 discussion also (b) have neglected the role of comressive heating, (i), in the trended data used for closure or constraint, (c) have ignored the thermal force-tye exchange terms while sometimes choosing ad hoc energy exchange rates, and (d) have noted, but otherwise ignored, the ossibility (Dmitruk et al. 004; Karimabadi et al. 014) that the turbulent cascades might be interdicted before deositing their energy in the roton heating and recent evidence (Marino et al. 008) that the cascades are not omniresent as initial modeling resumed. Branch and one of the 3rd branch aroaches of Phase 3 rely on a system of lumed arameter equations of turbulence transort that contain a substantial number of free arameters which are adjusted to reach their conclusions. Among these arameters are those which mock u the amount of shear resent in the system, its symmetries, the rate of local roduction of newborn ick-u ions feeding the turbulence levels, as well as boundary conditions for the domain being modeled. The ick-u effects according to this modeling seem to be most ronounced beyond 0 AU. The other driving mechanism of the turbulence icture is shear, which is most ronounced with the wraing caused by stream dynamics outside of 1 AU. Among the estimates of the second branch is a semi-emirical digest between 3 and 4 AU that estimates the energy transfer rate of the cascade to the lasma (Marino et al. 008). In addition to suggesting that the cascade could not comletely suort the simultaneously measured roton temerature (even if it all went directly there), they also indicated that the energy transfers attributed to the cascades were not always resent. The most detailed and extensive radial analysis has been erformed for the Voyager intervals, where incidents of ick-u ion effects have been identified beyond 0 AU, where T (r) is seen to increase with radius, using careful baselining with 1 AU observations. The ick-u ions drive turbulence in this regime, but turbulence caused by shear would aear to be inadequately sulied in the inner helioshere to be effective and is thus most effective after the stream stream dynamics start wraing u their characteristics, and successive streams and ejecta start overtaking one another beyond 1 AU. There aears to be a desire in the literature to homogenize all of the radial domains of the solar wind s roton rofile as being determined by wave-driven turbulence, desite the almost certain diminution of the effectiveness of ick-u ions and stream driven shears inside of 1 AU. In the inner helioshere alternate mechanisms have been examined (e.g., Cranmer et al. 007; Isenberg & Vasquez 007; Chandran et al. 011), usually assuming a osited wave sectrum of wave ower at the base of the corona. While the roton heat flux (effect (iii)) is formally resent in Chandran et al. (011), it is based on a low-order truncation much in the same sirit as Sitzer s exansion, excet here it is about an anisotroic ion distribution. Although the roton heat flux is...formally..

3 The Astrohysical Journal, 809:16 (1), 015 August 0 resent, its hysical accuracy is unclear for the same reasons that Sitzer roton transort would be inaroriate. In other references cited (including in the outer helioshere), this roton heat flux cometition has not even been considered as a ossible cometitor for the turbulence exlanation via effect (ii). Inner elioshere Phase work merged theoretical arguments and radial ower-law fits to observed lasma moments in order to investigate the balance of the roton energy equation and infer, by subtraction, the missing ingredients that such an analysis suggests must come from DQ mechanisms that involve heating or cooling of the rotons. These calculations aear at first insection to involve a comlete attemt (excet thermal force effects and comressive effects) to look at the balance of the roton internal energy equation. owever, as with any exerimental arguments, all of these conclusions are arametric in the assumed accuracy of the key observable quantities that rovide the leverage for the final scientific evaluation. The roton heat flux q (r) lays this role in the conclusions reorted by Marsch et al. (1983) and ellinger et al. (011, 013). As argued in this aer, the systematically low value of the elios determination of the roton heat flux, q () r, leaves even the structurally comlete forms of Phase analysis with no clear conclusions... Phase 4 T () r Determined by the Proton eat Flux? While the Sitzer Braginskii formulae for heat flow are readily available for electrons and ions, it has taken some time to internalize their not being alicable anywhere in the elioshere. Several models retain the Sitzer form for low altitudes in the corona, which arguably even has the wrong sense; in so doing, it builds in a significant conduction loss from the coronal maximum layers back down to the transition region. Even though these redictions of heat flow are now known to be unreliable, there remains the imression that their magnitudes can still be used to estimate their imortance. The theoretical assumtion underlying Sitzer Braginskii is that the Knudsen number, K n, of the secies is erturbatively small: Kn 1, which is vacated within DR 0.05R above the transition region ( & Karimabadi 013). Sitzer s estimate for q e is structurally as well as quantitatively inadequate. Sitzer s estimate of the size of q is much more inaroriate and inaccurate, since the the roton K n, is so much larger than even the electron K ne,, and thus further away from the erturbative values required for Sitzer s analysis. Nonetheless, the small value of Sitzer s formula for the roton heat flow has caused many to ignore the otential role of the ion heat flux in the variation of the ion temerature rofile in the solar wind. Others have demonstrated that the roton heat flow in a collisionless suersonic flow might be negligible, starting with Maxwellian boundary conditions (ollweg 1971; Schulz & Eviatar 197). Still others have looked at the role of the seed deendence of binary collisions in controlling the skewness (Livi & Marsch 1987), and others have ignored Coulomb effects altogether considering other effects to dominate the mean free ath for rotons, esecially in the far reaches of the elioshere (Williams 1995). This aer exerimentally looks at the ossibility that the roton heat flux imlied by the well-documented elios temerature variation summarized in Figure 1 could determine the size of the roton heat flux, assuming that divergence of the roton heat flux (effect (iii)) is the only cometition for the adiabatic cooling of effect (i), while exlicitly assuming that there is no DQ effect involved and that collisional exchange effects (iv) for the rotons are negligible. The likelihood that nature behaves in this way in the inner elioshere is then tested in the three distinct ways discussed below. While TE comaring the resent solutions q () r with the emirical rofile reorted by elios, q () r,astrong systematic shortfall was identified in the elios determinations of q () r. This systematic finding concerning the elios heat flux imlies that those arguments constructed under Phases or 3 that relied on the insufficiency of the elios-determined q (or ignored its contribution) to reach their ublished conclusions concerning heat addition are no longer valid. 3. MODEL FOR TE SOLAR WIND VARIATION OF T (, r U) To illustrate the ossible effects of roton heat flux, in this section, we develo a model for the ion heat conduction and illustrate its use in the ion energy equation. The model for the ion heat flux takes the form q = nmw bˆ, (1) 3 where, n, M, and w are the skewness, the roton mass, the density, and their rms thermal seed, resectively. This form reresents an energy density of the order of the ion ressure making rogress in the rest frame with a roagation seed of the ion thermal seed as amlified by the skewness. Some might refer to Equation (1) as the saturated heat flux form, however, the situation is actually the reverse: the saturated form starts from a dimensional argument like Equation (1) and then attemts to suggest the size of based on various surmises and, in some cases, calculations concerning instabilities in the resence of certain model forms for the distribution functions. Any roton heat flux can be written in this form with being a dimensionless number that is to be determined. By adoting this form, there is no resumtion of which stochastic effect(s) control the size of q. It is clear that 0 is the condition that attends thermodynamic adiabaticity. Geometrically reflects the average skewness of the velocity distribution in the roer frame where there is no mass flux. When the heat flow aroaches zero, the skewness, or earshaed nature, of f ( w = v - U) goes to zero. Inserting the general moment form for the heat flux into that of Equation (1), we obtain ( ) 1 q M v U bˆ = - v - U º Mnw 3, () which leaves our dimensionless skew,, given by ( ) 3 æ w öæ v U bˆ v U v n f = ò () èç ø w èç öæ 3 dv ö. (3) 3 w ø èç ø Equation (3) shows that the motivation for the form of Equation (1) is to measure arts of the integrand in dimensionless thermal seed units, that is, h = v - U w. If the seed range of suort for f ( w) is of order w,asina Gaussian of half width w, the skewness can be exected to be 3

4 The Astrohysical Journal, 809:16 (1), 015 August 0 small, since the integrand is then the roduct of several factors that are all small relative to unity. In textbook kinetic regimes, is small because it is roortional to the small exansion arameter of these theories, which invariably exand f ( v) about local Gaussians. For future consideration, it is well to notice that in such regimes where Sitzer formulations for q j are valid, it may be rewritten in the form of Equation (1): q nkt t B = 3.9 T = O(1) KnMnw 3, (4) M where t is the collision time, Kn º lmf L is the Knudsen number for the collisions that isotroize the distribution, and lmf = tw. Equations (1) and (4) suggest the close association between skewness and Knudsen number that can exlicitly be checked in the Sitzer Braginskii limit: µ K. (5) n This insight is more general than the Sitzer Braginskii formulae, suggesting that the controlling asects of skewness are the controllers of the Knudsen number. Even the collisionless solar wind has some binary collisions that limit its Knudsen number until other, more effective scattering agents are identified. Whatever sets the realized free ath for momentum transort will shae the skewness, and hence the heat flux. Because the roton Coulomb Knudsen number can be large, there is every reason to exect that the skewness can be much larger than those regimes of textbooks that insist (for their mathematics) on Kn 1. Rather than solve the kinetic equation to redict, we adot Equation (1) ensuring that its units are correct and the roduct of two terms: one that summarizes collisionality via the skewness, and the other that aroximates the energy flux density available to carry the heat. In this sense, this modeling does not have a secific transort agent in mind, just that it is very hard to sto such an energy flow, other than when the scattering hysics effectively makes the mean free ath very short comared to the scale lengths of the fluid. It is also ossible that a mixture of scattering rocesses, including waves, set the size of. owever, a baseline maximum of K n is rovided by Coulomb collisions (that never disaear) with ossible effects from waves suerosed, not necessarily as suliers of heat but as scattering agents moderating the heat flow that would otherwise occur with articles following only Coulomb interdicted collisionless trajectories as discussed by Livi & Marsch (1987). The emirical aroach of this aer u to this oint, using Equation (1) and the elios rofiles, does not commit to a secific rocess. We show below that when is inferred from the 5 year averaged elios data, it is an orderly function of bulk seed (see Figure 4) After we demonstrate that the roton heat flow might be attractive to exlain the reorted radial variations in Section 4, we discuss in Section 5 the theoretical and measured indications that q is too small to be of consequence. 4. SOLUTIONS We model the wind in the elios radial range as having a constant seed U o. By ignoring couling to electrons and external ion energy sources in the asymtotic uniform flow Figure. Numerical solutions of solutions to Equation (7) for a range of values for a fixed assumed thermal mach number = 10 at 60R. Numerical solutions closely follow the small argument ower law behavior of Equation (6). Digests of ower law behavior for all and variations are dislayed as searate symbols in Figure 3 below. regime, the ion internal energy equation takes the form dt dr 4 T =- - q, (6) 3 r 3nk U where k B is Boltzmann s constant and U o is the assumed asymtotic wind seed. After using Equation (1) for q and treating as constant (that ossibly deends on U o ), and the magnetic field in the radial direction, Equation (3) has the solution T () r ln T ( b) where o and where B o é æ T r r ex 1 ln () ö 4 + b 1 ln, (7) ëê èç To( b) ø - ù =- ûú 3 ro = Uo wo b º, (8) is the ion thermal mach number at ro, where wo º w( ro). We consider below the range of thermal mach numbers Since and are both assumed constant along a U o stream line, the only factor that controls the solution of Equation (6) is b. From Equation (6), it is clear that 0 yields the collisional fluid s adiabatic r -43 rofile, while the heat flow imlied by a finite ositive roduces shallower radial decreases. In the exonential s small argument regime, we obtain the near ower-law form that most roton modelers have used to fit the elios observations: 4 æ ro ö3+ 3b T() r ~ To( b). (9) èç r ø The radial rofiles for the full nonlinear solutions of Equations (6) and (7) are shown in Figure in a log log format that emhasizes the near ower-law behavior of this solution over the relatively narrow radial range surveyed by elios; searate curves have been determined for in the range 0 < < 4.0 assuming = 10. As exected, the rofiles with very small tend toward adiabatic behavior, 4

5 The Astrohysical Journal, 809:16 (1), 015 August 0 Figure 3. Diamonds reresent inverse radial ower law exonents for the theoretical solution to Equation (7) with henomenological heat fluxes, for different combinations of assumtions of and. Diamonds in smooth arcs are results of different for fixed, assumed. A modest range of can exlain the range of reorted roton radial exonents, in the resence of roton heat flow modeled as in Equation (1). while rofiles with larger values of have increasingly flatter rofiles with radius. The diamonds in Figure 3 reflect the ower-law sloe for individual solutions to Equation (7) versus assumed values of ; seven choices of [4, 5, K9, 10] at ro = 60R generate seven families of curves. The colored horizontal lines reflect the best-fit owerlaws(by seed range) determined from the fits to elios data (Loez & Freeman 1986, 1987). Each horizontal line is labeled with the center solar wind seed used for the fit; fit ower-law uncertainties are indicated with flags of the same color as the horizontal line for their best-fit value. The henomenological heat flux of Equation (1) can reroduce the exerimental elios ower laws at different wind seeds rovided a necessary deendence of ( Uo ) is chosen; its variation (averaged over the Mach number at fixed seed state) is shown with red diamonds in Figure 4. The horizontal error flags reflect the seed ranges of the fits done by elios. The vertical error flags reflect the imact on the skewness estimates caused by suosing different roton mach numbers at 60R, together with the uncertainty of the best-fit elios exonents. As exected, the slow wind is almost adiabatic (with (50) 0), while the higher wind states are suggested to have increasingly skewed distributions. It is imortant to remember that is an emirical arameter determined from assuming Equation (1) and solving the energy equation so that the redicted rofile of T TE () r agrees with the logarithmic derivatives of Figure 1 inferred from elios data. Other than assuming to be constant along the stream lines of constant bulk seed, no other assumtion has been made about the scattering agents that determine its size Does Emirical Make Sense? As a baseline for interreting Figure 3, we consider the scaling of if Coulomb rocesses were controlling the Knudsen number. The temerature and density control lmf and the scale length L is roortion to the radius. It is well known that the roton temerature and bulk velocity are strongly correlated as T 1 µu (Burlaga & Ogilvie 1970), while in the asymtotic wind nr = C so that for Coulomb Figure 4. Diamonds: inferred deendence ( U) vs. solar wind seed state required to reroduce the elios 5 years average radial variations (Loez & Freeman 1986) within their errors (Loez & Freeman 1987). Diamonds reflect average values of considering a range of ossible roton mach numbers assumed at 60R ranging from 4 10 and the reorted statistical uncertainty of the ower law exonents. Blue curve shows the indeendently estimated scaling of ( Uo ) imlied by the Coulomb scattering l mf µ T n and conserved mass flux deendences: l µ. scattering we can estimate K n, mf Uo 4 lmf T 4 = µ µ ru o. (10) L nr Since U o varies by more than a factor of three across the elios data set, it, rather than r, controls the variation of K n, across the elios orbit by a ratio of 7:1. If only Coulomb effects were resent across all seed states, then these arguments would suggest that a strong U 4 o deendence should be recovered from the theoretical modeling discussed above. Motivated by our discussion of K n, in Figure 4 we have over lotted the Coulomb form from Equation (10) via the relation µ ( UU * )4 over our direct determinations of ( Uo ) (red diamonds), which were obtained from the energy equation and elios data without committing to the actual scattering agent. The lowest four seed states of the six examined fit this relation nearly erfectly (with U * 350 km s 1 ), comatible with the rogression toward adiabatic behavior as U o 50 km s 1. In the two highest seed states that have been summarized, a reduction from the Coulomb allowed skewness, or at least a leveling off, is required to exlain the radial gradients. This might be a regime where the waves that are resent lay a role in throttling the otherwise large skew by the scattering they might cause. Note that such scattering need not be interchangeable with suggesting the waves as an energy source. A ossible caveat must be considered since Loez & Freeman exlicitly make note of strong oversamling in the data between km s 1 ; one thing that can occur if many of the samles in these radial buckets are from a few of the 34 elios I and II traverses is that the reorted radial ower law fits (cf. Figure 1) may not be as free of comressional PdV effects or other transient henoma as in the more oversamled regions (cf Figure 1 in Freeman 1988). These ossibilities do not detract from the strong and nearly erfect suort of the heat flux hyothesis for Coulomb moderated skewness shown by the U o < 600 km s 1 regime, which is the most oversamled 5

6 The Astrohysical Journal, 809:16 (1), 015 August 0 Figure 5. Comarison of radial ower law exonents (a) and amlitudes (b) for the elios heat flux rofiles and those found to be needed via Equation (1) to exlain the long term elios radial ower law behaviors of roton temerature. and, arguably, the elios data most free from comressive and transient henomena that our sherically symmetric time stationary modeling would not accommodate. 4.. Do Theory and Observations of q () r Agree? dlnt If the observed temerature rofile has g ( Uo ) º- dr, then the roton heat flux imlied by Equation (1) should have a radial variation 3 (3.:4) - q ( U, r) µ ( U) nt µ ( U) r, (11) where the shallowest gradient goes with the higher seed wind and 4 goes with the adiabatic slow wind. Note that the size of the heat flow deends both on the roduct of the seed deendent skew and the radial variation of the ion thermal seed. In the adiabatic regime, the vanishing of the heat flux comes from the extreme reduction of skewness caused by K n 0. As will become clear below, we are interested in searately comaring the amlitudes Q ( Uo ) and exonent gq when reresenting q () r as q ( U r) Q( U ) r gq ( Uo æ ) o ö o, = o. (1) èç r ø We will show that the logarithmic derivatives of q, i.e., gqs, are nearly the same between elios reorted values and the theoretical curves determined from the energy equation fits above. At the same time, we will document that the elios determinations of Q ( Uo ) s are significantly smaller than those from the model exlored in this aer Logarithmic Derivatives We fit the reorted elios binned variation of the roton heat flux with results shown in Figure 5(a) and (b). These owerlaw fits have been erformed using the generalized leastsquares aroach for errors in ordinate and abscissa discussed by Press(199). The log log fit addresses the bucket widths in radius for the elios resentation; as no errors are resented for the roton heat fluxes or their standard deviations (as they are bucket averages), we have assumed that the fractional errors for the heat flux are at least as large as those acknowledged for the temerature (Marsch et al. 1983). Actually, the heat flux determination as a cancellation is robably not as accurate as the even moments associated with the ressure. The temerature fractional error was reorted to be 30%, and for these fits we assume 50% uncertainty for the reorted roton heat flux. Below, we discuss the fact that the elios roton heat fluxes are systematically low by factors of 4 1 with tyical values of 7.8. This rescaling does not affect the best-fit ower-law index. At this adjusted level, the determination is assumed to be 50% accurate. The entire fit sace is maed and the minimum of c n is determined. The 68% errors for the intercet and sloe errors were determined by the extremities of the level surface of n c =.3. The stee sloe of the fractionally uncertain heat flow s variation with r, couled with the half decade of variation in log r and the uncertainty of the abscissa, consire to give the best-fit ower-law exonents a rather wide range of equivalent ossibilities. For the resent, we focus on Figure 5(a), which comares the best-fit ower-law indices from elios data and those that are required by our model in Equation (1). The fit sloes (i.e., exonents) are consistent with equality, with the best agreement occurring in the higher seed states where the modeled skew was higher. Figure 5(b) shows two traces for the amlitude Q o from fitting the elios data versus the amlitudes Q o which Equation (1) imlied from the heat flow that reroduces the observed radial gradient of the roton temerature. The dotted lines connecting the black diamonds are our best-fit amlitudes from fitting the unmodified elios heat flux integrals summarized in the literature (Marsch et al. 1983). These amlitudes are much lower than those exected for Equation (1) to exlain the temerature gradients. As we discuss in Section 5, there is a very strong systematic tendency for the elios heat flux to be smaller than what is actually resent in the medium. The shortfall is estimated to be of the order of 780%; that is, the true ambient heat flux is of the order of 7.8 times the inventoried value as a result of this systematic effect. The blue diamonds with orange error flags in the abscissa and ordinate in Figure 5(b) reflect the suggested revised values for the elios information, rovided the study resented next is truly reresentative of all of the situations surveyed by the resented elios data. While the most robable value is 7.8, this factor ranges from 4 1. With these considerations, the Q o amlitudes from the corrected elios data and the imlications of Equation (1) would be consistent. 5. MEASURING TE ION EAT FLUX The original elios analysis considered the role of the ion heat flux, q () r, as determined by numerical integration of the roton-extracted distributions and concluded that it was inadequate to exlain the observed rofiles of the ion temerature (Marsch et al. 1983). According to their resented best-fit curves with unknown treatment of errors or ambiguities in ln r, the heat flux was within a factor of 3 5 of being adequate to exlain the temerature variations and an order of magnitude deficient in low-seed winds. As we have just shown in Figure 5, the errors on the best-fit ordinates and abscissae are substantial and high-quality fits require that we inform the otimization rocess of the errors known to be in the data; these uncertainties were aarently not used in characterizing the elios assessment of the imortance of the heat flux. Nonetheless, as we have shown in Figure 5, more aroriate estimates of the heat flow rovide amlitudes (black diamonds 6

7 The Astrohysical Journal, 809:16 (1), 015 August 0 connected by dotted line) that are substantially below that needed to exlain the temerature variation roosed in this aer. owever, these fits did show that the radial variation needed for the resent aroach were consistent with the radial trends of the data Figure 5(a). We now look carefully at the elios aroach for determining the roton heat flux from their EZmeasurement by numerical integration EZMeasurements for Ions: Systematics that Imact Proton eat Flux Most 3D roton lasma detectors, like the elios sensors, are variants of detectors that filter charged articles by their energy er unit charge, EZ, and direction of arrival, rather than filtering the flux based on their mass or the charge state of the articles, such as is done with time of flight mass sectrometry. In the solar wind, the dominant ions are rotons and α articles with four times the roton mass and carrying two charges. When the rotons are measured in the solar wind within a narrow range of energy er unit charge E Z k of an electrostatic analyzer, there is an intrinsic degeneracy whereby different ions with different charges and otentially different masses can be collected in the same telemetry window of D( EZ) k, rovided that the ions arrive from the selected common bundle of directions in velocity sace within the field of view set by entrance aertures and deflectors. If the energy er charge is the same for two different charge states, then the acceted velocities of the two secies that are found in the same EZ bucket have the algebraic vector relationshi given by MZ j U + w = U + w MZ j ( j j), (13) where Uk and wk are the bulk velocity and random velocity vectors of the k-th secies. Since the bulk motions of different secies can only drift differentially along bˆ, this condition may be restated as MZ j U + w = ( U + D j, + w j), (14) MZ j where D j = Uj - U = z jbˆ is the field aligned sliage between commonly observed secies. Focusing on contamination from α articles, this condition becomes ( Da a) U + w = U + + w (15) or, assuming the roton bulk velocity is in the radial direction, and If w ( ) w = - 1 U rˆ + + w. (16) D a, = tw b ˆ, then the matching conditions become o ( ) tw = - 1 U cos c + z + w (17) o a, a, ( - 1)U bˆ = wa bˆ, (18) where cos c = rˆ b ˆ and α article gyrotroy has been assumed. Assuming the roton and α thermal seeds are equal to wo () r and isotroic, we can write these conditions in terms of a dimensionless thermal seed variables, n = and rotons, that is, and w j wo, of the alhas na, ^ 0.98 ( r)sin c( r) (19) n 0.707t ( r)cos c( r) - a, b () r n n a, (0) where gyrotroy has been assumed and the sliage between secies is assumed to be at the local Alfvén seed, VA = B (4 m( n + 4 na )) and b º 8nkBT B. The act of observation of solar wind ions through an EZ lens induces ambiguity in the information about the lasma obtained with an EZsensor. The basic telemetered information of article counts C S arriving from a given direction qv, fv and collected in a narrow window about E Z k is an irreducible, entroy roducing sum: C = C + S C, (1) S j j where the summation over all counts from secies other than rotons are from those different hase sace locales, the mass and charge states, that satisfy Equation (13). Two different aroaches to this roblem have been emloyed: (a) the forward modeling aroach, which assumes f ( v j ) for all of the modeled secies and then redicts the counts that should be collected as a function of the free arameters of such a model; and (b) the other aroach seeks to determine the moments of f () v from a reasonable aroach that first searates the telemetered counts and assigns them to different secies. From the ersective of this aer, both aroaches have liabilities; the first roduces answers that are otimizations for the fit sace assumed, while the second aroach determines the answers by numerical integration, rovided the count rate matrix is artitioned aroriately. As we have argued above, the rocess of assing through the E/Z detector is the source of some essential ambiguities in the count rate matrix. Neither method affects a lossless reversal of this entroy caused by this form of data collection. 5.. Proton α Particle Secifics In this brief section, we focus on the two most common charge states, rotons and α articles, e ++, which tend to have the least overla in EZ. Figure 6 illustrates the nature of the measurements for an EZdetector. The tyical situation in the solar wind is that the α articles are observed to sli along the magnetic field and lead the rotons as they leave the Sun. In addition, the α articles and rotons are commonly observed to have comarable thermal seeds w wa leading to temeratures roortional to their mass. Tyically, the α articles reresent 4% by number, with a range of % 10%. As concetually shown in insets A and B of Figure 6, the rotons and α articles occuy their own velocity sace. owever, the nature of the EZsensor is that arts of the α hase sace and those of the rotons overla in energy er unit charge, which is the basic tuning arameter that the electrostatic analyzer differentiates according to Equations 7

8 The Astrohysical Journal, 809:16 (1), 015 August 0 Figure 6. Insets (A) and (B) dislay count rate matrices for searate roton and alha article hase saces. Bottom inset illustrates how these two concetually searate hase saces are mixed by an EZdetector, creating a count rate matrix (C) that is laced in telemetry and must be searated into its searate constituents before clear moments can be determined from either secies. Carrying a heat flux, the roton distribution in (A) is skewed. This asymmetry along the magnetic field is referentially submerged in the saddle oint region under the α article counts where the two searate count rates overla (inset (C)). This saddle oint is about.5 roton thermal seeds above the roton bulk velocity eak. The crease mentioned in the text is the ath of steeest descent down from the saddle oint, shown here in red. In the elios data rocessing, counts below the crease are labeled rotons and those above the crease are alhas, which effectively dros the roton velocity distribution to zero above the crease along bˆ. Examles of the effect of this rocedure are shown in Figure 8 below. (14) and (16). As shown in inset (C) of this figure, the telemetered count rate matrix is an unknown mix of hase sace contributions in each ixel of áezñ and direction of arrival. Oerations must be undertaken on the comosite C S matrix to differentiate the likely contributors that went into each element of the matrix. Thus, as an inverse roblem, one has the numerical matrix of inset (C) and would like to extract the resective numerical matrices for the constituent masses and charges corresonding to insets (A) and (B). Like most inverse roblems, the answer is not unique but rocedures have been described to oerationally unack the third inset into candidate inut matrices. As one can see from this figure where the rotons are modeled as ossessing a heat flux shown by the skewness in their counting rates, this feature of the rotons is found under the αs and may become lost in the inverse roblem under discussion. The suersonic nature of the wind (relative to ion thermal seeds) imlies that the seed v* of the eak counting rate for each suersonic secies is nearly at its bulk seed (v* - = U(1 + 4 )), or at its imlied energy er unit charge seed in the lower anel. The α eak and roton eak will be searated by ( - 1) U, as seen in the lower inset of this figure. owever, the ion velocity distribution functions are not Dirac delta functions, but have robability distributions that cause counts from both secies to be found in the saddle oint region between the two bulk velocity eaks. The number of articles in the vicinity of these thermal sreads determine the anisotroy and heat fluxes of the resective secies. As also shown in inset (C), the thermal sreads of the αs and rotons overla, and there are many ixels where the entroy caused by the sum created by the detector must be undone by data rocessing on the ground. Some have aroached this roblem by fitting assumed model forms (such as a suerosition of convecting, anisotroic Gaussians) to the underlying distribution functions (Feldman et al. 1973). These techniques must secure their model s arameters for f () v under the α articles as constrained by observations elsewhere, and rely heavily on the aroriateness of the model chosen for that occurring in nature. The elios team avoided fitting functions altogether, wishing to determine the moments of the constituents by numerical integration after first assigning a artition of the count rate matrix. Aside from semantics, it should be clear that such an assignment involves the same kind of judgement, (different) ambiguities, and lack of uniqueness for the information determined in this way as found by those using exlicit modeling in the forward modeling aroach Estimating Tyical Values for Crease Seed Oerationally, there is usually a crease in the comosite count rate sectrum (indicated in red in Figure 6(C)) that asses through the saddle oint in the count rates, which gives some idea of the cross over location of dominance between secies count contributions. This crease must be located (in three-dimensions) and a first aroach (Marsch et al. 198) assumes that the makeu of C S is all rotons on the low EZ side of the crease and all α articles on the other side, closer to the second eak. (This general aroach was checked by comaring this searation with an electrometer that measured the current, rather than the counts, and showing that this ratio was close to two for that art of the counts identified as redominantly α articles. It should be noted that this measurement was designed to differentiate from 1, but not 1.85 from, for examle. Thus, the extensions of the roton hase sace under the alhas, even with these checks, could still have occurred.) It should be carefully noted that this rocedure is equivalent to making the f ()for v the two secies zero on the corresonding other side of this dividing crease. This will have imortant imlications below. Both fitting and moments have difficulties with the recovery of f () v in the vicinity of the crease, esecially when determining how the robability for the rotons behaves under the dominant region of the other secies. Conversely, the arallel temerature of the α articles is also otentially affected by the unacking of the count rate matrix. Ordinarily in the moment aroach, an examination of the artial sums contributing to the integrals can be used to confirm the oerational convergence of the moment in question; the effective truncation of the moments above v crease severely interferes with this internal consistency check for convergence, since the truncation de facto makes the integral convergent to its value obtained when the uer limit exceeds h w. Fitting rocedures to narrow eaks are well disosed toward obtaining the suersonic flow velocity and the density, but even the trace of the ressure or the ressure anisotroy can be influenced by the adoted oerational aroach to handling the crease area. 8

9 The Astrohysical Journal, 809:16 (1), 015 August 0 Figure 7. (a) Distribution of h, the average dimensional distance (in thermal seed units) from the eak of the roton velocity to the crease along bˆ in the EZcount rate distribution as a function of alha concentration na. Symbols reflect the average and flags the standard deviations at fixed alha concentrations over a wide range n of external arameters that control the overla and ensuing entroy in the count rate matrix as discussed in the text. This figure summarize one half billion combinations of arameters that cover their exected ranges and combinations over the elios orbit. Blue traces are the boundary including 90% of all outcomes; red curves are the most extreme values (high and low) in the samle. (b) Distribution of h, the average dimensional distance (in thermal seed units along bˆ) from the eak of the roton velocity to the crease in the EZcount rate distribution as a function of roton thermal mach number. Flags reflect the standard deviation about the means shown. The heat flux is an extreme examle of this tye of roblem already alluded to by Marsch et al. (198). We have used the relations in Equations (17) and (18) to determine the variation of the count rates into an EZdetector as a function of the increasing roton seed along the magnetic field. At resent, we are interested in where the crease intersects the uer limit of w for the rotons and the overla of α articles starts to corrut the clear observations of this art of the rotons velocity sace. Moving along the ray 1 v = U w bˆ o + h á ñ, we can find where zero itch angle articles cross the crease ; its dislacement from the roton bulk velocity scaled by the roton rms thermal seed yields the dimensionless offset, h, of the crease from the roton bulk velocity, that is, w h = D w. () o This number is of interest since it informs what is the uer, roer frame limit of integration that would have been used in the heat flux integration that we seek to use in Equation (5) below. As may be seen from Equations (17) and (18), h deends on a rather large number of background arameters: the α-roton concentration ratio, the flow seed, the mach number, the roton beta, and the angle that the magnetic field makes to the radial. We have surveyed nearly half a billion combinations of these arameters for bulk seeds in the range 50 U 850 km s 1, roton beta in the range 0.1 < b< 1, roton thermal mach number in the range 4 11, the angle c between b ˆ and the radial in the range 0 c 45, and variations of the α concentration from % to 10% to determine the likely distribution of the h crease locations in roton rms thermal seed units. This range of arameters includes those that would be resented to the elios sacecraft between erihelion and ahelion. This crease estimate study assumes Gaussians for the ions, and its size could be influenced by yet another quantity how large is the roton heat flux and in what manner do the rotons suort this skewness. Since there is at resent no clear model for the roton hase sace distribution when carrying a given amount of heat, one can generally say that the crease seed will increase slightly above our estimate deending on how the heat flow signatures occur in the roton velocity distribution and how much α article field aligned anisotroy is resent in the data. Figure 7(a) summarizes over a half a billion combinations of arameters that could influence h as a function of the α concentration (which scales the relative heights of the two eaks in Figure 6), and the lotted standard deviations reflect the variability of the crease seed that can be exected. The α concentration roduces an orderly, but slight trend for the mean value h indicated by the black diamonds in this figure. The disersion about this curve is nearly constant, but the range of h about h is slightly skewed toward higher values. The red curves indicate the absolute observed extremes in this survey for the crease seed. Nonetheless, this extensive survey shows that tyical values for the crease seed are extremely unlikely outside of.1 < h < To routinely obtain convergent heat fluxes by numerical integration, we show in Section 5.4 that uncomromised distribution function coverage until at least h > 5.5 would aear to be mandatory. When h crease < 5.5, the contributions to the integral on [ n crease, 5.5] are (i) substantial and (ii) uninventoried in the elios model indeendent determination of the heat flux moment with limits set by the crease determination Imact of n on eat Flux Determination by Numerical Integration We now wish to illustrate the nature of this systematic loss of information and what imact it has on model indeendent numerical integration for the roton heat flux. A well-known 9

10 The Astrohysical Journal, 809:16 (1), 015 August 0 roerty of the moments is that their integrands are systematically dominated by contributions that migrate to larger and larger random seeds about the bulk seed as the moment number increases. For our resent uroses, we will discuss the density, momentum, ressure, and heat flux as having moment numbers of 0 3. Thus, the heat flux integral ordinarily derives its rincial contributions further from the origin of velocity sace than the temerature, the momentum, or the density. The recise location of these maxima deends on the details of the distribution function. If Sitzer Braginskii transort is credible, then the maxima for these moments can be worked out analytically (see & Karimabadi 013). The main effect can be seen from the form of the moments: M n = n+ ò ( f ( w ) + df ( w)) w Pn(cos( qw)) dwdww, where w = v -U is the velocity sace coordinate in a frame at rest with the secies bulk seed, where P n are the Legendre olynomials of the order of n, and df are the transort induced corrections to the distribution function in the resence of satial nonuniformity and microscoic forces. Thus, the integrands leave the origin with f ( w) multilied by w n+, which increasingly detunes the value of the integrand to the features of the robability distribution near the origin (because it is flatter and flatter with increasing n). Eventually, the growth of this olynomial in w will comete with the decrease in f ( w ) + df ( w) as w. For a given distribution function, the location of the maximum contribution may be comuted. For the same distribution function, the location of the maxima moves to higher seeds as n increases. For the odd heat flux there are two minimaxes, occurring at different and oositely signed w. To illustrate a consequence of the crease artition of CS at v = U + hwo, we have modeled (Figure 6(A)) a simle roton distribution in Figure 6 that contains a core halo model for heat flow (with indicated skewness; Feldman et al. 1973). The urose is not to endorse such a model, but to have a concrete distribution with a known heat flux; with such a concrete examle, the imact of the crease uer limit for the heat flux can be quantified. As with all skewed distributions, this model form redicts counts more asymmetrically sread in velocity sace than does a Gaussian with no skewness. (The counting rate eaks of Figure 6 do not recisely retain the symmetries of their arent robability distribution functions f (), v since the counting rates are integrals over seed with weights of v 4, which bias the high seed side counts from a symmetric Gaussian f ()to v be slightly asymmetrical.) For suersonic distributions, this effect, while resent, is small (see Figure 6(B)) and should contrast with the markedly skewed skirt on the modeled roton distribution in Figure 6. When merged in an EZdetector (inset (C)), this skew signature is artially obscured under the location of the α distribution with non-zero width. The roton heat flux in magnetic field aligned coordinates, takes the form of where ò q = Q( w ) dw, (3) ò - 1 ( ) ( ) Q( w ) º f w, w^ mw w + w^ w^dw^. 0 (4) Figure 8. Variation of the fraction f ( h* w ) of the actual heat flux obtained by integration between [-, v * ] for a core halo skewed roton distribution. For alha concentrations of 4% the statistical zone of the crease at hw o found in Figures 7(a), (b) is indicated by the lower robability histogram, P ( h ). This statistically defined regime of h determines a robability distribution of the fraction of the true heat flux, Pf ( ( n )), that imlies an average short fall of ásñ= 780%. This suggests that if this modeled velocity distribution were realistic that the elios heat flux reorted by numerical integration would be too small by an average factor of ásñ= 7.8. This systematic effect is in the summarized roton heat flux moment of the elios data set. We then examine the fractional value of the roton heat flux s convergence in the form f ( * wo) h = ò h* w - o Q( w ) dw q. (5) We note that the negative arallel half of the random roton velocity sace is unencumbered by other counts from different comosition, so that starting the integral for f at large negative values of w should initially obtain the correct artial integrated values, Q ( w < 0). The crucial question then becomes what value does f ( hwo ) take on comared to unity? If this number is small, then the elios heat flux would seriously have underestimated the true heat flux by a factor of the order of f -1 ( hwo ). The results of this exercise are indicated in Figure 8. For values of h < 0, the artial integral is negative since the integrand of Q ( w ) contains the odd olynomial aw + b w 3. As the uer limit of integration goes through h = 0, the value of the integrand becomes increasingly ositive, adding to the total, and the fraction f starts reducing its negative size by getting successive ositive increments. As the sign changing curve for f ( * h wo ) indicates, the integral has converged to a ositive outward heat flux rovided h > 6. owever, the statistical location for h is rather comact, as shown by its robability distribution P ( h ) rising off the bottom horizontal axis. By interolating these results against the master curve for f ( * h w ), we can determine the robability distribution Pf ( ( h )) in the histogram anchored in the right vertical axis. The fractional convergence factor for the heat flux integral, f, is the indeendent variable of this second robability function. Its shae indicates that the most robable value of f ~ 0.1 and has a half width of aroximately While this robability is skewed toward higher values of the fraction (the most 10

11 The Astrohysical Journal, 809:16 (1), 015 August 0 extreme fraction (with robability ) is as high as 0.6), the dominant occuancy of this histogram is certainly below 0.. This indicates that this effect might suggest shortfalls instead, ranging between [1 /., 1 /. 06] = [5, 16.6], with the most likely shortfall being of the order of 10 fold. The actual -1 weighted average of áf ñ= 7.8 was used in constructing the blue diamonds in Figure 5(B), which nicely overla (with their errors) with the amlitudes suggested from Equation (1) which reroduced the roton temerature radial ower laws of the elios data. Since the α articles always lead the rotons along bˆ and because the maing of EZtends to lace the α eak at seeds U, the nature of the α article entroy systematically occurs on the high roton thermal seed side of the roton bulk seed eak of the distribution. Accordingly, the E/Z systematics (with truncations at v crease = hw ) will systematically tend to underestimate the roton heat flux. There are many variables involved in determining the size of this underestimate; among the rominent factors will be the actual size of the roton and α article thermal seeds, the actual α concentration, the actual sliage of the alhas along the magnetic field, and the concentration of the alhas relative to the rotons. All of these factors imact the recise determination of the crease. In constructing P ( hcrease) we have surveyed all of these effects excet w ¹ w a. The recise location also deends on how the roton distribution is skewed in velocity sace. It aears that short of a 3D mass-differentiated distribution function, the true size of the roton heat flux may not be yet known in a model indeendent way, certainly not as function of radius from the elios data set. 6. DISCUSSION AND SUMMARY The resent aer has exlored the ossibility that the roton heat flux and its imlicit redistribution of internal energy can exlain the observed radial variations of the roton temerature as a function of solar wind seed state. Using Equation (1), we have determined the average skewness required to relicate the long-term elios roton radial variations reorted reviously; this has been done without any redetermination of the stochastic agents involved in regulating the transort. Indeendently, we have also estimated the variation of that would be exected if it were rincially determined only by the Coulomb mean free ath l mf for scattering; its scaling with bulk seed in the asymtotic wind is (i) strong and (ii) strikingly similar (see Figure 4) to the emirical skewness determined by comaring the radial temerature rofiles roduced by Equation (1) in the energy equation, being esecially recise below Uo < 600 km s 1. The deduced variation of skewness, determined by the observed gradients of T (r), recovers the exected strong bulk seed deendence that would attend the skewness being set by the Coulomb collisional mean free ath that is over lotted in Figure 4. Emirically there seems to be some uer limit set for the skewness of the rotons in the highest seed winds that reverse this trend of growth of skewness with roton binary mean free ath. This might be a regime where wave article scattering begins to limit the mean free ath of the rotons involved. Alternatively, this highest seed regime of the elios catalog is the least well samled multile times (see Figure 1, Freeman 1988) and may have more transient comression hysics in its average rofile. More needs to be done in this area as well. We have recovered the adiabatic behavior of the lowest seeds in the wind within this frame work by inferring a small skewness in such a low-seed state that is consistent with our overall scaling. By way of contrast, to date, the alternate view of external heating for the wind rotons has not reroduced the elios radial variations of the rotons as a function of seed state. The roton internal energy budget for the wind is rather small, being - of the ram energy; accordingly, a theory that exlains its variation must be fairly accurate and consider carefully all ossible contributions. Thermodynamic adiabaticity does not just haen, it requires collisions to ensure that heat conduction does not disrut this ostulated thermal isolation. As shown in Figure 4, the collisionality that enforces very small skewness in the ultra-slow wind roduces very weak q and induces adiabatic behavior in the resence of the flux tube s exansion. As shown by the same grah, this argument cannot be sustained in the higher seed winds with the lower densities and higher temeratures which both cause a raidly growing mean free ath for Coulomb collisions that allows heat to flow generally. Proton heat will flow because there are insufficient collisions to imede it from flowing; ostulating adiabaticity requires a rationale that is generally not resent in the solar wind s exansion. When considering the collisionless solar wind, U 350 km s 1, the roton heat flow will generally be resent and disrutive of thermodynamic adiabatic conditions. Arguments for the necessity of DQ in such flows that have ostulated adiabatic exansions are inconsistent. When modeling the effectiveness of coronal heating, suitable heat flow models must be included. As shown with the resent calculation, q is a significant entroy source and is caable of removing the need for DQ addition. Early arguments that the observed T disagreed with the adiabatic solar wind solution do not actually imly that heating of the DQ tye is required; instead, they imly that one of the four tyes of entroy addition discussed in this aer is involved in changing the temerature rofile. Unless and until an argument is made that collisional hysics suorts an insulating adiabatic descrition, the role of a relevant heat flux must be inventoried which is valid for the Knudsen number of the medium being discussed. A relevant model must also redict the direction of the heat flux. The accuracy of modeling heat flowing back toward the transition region from coronal maximum clearly imacts the amount of heat required to maintain the corona; however, retaining Sitzer s invalid heat descrition with its q =-k T rediction in the low corona while studying the effectiveness of wave daming is just such a disconnect. There are several aers discussed above that do just that! The radial variations for q required by Equation (1) are not inconsistent with the ower laws for q discussed by elios researchers Marsch et al. (198). We have refit their resented data assuming that the heat flows have 50% errors, treated the bucketed errors in ln r roerly, and determined heat flux radial exonents with errors that are comared with those required from Equation (1) in our model in Figure 5. As shown in Figure 8, the arallel velocity sace integral for contains a strong negative contribution from below the solar wind velocity before acquiring its final ositive value, if all of velocity sace is in view. That figure shows with P ( h ) that this moment was routinely interdicted well before it converged. This interdiction at + w occurs for nearly all of the conditions surveyed for the elios mission. Accordingly, the q, 11

12 The Astrohysical Journal, 809:16 (1), 015 August 0 reorted heat flux values from the elios numerical integrations have three roerties: (i) though numerically incorrect, they are aligned with bˆ; (ii) though numerically incorrect, they are roortional to the correct value (by the fixed dimensionless cutoff), so that their radial variation dlnq dlnr should nearly be correct, since this quantity is indeendent of the roortionality constant; and (iii) the size of q does leave elios q too small in the ion entroy equation for its true contribution. Given that the α osition in E/Z rather regularly clis the q integral at a comarable number of thermal seeds above the roton eak, the radial variation of q might be better than its overall estimate of the size of the heat flux. We have shown (Figure 5(a)) that the radial exonents of elios and this aer are comatible, and that, after correcting for an estimate of the shortfall indicated by the α article entroy, that the magnitude of the heat flux required by Equation (1) aears to be consistent with the observations (Figure 5(b)). These findings have imlications for the significant body of research that has relied on the elios heat flow determinations and their radial variation to sustain their conclusions (Marsch et al and ellinger et al. 011, 013). Other mass-resolved roton sectrometers have been flown in the solar wind on ACE, Ulysses, and Stereo, but determinations of the roton heat flux measurements have not yet been discussed in the literature. With Solar Probe Plus there will be a 3D mass-resolved sectrum (J. alekas 015, rivate communication) for rotons, at least at times, that can evaluate the idea that the roton heat flux exceeds what can be determined by generalized crease artitions of E/Z hase sace, which will also be available. The unusual circumstance of asymtotic wind conditions and many elios asses through a fixed radial regime has allowed the calculation of the roton heat flow necessary to exlain the observed temerature rofiles while simultaneously ignoring comressive heating, wave daming, or exchange terms. Arguments have been advanced in connection with Figure 4 of this aer that Coulomb mean free ath effects are resent in the inner elioshere data as the nearly exclusive limiters of the roton heat flux that flows for U 600 km s 1. For even less collisional, higher-seed winds, other agents (erhas wave scattering) could lay a role in keeing the mean free ath from going as high as the roton mean free ath would suggest, although the quality of the radial rofiles is not as good for these seed ranges as those between 300 and 650 km s 1. These data reenforce the imortant idea that by weakening collisions one does not cause the heat flow to go to zero; rather, the reverse occurs: the heat flow becomes more imortant with enhanced skewness (Figure 4) as the system becomes more collisionless. These data also seem to suort the ro temore aroach for estimating heat flows in large Knudsen number regimes by using a form like Equation (1) that is self similar to, but retains the same structural scaling K n that can be verified in detail with Sitzer Braginskii transort in the form of Equation (10) when K n, is suitably small. We thank D. McComas, G. 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