Issue 
A&A
Volume 571, November 2014



Article Number  A78  
Number of page(s)  7  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/201424421  
Published online  14 November 2014 
Traveling solarwind bulkvelocity fluctuations and their effects on electron heating in the heliosphere
^{1} Argelander Institute for Astronomy, University of Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
email: hfahr@astro.unibonn.de
^{2} Lebedev Physical Institute, Leninskii Prospect 53, 117924 Moscow, Russia
^{3} Space Science Center, University of New Hampshire, 8 College Road, Durham NH 03824, USA
Received: 17 June 2014
Accepted: 12 September 2014
Ambient plasma electrons undergo strong heating in regions associated with compressive bulkvelocity jumps ΔU that travel through the interplanetary solar wind. The heating is generated by their specific interactions with the jumpinherent electric fields. After this energy gain is thermalized by the shock passage through the operation of the Buneman instability, strong electron heating occurs that substantially influences the radial electron temperature profile. We previously studied the resulting electron temperature assuming that the amplitude of the traveling velocity jump remains constant with increasing solar distance. Now we aim at a more consistent view, describing the change in jump amplitude with distance that is caused by the heated electrons. We describe the reduction of the jump amplitude as a result of the energy expended by the traveling jump structure. We consider three effects: energy loss due to heating of electrons, energy loss due to work done against the pressure gradient of the pickup ions, and an energy gain due to nonlinear jump steepening. Taking these effects into account, we show that the decrease in jump amplitude with solar distance is more pronounced when the initial jump amplitude is higher in the inner solar system. Independent of the initial jump amplitude, it eventually decreases with increasing distance to a value of about ΔU/U ≃ 0.1 at the position of the heliospheric termination shock, where ΔU is the jump amplitude, and U is the average solarwind bulk velocity.The electron temperature, on the other hand, is strongly correlated with the initial jump amplitude and leads to electron temperatures between 6000 K and 20 000 K at distances beyond 50 AU. We compare our results with in situ measurements of the electroncore temperature from the Ulysses spacecraft in the plane of the ecliptic for 1.5 AU ≤ r ≤ 5 AU, where r is the distance from the Sun. Our results agree very well with these observations, which corroborates our extrapolated predictions beyond r = 5 AU.
Key words: plasmas / Sun: heliosphere / solar wind
© ESO, 2014
1. Introduction
The electron temperature in the solar wind is expected to rapidly decrease with increasing distance r from the Sun, as soon as the electron heat conduction ceases that serves as the prime energy source (Feldman et al. 1975; Pilipp et al. 1987; Scime et al. 1994). At distances smaller than 5 AU, electron distribution functions have been identified as corehalostructured distributions with an electron heat flux that decreases with a power law according to ∝r^{2.36} (see McComas et al. 1992). On the basis of electron data taken from the Helios, Wind, and Ulysses spacecraft, Maksimovic et al. (2005) have carefully analyzed the radial change of the corehalostrahl structure of the electron distribution function with distance from the Sun in the range between 0.3 AU and 1.5 AU. These authors found that while the relative abundance of core electrons remains fairly constant with distance, the relative abundance of halo electrons increases and that of strahl electrons decreases, suggesting that the relative increase in halo electrons is connected to the relative loss in strahl electrons. Interestingly, however, both the core electron temperature and the halo electron temperature decrease with distance. This effect can be represented by kappa distribution functions with decreasing kappaindices and will be best fitted by the decrease of the electron kappa index from κ = 6 at r = 0.5 AU to κ = 3 at r = 1.5 AU. The increase in the relative abundance of the halo population is interpreted as the consequence of an isotropization of the strahl population, which leads to a conversion into the halo population (see also Štverák et al. 2009).
Beyond the outer ranges of the Ulysses trajectory (i.e., at solar distances beyond 5 AU), measurements of lowenergy solarwind electrons are not available. Up to now, electron temperatures have been expected to decrease to negligible values in this region for theoretical reasons. At such large distances from the Sun, processes such as whistlerwaveturbulence generation caused by instabilities that are driven by the electron heat flux (see Scime et al. 1994; Gary et al. 1994) become unimportant. Pitchangle scattering and energydiffusion processes can likewise be neglected at these distances (Schlickeiser et al. 1991; Achatz et al. 1993). However, more recently, Breech et al. (2009) have presented a theoretical study of the heating of solarwind protons and electrons by dissipation of magnetohydrodynamic (MHD) turbulent energy. While their study shows that the theoretically obtained proton temperatures fit the Ulysses data, the theoretically obtained electron temperatures (see Fig. 3 in Breech et al. 2009), fail to fit almost all the data because they are too low.
As a remedy of that failure, we most recently conjectured that the interaction between electrons and the electric fields associated with traveling fluctuations in the solarwind bulk velocity (i.e., traveling shocks) can provide an energy source for electron heating in this part of the heliosphere (Chashei & Fahr 2014). All solarwind properties, including the solarwind bulk velocity U, show strongly pronounced variations on many time scales as well as shocklike structures (Feng et al. 2009; Yue & Zong 2011; Janvier et al. 2014). We show a time line of the measured solarwind bulk velocity in the plane of the ecliptic at 1 AU in Fig. 1 (cf. Echer et al. 2005; Lai et al. 2010; Sokół et al. 2013). In agreement with these observations, we find a typical occurrence rate of about 30 jumps with a significant amplitude per year. These jumps are convected over the spacecraft with an average solarwind bulk velocity of U ≈ 400 km s^{1}, which leads to a typical distance of about L_{j} = 3 AU between subsequent shocks.
Fig. 1 Solarwind bulk velocity as a function of time in the plane of the ecliptic at 1 AU. We show OMNI2 data (from ftp://spdf.gsfc.nasa.gov/pub/data/omni/low_res_omni/) from a combined set of spacecraft observations to demonstrate the ubiquitous fluctuations in the solarwind bulk velocity. 

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Fluctuations ΔU(t) ≡ U(t) − ⟨U(t)⟩, where ⟨·⟩ is the time average and U is the solarwind bulk velocity, persist to large distances from the Sun up to 20 to 40 AU, as clearly demonstrated by Voyager observations (Richardson et al. 1995). Figure 2 in Richardson et al. (1995) shows that while the bulk velocity fluctuations survive up to large solar distances, the average bulk velocity U = ⟨U(t)⟩ appears to be constant, implying that differential kinetic energy is not converted into kinetic energy of the wind, but into thermal degrees of freedom of the plasma system.
We suspect that these bulkvelocity fluctuations are responsible for the still poorly understood heating of electrons at larger distances from the Sun. We have recently proposed in a quantitative discussion that compressional bulkvelocity waves heat solarwind electrons (Chashei & Fahr 2014). We determined the fraction of the differential kinetic energy of the traveling shocks and quantified the energy that is transferred to thermal energy of the solarwind electrons by means of the Buneman instability as a function of the bulk speed U_{2} downstream of the velocity jump. The joint bulk speed of electrons and protons, after passing the jumpassociated electricfield jump, is given by (1)where m_{e,p} denotes the mass of the electron and of the proton, and U_{2} denotes the bulk velocity of the downstream centerofmass system. With m_{e} ≪ m_{p}, this expression leads to (see Chashei & Fahr 2014) (2)where s ≡ (U + ΔU)/(U − ΔU) is the jump compression ratio. The difference U_{2} − U_{2p} is much smaller than U_{2p}, and hence the overshoot energy of the electrons in the downstream bulk frame is given by (3)If this kinetic energy ΔW_{e} of the overshooting electrons can be locally converted into electron heat, this process leads to an electron temperature increase ΔT_{e} after each jump passage given by (4)where k is the Boltzmann constant.
This process describes an average gain of thermal energy that leads to a systematic heating of the solarwind electrons per radial increment dr as a result of repeated shock passages. The resulting radial dependence of the electron temperature can be described by a transport equation for the thermal energy. We expect that the electron heating due to accumulated jump passages in the heliosphere beyond about 5 AU is statistical in nature. We denote the average distance between consecutive jumps as L_{j} and define the average jump occurrence rate as ν_{j} ≡ U/L_{j}. With these definitions, the equation for the radial electron temperature is given in the following differential form (see Chashei & Fahr 2014): (5)where ΔX ≡ ΔU/U. The term on the righthand side of Eq. (5) describes the electron heating induced by jump passages. When they solved this equation, Chashei & Fahr (2014) assumed that ΔT_{e} = ΔT_{e}(ΔU) is a constant. This assumption is true if ΔU is independent of the distance r. In that case, the radial profile of the resulting electron temperature is given by (6)where x ≡ r/r_{0} is the dimensionless spatial coordinate and T_{e0} denotes the electron temperature at r = r_{0} = 1 AU (solution shown in Fig. 1 of Chashei & Fahr 2014). This solution suffers from the inconsistency that the jump kinetic energy is assumed to be constant even after transferring energy to the electrons. We make this earlier approach more consistent in the sections below by taking into account the energy consumption at the passage of each jump during this process.
2. Change of the jump amplitude with distance from the Sun
To increase the consistency of our approach, we now include higherorder corrections to the electron heating due to the variation of the jump amplitude ΔU with distance r. This amplitude is assumed to be the primary physical reason for the gain in thermal energy of the electrons. Therefore, we have to adequately describe the change of ΔU caused by energy expended by the excess kinetic energy of the jump structure. In situ observations by the Voyager 2 spacecraft at distances between 10 AU and 40 AU from the Sun (Richardson et al. 1995) showed that compared with solarwind bulkvelocity measurements carried out simultaneously at 1 AU by IMP 8, the average solarwind speed does not change with distance. On the other hand, the amplitude of the speed fluctuations strongly decreases with distance from the Sun (see Fig. 6 of Richardson et al. 1995). This observation indicates that these fluctuations do work, while the bulksolarwind outflow does not. Our theoretical approach is based on these observations, adopting that the average solarwind speed U is constant with distance from the Sun. Based on this observationally supported assumption, we consider three effects that determine the change of ΔU with distance r:

a)
heating of electrons,

b)
work done against the slower side of the jump with its higher pickupion pressure, and

c)
steepening of the jump profile by nonlinear superpositions of smallscale bulkvelocity fluctuations.
In the following, we separately discuss these three different effects.
2.1. Reduced compression due to electron heating
We consider the spatial divergence of the jumpassociated flow of excess kinetic energy on the highvelocity side of a jump with the amplitude ΔX = ΔU/U. This jump acts as a local source of electron thermal energy, and this heating reflects a local energy sink for the excess kinetic energy that is represented by the compression profile ΔX(r). Using Eq. (5) for the electron temperature, we can formulate an expression for the energy sink associated with this jump as the divergence of the excess kinetic energy flow: (7)where n_{e} = n_{p} = n is the local solarwind electron/proton number density and ΔT_{e} is the electron temperature increase per jump passage as given by Eq. (4). We assume that the mean bulk velocity U = (1/2)(U + ΔU + U − ΔU) is constant and find (8)Supported by Voyager data, we assume that the traveling jumps in bulk velocity have a small amplitude (ΔU ≪ U, which is equivalent to ΔX ≪ 1). This observation allows us to approximate the electrontemperature increase per jump passage in Eq. (4) using the linearizations (9)and (10)We can then rewrite Eq. (4) as (11)and obtain from Eq. (8) (12)Assuming a spherically symmetric decrease in density of the average solar wind flow with n_{e} ∝ r^{2}, we then obtain (13)and find (14)From this relation, we derive in a first step (15)which delivers a solution of the form (16)We show ΔX as a function of distance r from the Sun for five different values of ΔX_{0} in Fig. 2. We use r_{0}/L_{j} = 1/3 in agreement with observations at 1 AU (see Fig. 1). The jump amplitude ΔX remarkably decreases with increasing distance from the Sun. The decrease is even more pronounced for cases in which the initial value ΔX_{0} is higher in the inner solar system. However, independent of ΔX_{0}, the jump amplitude assumes values of ≲0.1 at the position of the heliospheric termination shock (i.e., at r ≈ 90 AU).
Fig. 2 Compression ΔX as a function of distance r from the Sun for five different values of ΔX_{0} at r_{0} = 1 AU with r_{0}/L_{j} = 1/3. The compression decreases with distance and approaches a value of about 0.1 at the position of the termination shock. The dependence on the jump occurrence r_{0}/L_{j} is discussed in Chashei & Fahr (2014). 

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With this result for the dependence of ΔX on r, we solve the earlier differential equation, Eq. (5), for the resulting electron temperature and obtain (17)where λ ≡ L_{j}/r_{0}. The solution of this inhomogeneous differential equation is given by (18)with . We show the result of Eq. (18) for five different values of ΔX_{0} in Figs. 3 and 4. We use T_{e0} = 2 × 10^{5} K and U = 400 km s^{1}. In Fig. 3, we show our results for 1.5 AU ≤ r ≤ 5 AU and compare them with in situ Ulysses measurements of the electroncore temperature during the spacecraft’s first orbit in the plane of the ecliptic (from December 28, 1990 until December 31, 1991). The data were taken with the Ulysses SWOOPS experiment (Bame et al. 1992). Our predictions and in situ measurements agree very well. The modeled and observed electron temperatures are significantly higher than predicted for an adiabatically expanding gas (i.e., T_{e} ∝ r^{− 4/3}). We achieve the best agreement for values of ΔX_{0} between 0.25 and 0.3. In Fig. 4, we extrapolate our results beyond 5 AU and show our predictions for 1 AU ≤ r ≤ 100 AU. The electron temperature in our model is significantly higher than predicted from adiabatic expansion beyond 10 AU for all values of ΔX_{0}. It assumes values of about T_{e} ≈ 10^{4} K at the position of the heliospheric termination shock.
Fig. 3 Electron temperature T_{e} as a function of distance r from the Sun for five different values of ΔX_{0} with r_{0}/L_{j} = 1/3 and U = 400 km s^{1}. The electron temperature is greater than adiabatic. In addition, we show Ulysses observations of the insitu electroncore temperature in the plane of the ecliptic and the prediction from adiabatic expansion. 

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Fig. 4 Electron temperature T_{e} as a function of distance r from the Sun for five different values of ΔX_{0} with r_{0}/L_{j} = 1/3 and U = 400 km s^{1}. The electron temperature beyond 10 AU is higher than predicted from adiabatic expansion. 

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2.2. Change of compression due to work done against entropized pickup ions
In this section, we consider another effect that may also contribute to a decrease in the jump amplitude ΔX: the work done by the faster front against the difference in ion pressure over the traveling shock front. The faster regime (U_{1} = U + ΔU) is running into the slower regime (U_{2} = U − ΔU) with a differential velocity ΔU. During this process, the plasma has to do work against the pressure difference between the two regimes to adapt the flow to the slower regime (i.e., U_{2}). We estimate the work done per unit time as (19)where D is the transit distance over the shock profile, and are the adaptive pressures (i.e., the entropized kinetic energy densities) on the upstream and downstream sides of the jump. Especially in the outer heliosphere (r ≥ 5 AU), the ion pressure is dominated by the pickupion pressures on either side of the jump. Under this assumption, the expressions for the ion pressures are significantly simplified (see Fahr et al. 2012) for a perpendicular jump (ΔU ⊥ B), leading to (20)where the remaining pressure adaptation function A_{⊥}(s) for a perpendicular shock is simply given by A_{⊥}(s) = s with s ≃ 1 + 2 ΔX . This leads to (21)which allows us to formulate the ioninduced energy change as (22)With this additional term, we now obtain the following corrected differential equation for ΔX: (23)We write the upstream pickupion pressure in the form P_{1,pui} = ζn_{e}kT_{pui} with the pickupion abundance ratio ζ = n_{pui}/n_{e}. This leads to the new differential equation (24)Again, taking n_{e} ∝ r^{2}, this relation then simplifies to (25)We introduce the pickupion Mach number and assume that this number be constant in the outer heliosphere (see, e.g., Fahr & Ruciński 1999; Fahr 2007). We then find (26)Expansion of the exponential term in Eq. (26) for D ≪ r_{0} leads to Eq. (16). Since the shock transit distance is much smaller than 1 AU, the corrections due to the pickupion pressure lead to profiles that are qualitatively very similar to the curves shown in Fig. 2.
2.3. Increased compression due to nonlinear wave steepening
There may also be processes that counteract those described in Sects. 2.1 and 2.2: processes that support a pileup of the bulkvelocity jump amplitude. For instance, fluctuations in the bulk velocity may cause such a pileup by nonlinear superposition. Therefore, we consider wave steepening in the system in addition to the previously discussed processes. Smallscale velocity fluctuations described by δU(x,t) = δU(k)cos[k(x − Ut)] can pile up into a largescale fluctuation with L ≃ L_{j} = 2π/k_{min} by nonlinear wavecoupling and dissipation processes. For onedimensional waves, this situation is described by the following equation (see Infeld & Rowlands 1990 or Treumann & Baumjohann 1997): (27)where F denotes a dissipation force that counteracts the nonlinear term on the lefthand side and compensates for catastrophic wave steepening and wave breaking. For the socalled Burgers’ equation (see Treumann & Baumjohann 1997), a particular dissipative force is introduced in place of F that is proportional to the second derivative of the velocity perturbation, which leads to the following differential equation: (28)where α is a positive dissipation coefficient that acts like a diffusion coefficient (assumed to be constant with distance r). The background plasma moves with the velocity U, and δU represents the superposition δU = U + ΔU. If the nonlinear steepening of δU (second term on the lefthand side of the Burgers’ equation, Eq. (28)) increases, the dissipative term on the righthand side can compensate for the nonlinear term and can allow for a stationary solution in the system that comoves with the nonlinear wave profile. We assume that this developing nonlinear wave asymptotically moves with the velocity ΔU. This allows us to write the Burgers’ equation in this particular comoving system, where the first term of the lefthand side disappears (i.e., ∂δU/∂t = 0) when we transform the equation to space coordinates y = x − ΔUt. This procedure then leads to (29)The solution of this equation is easily obtained in the form of a velocity shock ramp given by (30)which can be rewritten in the form (31)To estimate the appropriate value of α (which has the dimension of cm^{2}/s), we return to the original Burgers’ equation and estimate the time scale for steepening (or in the opposite case: for dissolution) of the wave profile by the pure diffusiontype equation (i.e., domination of the dissipation term) given by (32)We find the solution of this equation within the system [− D; + D] by (33)The kineticenergy density of the velocity fluctuations within the two flanks [− D; + D] of such a velocity structure with the structure scale D is given by (34)Assuming free diffusion, its temporal change is given by (35)Taking this expression for for nonlinear diffusion or steepening per unit volume, we obtain (36)Evaluating this integral expression further leads to (37)and finally to (38)According to the profile in Eq. (31), D ≃ α/ ΔU. On the other hand, the characteristic time τ of the shock passage is given by τ = D/ ΔU = α/ ΔU^{2}. Evaluating now the above expression for this characteristic time τ leads to the following expression: (39)or finally (with ΔU ≈ δU_{0} as suggested by Eq. (33)) to (40)We then obtain the following transport equation with the newly found term for : (41)The last term on the righthand side represents the energy that is required to maintain the jump profile. Free diffusion would instead destroy the profile, according to Eq. (35). The transport equation is then given by (42)With the definition (43)we find (44)Inserting ΔT_{e} from Eq. (11) then leads to the following differential equation: (45)or equivalently to (46)and (47)For a radially symmetric density decrease, we rewrite the transport equation including terms that decrease (first term) and increase (second term) the compression as (48)Combining decreasing and increasing factors can lead to a vanishing gradient and thus to a constant compression ΔX if (49)Therefore, the newly derived term for structure steepening will only compete with the first term if the jump amplitude has decreased to values of ΔX ≲ 0.06 (L_{j}/D). Looking for a solution of the full equation, we start from the solution of (50)As shown in Sect. 2.1, the solution is given by (51)The solution of the other part, (52)is derived from (53)and yields (54)This leads to (55)According to these considerations, the general solution is given by the superposition (56)The corrections as a result of nonlinear wave steepening are small as long as ΔX ≳ 0.06. In those cases, we only need to consider the first term and hence retain the earlier solution we derived in Sect. 2.1 and displayed in Figs. 3 and 4.
3. Conclusions
We have shown that traveling solarwind bulkvelocity jumps effectively process solarwind electrons in energy at their propagation outward from the Sun through the inner heliosphere. These fluctuations in the solarwind bulk velocity are ubiquitous, as shown in Fig. 1. In an earlier paper, we have shown that this energization can be expressed in terms of a substantial temperature increase of the solarwind electrons at larger distances from the Sun of about 50 AU to 90 AU. Assuming that the jump amplitude ΔX = ΔU/U of the propagating bulkvelocity structure does not change with solar distance r, the previous study predicts electron temperatures of more than 10^{5} K at 90 AU (i.e., at about position of the solarwind termination shock). In this study, we introduced higherorder corrections that take into account that the energy for the energization of solarwind electrons is taken from the kinetic excess energy of the propagating jump structure. We found that the previous assumption of a constant jump amplitude ΔX is most probably unrealistic. In addition, the jump structures do permanently work against the ion excess pressure on the downstream side of the shock structure. Taking into account these two physical processes allowed us to quantitatively show that the jump amplitude ΔX = ΔX(r) decreases with distance from the Sun, eventually reducing ΔX independent of the initial value ΔX_{0} of the jump amplitude to values of ≲0.1 at the termination shock, as shown in Fig. 2. The nonlinear pileup of bulkvelocity fluctuations counteracts these two mechanisms with the tendency to reform the solitary jump structure by forming waves at larger scales. We found, however, that this mechanism is most likely not effective enough to compensate for the reduction of ΔX with distance, unless ΔX ≲ 0.06.
Although the described mechanisms reduce ΔX with distance from the Sun, as shown in Fig. 2, the jumpinduced heating mechanism still leads to higher electron temperatures than anticipated from adiabatic cooling at solar distances beyond 10 AU. We predict values above 6000 K to 20 000 K (strongly depending on the initial value of the jump amplitude ΔX_{0} in the innermost heliosphere at r = r_{0} = 1 AU) at distances beyond 50 AU with the solarwind electrontemperature profiles T_{e}(r) shown in Figs. 3 and 4. Our results agree very well with in situ measurements of the electroncore temperature in the plane of the ecliptic from the Ulysses spacecraft. We achieved the best agreement for values of ΔX_{0} between 0.25 and 0.3, which suggests that these values describe the realistic initial jump amplitude in the plane of the ecliptic. In situ observations of the electron temperature are not available for heliocentric distances beyond 5 AU, so that our results are a predictive extrapolation beyond the explored range.
We conclude that solarwind electrons do not rapidly cool with distance from the Sun, as has been generally assumed up to now. They cannot be considered cold beyond 10 AU. Instead, they continue to be correlated with solarwind ion temperatures at large distances (see Richardson et al. 1995).
Acknowledgments
D.V. is supported by NASA grant NNX12AB27G. We used data from NASA’s OMNIWeb Service provided by the Goddard Space Flight Center Space Physics Data Facility (GSFC/SPDF), as well as Ulysses data provided by NASA’s National Space Science Data Center (NSSDC).
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All Figures
Fig. 1 Solarwind bulk velocity as a function of time in the plane of the ecliptic at 1 AU. We show OMNI2 data (from ftp://spdf.gsfc.nasa.gov/pub/data/omni/low_res_omni/) from a combined set of spacecraft observations to demonstrate the ubiquitous fluctuations in the solarwind bulk velocity. 

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In the text 
Fig. 2 Compression ΔX as a function of distance r from the Sun for five different values of ΔX_{0} at r_{0} = 1 AU with r_{0}/L_{j} = 1/3. The compression decreases with distance and approaches a value of about 0.1 at the position of the termination shock. The dependence on the jump occurrence r_{0}/L_{j} is discussed in Chashei & Fahr (2014). 

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In the text 
Fig. 3 Electron temperature T_{e} as a function of distance r from the Sun for five different values of ΔX_{0} with r_{0}/L_{j} = 1/3 and U = 400 km s^{1}. The electron temperature is greater than adiabatic. In addition, we show Ulysses observations of the insitu electroncore temperature in the plane of the ecliptic and the prediction from adiabatic expansion. 

Open with DEXTER  
In the text 
Fig. 4 Electron temperature T_{e} as a function of distance r from the Sun for five different values of ΔX_{0} with r_{0}/L_{j} = 1/3 and U = 400 km s^{1}. The electron temperature beyond 10 AU is higher than predicted from adiabatic expansion. 

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In the text 
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