Solar wind charge exchange in cometary atmospheres. II. Analytical model
Cyril Simon Wedlund, E. Behar, E. Kallio, H. Nilsson, M. Alho, H. Gunell, D. Bodewits, A. Beth, G. Gronoff, R. Hoekstra
AAstronomy & Astrophysics manuscript no. ICA_solar_wind_charge_exchange_PaperII © ESO 2019January 24, 2019
Solar wind charge exchange in cometary atmospheres
II. Analytical model
Cyril Simon Wedlund , Etienne Behar , , Esa Kallio , Hans Nilsson , , Markku Alho , Herbert Gunell , , DennisBodewits , Arnaud Beth , Guillaume Grono ff , , and Ronnie Hoekstra Department of Physics, University of Oslo, P.O. Box 1048 Blindern, N-0316 Oslo, Norwaye-mail: [email protected] Swedish Institute of Space Physics, P.O. Box 812, SE-981 28 Kiruna, Sweden Luleå University of Technology, Department of Computer Science, Electrical and Space Engineering, Kiruna, SE-981 28, Sweden Department of Electronics and Nanoengineering, School of Electrical Engineering, Aalto University, P.O. Box 15500, 00076Aalto, Finland Royal Belgian Institute for Space Aeronomy, Avenue Circulaire 3, B-1180 Brussels, Belgium Department of Physics, Umeå University, 901 87 Umeå, Sweden Physics Department, Auburn University, Auburn, AL 36849, USA Department of Physics, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom Science directorate, Chemistry & Dynamics branch, NASA Langley Research Center, Hampton, VA 23666 Virginia, USA SSAI, Hampton, VA 23666 Virginia, USA Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG, Groningen, The NetherlandsJanuary 24, 2019
ABSTRACT
Context.
Solar wind charge-changing reactions are of paramount importance to the physico-chemistry of the atmosphere of a cometbecause they mass-load the solar wind through an e ff ective conversion of fast, light solar wind ions into slow, heavy cometary ions.The ESA / Rosetta mission to comet 67P / Churyumov-Gerasimenko (67P) provided a unique opportunity to study charge-changingprocesses in situ.
Aims.
To understand the role of charge-changing reactions in the evolution of the solar wind plasma and to interpret the complex insitu measurements made by
Rosetta , numerical or analytical models are necessary.
Methods.
An extended analytical formalism describing solar wind charge-changing processes at comets along solar wind streamlinesis presented. It is based on a thorough book-keeping of available charge-changing cross sections of hydrogen and helium particles ina water gas.
Results.
After presenting a general 1D solution of charge exchange at comets, we study the theoretical dependence of charge-statedistributions of (He + , He + , He ) and (H + , H , H − ) on solar wind parameters at comet 67P. We show that double charge exchange forthe He + − H O system plays an important role below a solar wind bulk speed of 200 km s − , resulting in the production of He energeticneutral atoms, whereas stripping reactions can in general be neglected. Retrievals of outgassing rates and solar wind upstream fluxesfrom local Rosetta measurements deep in the coma are discussed. Solar wind ion temperature e ff ects at 400 km s − solar wind speedare well contained during the Rosetta mission.
Conclusions.
As the comet approaches perihelion, the model predicts a sharp decrease of solar wind ion fluxes by almost one orderof magnitude at the location of
Rosetta , forming in e ff ect a solar wind ion cavity. This study is the second part of a series of three onsolar wind charge-exchange and ionization processes at comets, with a specific application to comet 67P and the Rosetta mission.
Key words. comets: general – comets: individual: 67P / Churyumov-Gerasimenko – instrumentation: detectors – solar wind, methods:analytical – solar wind: charge-exchange processes – Methods: analytical
1. Introduction
Over the past decades, evidence of charge-exchange (CX) re-actions has been discovered in astrophysics environments, fromcometary and planetary atmospheres to the heliosphere and tosupernovae environments (Dennerl 2010). They consist of thetransfer of one or several electrons from the outer shells of neu-tral atoms or molecules, denoted M, to an impinging ion, notedX i + , where i is the initial charge number of species X. Electroncapture of q electrons takes the formX i + + M −→ X ( i − q ) + + [M] q + . (1) From the point of view of the impinging ion, a reversecharge-changing process is the electron loss (or stripping); start-ing from species X ( i − q ) + , it results in the emission of q electrons:X ( i − q ) + + M −→ X i + + [M] + qe − . (2)For q =
1, the processes are referred to as one-electroncharge-changing reaction; for q =
2, two-electron or doublecharge-changing reactions, and so on. The qualifier "charge-changing" encompasses both capture and stripping reactions,whereas "charge exchange" denotes electron capture reactionsonly. Moreover, "[M]" refers here to the possibility for com-
Article number, page 1 of 17 a r X i v : . [ phy s i c s . s p ace - ph ] J a n & A proofs: manuscript no. ICA_solar_wind_charge_exchange_PaperII pound M to undergo, in the process, dissociation, excitation, andionization, or a combination of these processes.Charge exchange was initially studied as a diagnostic forman-made plasmas (Isler 1977; Hoekstra et al. 1998). The dis-covery by Lisse et al. (1996) of X-ray emissions at comet Hyaku-take C / i ) measure the speed of the solar wind (Bodewits et al.2004a), ( ii ) measure its composition (Kharchenko et al. 2003),and thus the source region of the solar wind (Bodewits et al.2007; Schwadron & Cravens 2000), ( iii ) map plasma interac-tion structures (Wegmann & Dennerl 2005), and more recently,( iv ) to determine the bulk composition of cometary atmospheres(Mullen et al. 2017).Observations of charge-exchanged helium, carbon, and oxy-gen ions were made during the Giotto mission flyby to comet1P / Halley and were reported by Fuselier et al. (1991), whoused a simplified continuity equation (as in Ip 1989) to de-scribe CX processes. Bodewits et al. (2004a) reinterpretedtheir results with a new set of cross sections. More recently,the European Space Agency (ESA)
Rosetta mission to comet67P / Churyumov-Gerasimenko (67P) between August 2014 andSeptember 2016 has provided a unique opportunity for studyingCX processes in situ and for an extended period of time (Nils-son et al. 2015a; Simon Wedlund et al. 2016). The observationsneed to be interpreted with the help of analytical and numericalmodels.As the solar wind impinges on a neutral atmosphere, either inexpansion (comets) or gravity-bound (planets), charge-transfercollisions e ff ectively result in the replacement of the incomingfast (solar wind) ion by a slow-moving (atmospheric) ion (Den-nerl 2010). Through conservation of energy and momentum,mass loading of the solar wind occurs and is responsible for thedeflection and slowing down of the solar wind ions upstreamof the cometary nucleus (see Behar et al. 2016a,b, for comet67P). For comet 67P, which has a relatively low outgassing rate,the atmosphere is essentially a mixture of H O, CO , and COmolecules (Hässig et al. 2015; Fougere et al. 2016). As a first ap-proximation, we only consider capture and stripping collisionsin H O, because this species represents the bulk of the cometarygas during the
Rosetta mission, except at large heliocentric dis-tances (above about 3 astronomical units or AU, see Läuter et al.2019). These reactions result in the production of energetic neu-tral atoms (ENAs, such as H and He), which continue to travelin straight lines from their production region, and further interactwith the ion and neutral environment.At comet 67P, evidence of solar wind charge transfer is read-ily seen in the observations of the Rosetta Plasma Consortium(RPC) ion and electron spectrometers. Nilsson et al. (2015a,b)and Simon Wedlund et al. (2016) have reported the detection ofHe + ions with the RPC Ion Composition Analyser (RPC-ICA,Nilsson et al. 2007), arising from incoming charge-exchangedHe + solar wind ions.Numerical and analytical models have been developed to ac-count for the detected ion fluxes. Khabibrakhmanov & Sum-mers (1997) developed a 1D hydrodynamic model of CX andphotoionization at comet 1P / Halley, concluding that the posi-tion of the bow shock shifted outward when taking into ac-count single-electron capture of protons in water. Ekenbäck et al.(2008) used a magnetohydrodynamics (MHD) model to produceimages of hydrogen ENA emissions around a comet similar tocomet 1P / Halley at perihelion. Simon Wedlund et al. (2016) pro- posed in a recent paper a simple 1D analytical model, usingonly one electron capture reaction (He + → He + ) to accountfor the He + fluxes that were routinely measured by RPC-ICAon board Rosetta . The authors showed that from the local mea-surement of He + / He + flux ratios in the inner coma, it was pos-sible to infer the total outgassing rate of the comet. Compari-son with in situ derived outgassing rates by the Rosetta OrbiterSpectrometer for Ion and Neutral Analysis Comet Pressure Sen-sor (ROSINA-COPS) (Balsiger et al. 2007) showed that withthese simple assumptions, month-to-month di ff erences betweenthe RPC-ICA-inferred and ROSINA-measured water outgassingrates remained within a factor 2 − > . Rosetta mission and while approaching perihe-lion, the solar wind experienced increasing angular deflectionwith respect to the Sun-comet line, defining a so-called "solarwind ion cavity" (Behar et al. 2017, noted SWIC for short). Thisis due to the increased cometary outgassing activity and massloading during that period of time, spanning April to December2015. As a result, and except for a few occasional appearancesdue to
Rosetta excursions at large cometocentric distances, noHe + and He + signal could be simultaneously detected in theSWIC (Simon Wedlund et al. 2016; Nilsson et al. 2017b; Beharet al. 2017).Charge-state distributions and their evolution with respect tooutgassing rate and cometocentric distance represent a proxy forthe e ffi ciency of charge-changing reactions at a comet such as67P, as sketched in Fig. 1. In our companion paper (Simon Wed-lund et al. 2018b, subsequently referred to as Paper I), we gaverecommended charge-changing and ionization cross sections forhelium and hydrogen particles colliding with a water gas.In this study (referred to as Paper II), we expand the initialapproach expounded in Simon Wedlund et al. (2016) to includeall six main charge-changing cross sections, and present a gen-eral analytical solution of the three-component system of heliumand hydrogen, with physical implications specific to comets. Theforward model expressions are given, and two inversions are pro-posed, one for deriving the outgassing rate of the comet, one forestimating the upstream solar wind flux from in-situ ion obser-vations. In Section 3, and using our recommended set of crosssections (see Paper I), we explore the dependence of the charge-state distribution at comet 67P on heliocentric and cometocen-tric distances, and solar wind speed and temperature. From geo-metrical considerations only, we finally make predictions for thecharge-state distribution at comet 67P at the location of Rosetta (Section 3.4).
2. Solar wind charge distributions at a comet
It is well known in the experimental community that charge-statefractions follow a system of coupled di ff erential equations thatcan be solved analytically: Allison (1958) (and associated erra-tum Allison 1959), and later Tawara & Russek (1973), for ex-ample, give expressions of the charge-state fractions of helium Article number, page 2 of 17yril Simon Wedlund et al.: Solar wind charge exchange in cometary atmospheres
Fig. 1.
Sketch of Sun-comet CX interactions. The upstream solar wind,composed of H + and He + ions, experiences CX collisions when im-pacting the comet’s neutral atmosphere, producing a mixture of chargedstates downstream of the collision. ENAs are depicted in pink. χ is thesolar zenith angle, r the cometocentric distance of a virtual spacecraft,and x points toward the comet-Sun direction, in cometocentric solarequatorial system coordinates. An increasingly deep blue denotes a cor-respondingly denser atmosphere. and hydrogen beams in gases for laboratory diagnostic in themeasurement of charge-changing cross sections. In these exper-iments, beams of incoming ions are set to collide, usually in avacuum chamber, through solid foils or in gases of known char-acteristics or neutral densities.In this section, we apply such a formalism to a cometary en-vironment. We give the equations in matrix form for an ( N + N + + , He + , He ) and (H + , H , H − ) solar wind projec-tiles and a cometary atmosphere, with N =
2. Inversions of theforward solution include the determination from local observa-tions of the neutral outgassing rate of the comet, as well as that ofan estimate of the solar wind upstream flux. Matrices and vec-tors are denoted in bold font. The nomenclature of the explicitsolution is loosely inspired by Allison’s, when needed.In the following, the CX forward model and its inversions aredescribed for the helium system. For completeness, the solutionfor the hydrogen system is given in Appendix A.
A solar wind plasma species X of initial charge i will un-dergo electron capture and loss reactions when interacting withcometary neutral species M:X i + fast + M slow −→ X ( i − q ) + fast + [M] q + slow , (3)X i + fast + M slow −→ X ( i + q ) + fast + [M] slow + qe − , (4)resulting in one reaction in the capture of q electrons by speciesX and the ionization of neutral compound [M], and in the other,in the loss of q electrons by species X. [M] denotes all possibledissociation, excitation, and ionization stages of species M. Indoing so, from the plasma point of view, fast, usually light, solarwind ions are depleted in favor of the production of slow-movingheavy cometary ions because the neutral gas has velocities ofabout 1 km s − (Hansen et al. 2016), which are added to the solar wind flow. This is one of the basic aspects of solar-wind massloading (Behar et al. 2016b). In the fluid approximation, the continuity equation for solar windspecies X i + of density n i along bulk velocity U i can be written as ∂ n i ∂ t + ∇ · n i U i = S i − L i , (5)with S i and L i its source and loss terms. To simplify this equa-tion, two assumptions can be made: ( i ) the upstream solar windis not time dependent, and so ∂ n i /∂ t = ii ) we assume that all particles of solar wind origin are mov-ing along the solar wind bulk velocity U i , with abscissa s in theSun-comet direction, with no deviation to their initial direction.Remarking that particle flux F i = n i U i , we obtaind F i ( s , T i )d s = S i ( s , T i ) − L i ( s , T i ) , (6)where T i is the ion temperature of ion species of charge i .Source and loss terms generally depend on the path d s = U i d t that the solar wind ions are having as a bulk (following bulkvelocity U i along streamline s ), but also on the ion temperature T i , that is, the path of the individual ion ( (cid:51) i d t ). We show belowthat the e ff ect of the temperature of the solar wind ions can betaken into account a posteriori , using for example Maxwellian-averaged cross sections at a given ion temperature (see Paper I),in order to mimic the change in e ffi ciency of the reactions. Here,we subsequently assume for simplicity that all ions of di ff erentcharge have the same temperature, and that T i =
0. Moreover,we have implicitly assumed that all charge states follow the samepath; rigorously, charged species will follow, depending on theirmass-to-charge ratio, a cycloidal motion driven by the solar windelectromagnetic field, whereas neutral species paths will be unaf-fected. For simplicity, we assume in the following that all chargestates of a solar wind species move with the same bulk veloc-ity (i.e., along solar wind streamlines). This assumption may in-troduce errors for example in the outgassing rate retrievals pre-sented in Section 2.3. In Paper III, our outgassing rate estimatesfrom ion spectrometer data match those from neutral measure-ments within a factor 2, implying a posteriori that to a first ap-proximation, this assumption may hold.For an initial system of N + N =
2, He + , He + , and He , or the multiple chargesystem of oxygen with N =
7, O + , O + , O + , etc.), source andloss functions for species of charge i can be rewritten as S i ( s ) = N (cid:88) j (cid:44) i σ j , i F j ( s ) n n ( s ) (7) L i ( s ) = N (cid:88) j (cid:44) i σ i , j F i ( s ) n n ( s ) , (8)where n n ( s ) is the cometary neutral density at coordinate s , σ j , i is the charge-changing cross section for processes creating a par-ticle of charge i , from a corresponding particle of charge j im-pacting a neutral species: for example, particle He + ( i =
2) iscreated from particle He + ( j =
1) through single electron loss.Similarly, σ i , j is the charge-changing cross section representingthe main loss from species of charge i to species of charge j : for Article number, page 3 of 17 & A proofs: manuscript no. ICA_solar_wind_charge_exchange_PaperII example, particle He + ( i =
1) is undergoing capture of one elec-tron, creating particle He ( j = i are performed over allother charge states j (with j (cid:44) i ).Posing that the column density element is d η = n n ( s ) d s anddropping the ( s ) dependence of the variables for convenience,equation (6) becomesd F i d η = N + (cid:88) j (cid:44) i σ j , i F j − N + (cid:88) j (cid:44) i σ i , j F i . (9)Assuming that the system is closed and the initial "undis-turbed" solar wind flux F sw of species X is conserved in astreamline cylinder, the sum of all charge states must remainequal to it: F sw = N + (cid:88) j F j . (10)If we express the lowest charge state l , in this case, the ( N + th state, of the initial system of coupled charged species X asthe sum of the other charge states, that is, F l = F sw − (cid:80) Nj (cid:44) l F j , theinitial system can then be rearranged and reduced to N coupledequations with N unknowns in matrix form, starting from thehighest (fixed) charge state k :d F ( η )d η = AF ( η ) + B , (11)where F and B are vectors of length N , and A is an N × N matrix: A = a k , k a k , k − . . . a k − , k a k − , k − . . .. . . . . . . . . a k − N + , k . . . a k − N + , k − N + , and B = F sw σ k − N , k σ k − N , k − . . .σ k − N , k − N + . Charge states ( i , j ) are here organized as row / column ele-ments a i , j of matrix A , in order to keep the generality on thecharge-state indices. Vector B contains the initial condition of thesystem, with the rate of production of each considered state fromthe lowest ( N + th state. Posing that the total charge-changingcross section (loss term) of charge state i is σ i = N (cid:88) j (cid:44) i σ i , j , (12)we can express the diagonal and non-diagonal terms of A : a i , i = − (cid:0) σ i + σ k − N + , i (cid:1) and (13) a i , j = σ j , i − σ k − N , i ∀ ( i (cid:44) j ) ∈ [ N charge states] , (14)for k the (fixed) highest charge state.We note that the charge-state distributions depend only onthe quantity of atmosphere traversed, and thus do not necessarilyimply a rectilinear trajectory along the Sun-comet line for theimpacting ions.However, when interpreting our results in Section 3, the pathof the solar wind ions is usually assumed rectilinear along theSun-comet plane, in the cometocentric solar equatorial system(CSEq) coordinate system (see, e.g., Glassmeier 2017). In thatcase, the model is valid for o ff - x CSEq -axis solar wind trajectories(as sketched in Fig. 1).
The solution of such a system is the sum of the particular so-lution to the nonhomogeneous system and of the complemen-tary solution to the homogeneous system (assuming B = F / d η →
0, system (11) simply becomes AF ∞ + B = F ∞ is none other than the charge distribution at equilib-rium (sources and losses in equilibrium), when the solar windhas encountered enough collisions so as to no longer change incharge composition (collisional thickness close to 1) (see Allison1958, for laboratory experiments). In cometary atmospheres, thisequilibrium can be reached in practice for high outgassing ratesand deep into the coma. Because matrix A is nonsingular (its de-terminant is non-zero because all charge-changing cross sectionsare di ff erent) and is thus invertible, F ∞ = − A − B is a particularsolution of system (11), which now becomesd Y ( η )d η = AY ( η ) , with Y ( η ) = F ( η ) − F ∞ . (15)The complementary solution to equation (15) with the initialcondition Y (0) = F (0) − F ∞ is Y ( η ) = e A η Y (0), with matrixexponential e A η ˆ = (cid:80) ∞ k = ( A η ) k / k ! a fundamental matrix of thesystem.Finally, the solution for the charge distribution column vector F function of the column density η is F ( η ) = F ∞ + e A η ( F (0) − F ∞ ) , (16)with: F ∞ = − A − B . The matrix exponential can be calculated by e A η = S e Λ η S − ,where Λ is the diagonal matrix of the homogeneous system(whose diagonal elements are the eigenvalues), and S is the ma-trix of passage (whose columns are the eigenvectors), so that A = S Λ S − .Result (16) is valid for any system of charged species, withdi ff erent charge states arising from charge-changing reactions(electron capture and loss) with the neutral atmosphere of anastrophysical body such as a comet or planet. This model mayinclude the calculation of the fluxes for high-charged states ofatoms in the solar wind, such as oxygen (O + , O + , . . . ), and car-bon (C + , C + , . . . ), responsible for X-ray emissions at cometsand planets (Cravens et al. 2009).This solution is also applicable to simpler charge-changingsystems such as solar wind helium particles (He + , He + , He ),for which we present an explicit solution below. For complete-ness, the similar solution for the hydrogen (H + , H , H − ) systemis also given in Appendix A. As previously, let projectile species be numbered by their charge,so that He + , He + and He have 2, 1, and 0 charges, respec-tively. Through a combination of limited column densities up-stream of the comet, expectedly small cross sections, and re-duced species lifetimes against autodetachment, lower chargestates of helium (such as the short-lived excited state of the He − anion, see Schmidt et al. 2012) are neglected. Article number, page 4 of 17yril Simon Wedlund et al.: Solar wind charge exchange in cometary atmospheres
For the three charge states of helium, the six relevant crosssections σ i j , here with i and j the starting and end charges, are σ : He + −→ He + single capture σ : He + −→ He double capture σ : He + −→ He + single stripping σ : He + −→ He single capture σ : He −→ He + double stripping σ : He −→ He + single strippingWe define for each charge state the total charge-changingcross sections, as in equation (12): σ = σ + σ for He + σ = σ + σ for He + σ = σ + σ for He (cid:88) σ i j = σ + σ + σ , with (cid:80) σ i j the sum of all six cross sections. With these notations, for N = η , matrix system (11) becomesd F ( η )d η = AF ( η ) + B , (17)with A = (cid:34) a a a a (cid:35) B = F sw (cid:34) σ σ (cid:35) and F (0) = (cid:34) F sw (cid:35) . The matrix elements a i j are, dropping the commas for clar-ity: a = − ( σ + σ ) , a = σ − σ , a = σ − σ , a = − ( σ + σ ) . In these new notations, we remark also that (cid:80) σ i j = − ( a + a ).Fluxes F will depend on the initial charge distribution of theincoming undisturbed solar wind. Far upstream of the cometarynucleus ( η = + ions only, so that F (0) = [ F (0) F (0) ] (cid:62) = [ F sw ] (cid:62) .A similar assumption can be made separately with protons. Wenow normalize our local fluxes to the initial solar wind flux bysetting F sw = F sw to obtainthe non-normalized quantities. The complementary solution of the homogeneous solution is ob-tained by solving the eigenvalue equation Av = λ v , with v theeigenvector associated with the eigenvalue λ . The characteristicpolynomial p ( λ ) = det( A − λ I ) = λ − Tr A λ + det A yields two real eigenvalues (Allison 1958), which are λ ± =
12 ( a + a ) ± q = − (cid:88) σ i j ± q , (18)when posing q = (cid:112) ( a − a ) + a a .Matrix A can then be eigen-decomposed into A = S Λ S − : Λ = (cid:34) λ − λ + (cid:35) , S = a (cid:34) t − q t + qa a (cid:35) , S − = q (cid:34) − a t + qa − t + q (cid:35) , with t =
12 ( a − a ) . The matrix exponential, expressed with the use of hyperbolicsine functions, is finally e A η = S e Λ η S − = q (cid:34) t sinh ( q η ) + q cosh ( q η ) a sinh ( q η ) a sinh ( q η ) − t sinh ( q η ) + q cosh ( q η ) (cid:35) × e − (cid:80) σ ij η . Extended to the charge fraction three-component columnvector F = [ F F F ] (cid:62) , the solution of system (17), a combi-nation of exponential functions, can then be written in the fol-lowing final form (equivalent to that of Allison 1958, for a nor-malized ion beam): F = F sw (cid:32) F ∞ + q (cid:0) P e q η − N e − q η (cid:1) e − (cid:80) σ ij η (cid:33) (19)with F ∞ = F ∞ F ∞ F ∞ = − A − B − (cid:80) i (cid:44) F ∞ i = D − a σ + a σ a σ − a σ σ ( a − a ) + a ( a + σ ) − a ( a + σ ) , P = P P P = ( t + q ) (cid:16) − F ∞ (cid:17) − a F ∞ a (cid:16) − F ∞ (cid:17) + ( t − q ) F ∞ − ( t + q + a ) (cid:16) − F ∞ (cid:17) − ( t − q − a ) F ∞ , N = N N N = ( t − q ) (cid:16) − F ∞ (cid:17) − a F ∞ a (cid:16) − F ∞ (cid:17) + ( t + q ) F ∞ − ( t − q + a ) (cid:16) − F ∞ (cid:17) − ( t + q − a ) F ∞ , recalling t =
12 ( a − a ) , q = (cid:112) ( a − a ) + a a and (cid:88) σ i j = − ( a + a ) , and A − = D (cid:34) a − a − a a (cid:35) , where D = det A = a a − a a . Article number, page 5 of 17 & A proofs: manuscript no. ICA_solar_wind_charge_exchange_PaperII
The equilibrium flux F ∞ = − A − B depends only on crosssections and is given here in full for convenience: F ∞ = F ∞ = ( σ + σ ) σ + σ σ ( σ + σ + σ ) ( σ + σ + σ ) + ( σ − σ ) ( σ − σ ) F ∞ = ( σ + σ ) σ + σ σ ( σ + σ + σ ) ( σ + σ + σ ) + ( σ − σ ) ( σ − σ ) . F ∞ = − F ∞ − F ∞ (20)In practice, it is convenient to normalize the fluxes to theupstream solar wind flux ( F sw = We present here two types of inversions of systems (19) andAppendix A.2 to retrieve from cometary observations some im-portant information on the neutral outgassing rate (as in SimonWedlund et al. 2016), and on the solar wind upstream conditions.
A first inversion of the helium system (19) or hydrogen sys-tem (A.2) consists of extracting the water outgassing rate Q from the species fluxes measured by an ion / ENA spectrometerimmersed into the atmosphere of a comet. The following devel-opment is applied to the helium system and the simultaneousdetection of He + and He + ions (as in RPC-ICA solar wind mea-surements), but can be easily extended to any species fluxes (e.g.,H / H + or H − / H + for hydrogen).Ideally, a normalized quantity should be used so that the ef-ficiency of the CX is taken into account without reference to theinitial solar wind flux. The ratio F / F fulfills this criterion (Si-mon Wedlund et al. 2016). In equation (19), we then set F sw = (cid:51) (m s − ) (Haser 1957)and a production rate Q (s − ) of neutrals n , is n n ( r ) = Q π (cid:51) r , (21)with r = (cid:112) x + y + z the cometocentric distance. We have ne-glected here the usual exponential term to account for the de-cay of neutrals at large cometocentric distances, e − ( r − r c ) k Tp / (cid:51) , with k Tp the total photodestruction rate of neutral species n (SimonWedlund et al. 2016), since it only accounts in the calculationof the column density for less than 2% di ff erence at the closecometary distances usually probed by Rosetta (i.e., for cometo-centric distances within a few tens up to 500 km or so). Moreself-consistent approaches (Festou 1981; Combi et al. 2004),taking into account the collisional part of the cometary coma,where the neutral gas moves at slower speeds (with parent, dis-sociated daughter and grand-daughter species having di ff erentejection speeds), give a di ff erent column density of neutrals thanthe Haser-like profile above, with respect to cometocentric dis-tance. However, for our demonstration, and given the uncertain-ties on several collisional parameters, a Haser-like model givesa reasonable first guess of the neutral distribution (Combi et al.2004).The outgassing rate appears as a variable in the column den-sity η . In the simple case of a rectilinear motion of the solar windions along the Sun-comet line, the column density depends on the solar zenith angle χ (Beth et al. 2016): η ( r , χ ) = (cid:90) + ∞ r cos χ n n ( s ) d s = Q π (cid:51) r χ sin χ = Q (cid:51) (cid:15) ( r , χ ) , (22)where (cid:51) is the average speed of the outgassing neutrals. χ (inunits of radians) is defined in the spherically symmetric case asthe angle between the local + x direction on the comet-Sun lineand the Sun, so that χ = arccos ( x / r ). The quantity (cid:15) ( r , χ ) thusonly depends on the geometry of the encounter, with the physicsof the gas production contained in outgassing rate Q and neutralvelocity (cid:51) .With these notations, system (19) can be rearranged as R = F (cid:16) − F ∞ / F (cid:17) F (cid:16) − F ∞ / F (cid:17) = (cid:16) P e qQ (cid:15)/ (cid:51) − N e − qQ (cid:15)/ (cid:51) (cid:17)(cid:0) P e qQ (cid:15)/ (cid:51) − N e − qQ (cid:15)/ (cid:51) (cid:1) , (23)which is of the form R = ( P y − N / y ) / ( P y − N / y ), posing y = exp( qQ (cid:15)/ (cid:51) ). The equation has two roots, for which weonly keep the positive one, since the discriminant of the equationis itself always positive: ∆ = − N − R N ) / ( P − R P ) ≥ − F ∞ / (1 − F ∞ ) ≤ R , which is always fulfilled.The solution for Q becomes Q = (cid:51) ln (cid:16) N −R N P −R P (cid:17) q (cid:15) ( r , χ ) . (24)In certain conditions, ratio R can be simplified to reflectthe direct in situ measurements made by an ion spectrometer,whereas avoiding reference to the initial upstream solar windflux, a piece of information usually out of instrumental reach.Thus, we can remark that R → F F , when F ∞ i F i (cid:28) , for i = , . This relation is in practice observed well for solar windspeeds below 400 km s − and for cometocentric distances be-tween 10 km and 500 km, which are the typical distances cov-ered by the spacecraft Rosetta while outside of the solar wind ioncavity (SWIC). The exact range of validity of this assumption isdiscussed later in Section 3.4.2.Following Simon Wedlund et al. (2016), it is interesting tonote that when only He + → He + (2 →
1) reactions are takeninto account (no electron loss or double capture) and He atomsare neglected, system (17) is greatly simplified, and leads to thefollowing expression of Q : Q = (cid:51) ln ( R + σ (cid:15) ( r , χ ) . (25)This expression is not self-consistent within the (He + , He + )system since the loss term from He + ions is not considered, andleads to an underestimate of the final outgassing rate. In practice,this expression remains useful in order to give a first indicationof the cometary outgassing rate (Simon Wedlund et al. 2016).A third expression of Q , when no electron losses are takeninto account, is proposed in Appendix B as a simple compromisebetween expressions (24) and (25). This is suitable for most ofthe Rosetta mission.
Article number, page 6 of 17yril Simon Wedlund et al.: Solar wind charge exchange in cometary atmospheres
At comets, the knowledge of the usptream (unperturbed) solarwind conditions when the spacecraft is deeply embedded in thecometary neutral atmosphere can be di ffi cult to estimate. Fromthe systems of equations presented above and a local observationof the ion fluxes, we show that it is possible, however, to retrievethe upstream solar wind flux, assuming no solar wind decelera-tion (consistent with the observations of Behar et al. 2016b, atcomet 67P) and a spherically symmetric outgassing. For a moreprecise approach, the trajectories of ions can be calculated us-ing, for example, a hybrid plasma model (Simon Wedlund et al.2017).An initial solar wind composed of α particles or protons willhave a flux F (0) = [ F sw i ] (cid:62) ( i = i = + or H + ). Fol-lowing a local measurement of the flux of α particles or protons F i ( r pos ) made at position r pos , systems (19) and (A.2) become F i ( r pos ) = F ∞ i + q (cid:0) P i e q η − N i e − q η (cid:1) e − (cid:80) σ ij η with P i = ( t + q ) (cid:16) F sw i − F ∞ i (cid:17) − a i , i − F ∞ i − , N i = ( t − q ) (cid:16) F sw i − F ∞ i (cid:17) − a i , i − F ∞ i − . (26)Solving for F sw i , the solar wind upstream flux is simply F sw i = F i ( r pos ) F ∞ i + q ( P i e q η − N i e − q η ) e − (cid:80) σ ij η . (27)The initial solar wind flux can thus be retrieved with the ad-ditional knowledge of the local cometary density, comprised in η . It is also useful to note, as in Sections 2.2 and Appendix A,that this formula is valid for any trajectory of the incoming solarwind ions because it depends only on the column density η tra-versed. However, deep inside the cometary magnetosphere, thesolar wind ions are strongly deflected, and owing to the changesin local magnetic field magnitude and direction, the normal cy-cloidal motion will be highly disturbed. This implies that inthe simplistic assumption of a rectilinear motion along the Sun-comet line of the incoming solar wind ions, the retrieved solarwind upstream flux will be underestimated by this method. Self-consistent modeling taking into account all charge-changing re-actions, using hybrid (Simon Wedlund et al. 2017) or multi-fluidMHD models (Huang et al. 2016), can overcome this caveat.
3. Results and discussion
Following the analytical expression of the solar wind charge dis-tribution in the case of a comet (Section 2), paired with the deter-mination of the cross-section sets in water (Paper I), we now turnto investigating the e ffi ciency of charge-transfer reactions withrespect to the solar wind proton and α particles. We do this fromthe point of view of equilibrium charge fractions, and the varia-tions in two solar wind-cometary parameters: the outgassing rate(depending on heliocentric distance), and the solar wind speed.In the following, we assume for simplicity a motion of thesolar wind along the Sun-comet line, that is, no deflection orslowing-down of the solar wind takes place, and no magneticpile-up region forms upstream of the nucleus. Consequently, theinitial solar wind along the Sun-comet line, containing solely(He + , H + ), becomes a mixture of their three respective chargestates. Moreover, unless otherwise stated, we adopt normalized quantities so that fluxes are comprised between 0 and 1 and theinitial solar wind flux is set to unity, that is, F sw = Figure 2 shows the equilibrium charge distributions for heliumand for hydrogen, that is, the charge fractions reached at equi-librium in case of the CX mean free path / n n σ CX (cid:28)
1. As shownin Section 2.2 and Appendix A (equations (19) and (20) for he-lium, and (A.2) for hydrogen), these fractions only depend on alinear combination of the cross sections, which themselves varywith impact speed and solar wind ion temperatures; they do notdepend on the initial composition of the impacting solar wind.They thus give insight into how e ffi cient the combined charge-changing processes are when energetic hydrogen or helium ionshit a dense atmosphere, or in a controlled environment in lab-oratory experiments such as charge-equilibrated Faraday cages,where a thin metal foil of thickness > . µ g cm − is typicallyused to achieve equilibrium (Tawara & Russek 1973).Calculations were performed for monochromatic solar windbeams (i.e., with an equivalent Maxwellian temperature of 0 K,black and gray lines in Fig. 2), and for a solar wind with aMaxwellian temperature of 3 . × K (thermal velocity (cid:51) th =
300 km s − for H + , 150 km s − for He + ions) that is represen-tative of a typical heating at a bow shock-like structure (bluecurves in Fig. 2). The equilibrium charge-state distributionswere calculated using the Maxwellian-averaged cross-sectionfits given in Paper I that are valid between 100 −
800 km s − impactor speeds.In the helium case, He + ions dominate at very high energies(speeds above 10 000 km s − ) but start to charge-transfer into amixture of He + ions and He atoms below. He + ions dominate ina narrow range around 3 000 − − impact speed whereHe and He + species make up only 20% each of the chargestate. In the typical impactor speeds of interest in solar wind-comet studies (100 −
800 km s − ), the beam is composed almostexclusively of neutral He species. The e ff ect of the solar windtemperature is marginal on the charge distributions (blue curvesin Fig. 2).Similarly, in the hydrogen case, H + ions dominate above500 km s − impact speed, whereas energetic neutral H atomsstart to dominate for all speeds below 200 km s − , including inthe solar wind speed region. H − anions make up below 10% ofthe total charge at any energy. In contrast to the helium case,however, the e ff ect of the solar wind temperature on the hydro-gen charge distributions becomes quite noticeable, especially be-low 500 km s − : compared to the monochromatic solution, theH fraction is 5% lower at 100 km s − solar wind speed, whereasthose of H − and H + increase by a factor 3 and 25 at the samespeed (although the proportion of H + to the total distribution re-mains very low). This behavior is due to the electron captureand loss cross sections of H , which peak at high energies, beingfavored over other reactions, e ff ectively populating H + and H − ions (see also Paper I). Overall, in both systems, most initial so-lar wind ions will have charge transferred to their correspondingneutral atom for velocities below 2 000 km s − by the time theyreach equilibrium. The composition of the beam with respect to cometocentric dis-tance in typical cometary and solar wind conditions is explored
Article number, page 7 of 17 & A proofs: manuscript no. ICA_solar_wind_charge_exchange_PaperII Impactor speed [km s -1 ]00.10.20.30.40.50.60.70.80.91 E qu ili b r i u m c ha r ge d i s t r i bu t i on Helium
T = 0 KT = 3.6 MK Impactor speed [km s -1 ] Hydrogen
T = 0 KT = 3.6 MK He He + He H + H H - Fig. 2.
Normalized equilibrium charge fractions of helium (left) and hydrogen (right) in H O gas as a function of solar wind speed. The fraction ofenergetic neutral atoms (He and H ) is indicated as a gray line. Corresponding Maxwell-averaged solar wind distributions for an ion temperature T = . × K are plotted in blue in the 100 −
800 km s − solar wind speed range. The grayed-out region above 800 km s − shows where thetemperature e ff ects are not calculated because of fitting limitations. below. Atmospheric composition and outgassing rates typical ofcomet 67P are used throughout. Charge-exchange reaction e ff ects are cumulative in nature, andas we showed, they depend on the column density of neutrals. Alight neutral species such as H or O (arising from photodissocia-tion of H O, OH, CO , or CO), will dominate the coma far up-stream of the cometary nucleus; such a minor species in the innercoma may thus play a non-negligible role in the removal of fastsolar wind ions a few thousand kilometers upstream of the nu-cleus because of its large-scale distribution. Above 200 000 kmfor a cometary outgassing rate of 10 s − , H may become themajor neutral species. This is especially relevant because reso-nant and semi-resonant reactions, such as H + + H (cid:10) H + H + ,have large electron capture cross sections. The resonant one-electron capture cross sections for H + + X at U sw =
450 km / simpact speed (1 keV / u) is about the same in X = H or in H O: σ (H) ≈ × − m (Tawara et al. 1985), compared to σ (H O) ≈ × − m (see Paper I). Moreover, becausecross sections for resonant processes continue to increase withdecreasing energies (Banks & Kockarts 1973), heating througha bow shock structure is expected to have a relatively small ef-fect on the e ffi ciency of CX reactions such as H + + H (see thediscussion on Maxwellian-averaged cross sections in Paper I).We now evaluate how much the solar wind proton flux de-creases as a result of proton-hydrogen CX. We first use a gener-alized Haser neutral model such as that of Festou (1981), takinginto account the photodissociated products of water, and applyit to comet 67P at 1 . ∼ × s − ) for maximum ef-fect. We then calculate the column density of hydrogen alongthe Sun-comet line up to 10 × km. We find that including Hand O and calculating the CX encountered by solar wind protonsdiminishes the expected solar wind flux at 1 000 km by 2% withrespect to the case where we include H O only. For lower so-lar wind speeds (200 km s − ), this decrease remains below 2 . + ions in He + − H reactions shows that the α particle flux decrease remains below 0 .
5% at 1 keV / uimpact energy.These results imply that at comet 67P, the inclusion of thephotodissociated products of H O has a very weak e ff ect onthe overall solar wind CX e ffi ciency and the conversion of solarwind protons and α particles into their ENA counterpart. Conse-quently, the hydrogen and oxygen cometocorona is neglected inthe following discussion. It is interesting to note, however, thatthis conclusion may di ff er between comets (because of di ff erentatmospheric composition and activity levels) and between stagesof their orbit. Because their outgassing rate is more than two or-ders of magnitude higher than that of comet 67P at perihelion,comet 1P / Halley and comet C1995 O1 / Hale-Bopp have an ex-tended hydrogen corona that does play a non-negligible role atlarge cometocentric distances (Bodewits et al. 2006).During the later part of the
Rosetta mission, CO , and to alesser extent CO, started to dominate the neutral coma over H O(Fougere et al. 2016; Läuter et al. 2019). At 1 keV / u solar windenergy, the He + -CO reaction has a one-electron capture crosssection of 5 × − m (Greenwood et al. 2000; Bodewits et al.2006), whereas that of He + -CO is about 6 × − m (Bode-wits et al. 2006). Because in H O, the σ cross section is about9 × − m (Paper I), the di ff erence in considering a H O-onlyatmosphere or a CO / CO one may lead to similar results, espe-cially when deriving a total neutral outgassing rate from the insitu measurements of the He + / He + ratio (see Section 2.3). Thisconclusion holds for H + as well, as one-electron capture crosssections between protons and H O and CO have the same mag-nitude, that is, about 20 × − m at 1 keV / u impact energy(Tawara 1978; Greenwood et al. 2000). The cometary water outgassing rate at a medium-activity cometsuch as 67P has been parameterized with respect to heliocen-tric distance by Hansen et al. (2016), using the ROSINA neutralspectrometer on board
Rosetta . Depending on the inbound (pre-perihelion) and outbound (post-perihelion) legs, the total H O Article number, page 8 of 17yril Simon Wedlund et al.: Solar wind charge exchange in cometary atmospheres neutral outgassing rate Q was Q in = (2 . ± . × R − . ± . s − (28) Q out = (1 . ± . × R − . ± . s − , indicating an asymmetric outgassing rate with respect to per-ihelion. R Sun denotes the heliocentric distance in AU. In order toobtain an estimate of the charge distribution of the solar windduring the
Rosetta mission, we chose to use the outgassing rate Q in determined at inbound, where H O dominates the neutralcoma. As mentioned above, close to the end of the mission, out-side of 3 AU, CO became predominant (Fougere et al. 2016;Läuter et al. 2019).We use in this section a Haser-like model (Haser 1957) in-cluding sinks, so that the density of water molecules n n at thecomet is given by n n = Q in π (cid:51) r exp (cid:32) − f d ( r − r c ) (cid:51) (cid:33) , (29)where r is the cometocentric distance, r c is the comet’s radiusand f d is the total photodestruction frequency (ionization plusdissociation) of H O as a result of the solar EUV flux. The ef-fect of the exponential term becomes important at large cometo-centric distances. f d depends on the heliocentric distance (Hueb-ner & Mukherjee 2015). In contrast to equation (21), which isvalid for close orbiting around the comet, the exponential termis kept because of the large cometocentric distances consideredhere and the cumulative aspect of charge-changing reactions. (cid:51) is the radial speed of the neutral species, typically in the range500 −
800 m s − at comet 67P (Hansen et al. 2016). Speed (cid:51) iscalculated using the empirically determined function of Hansenet al. (2016): (cid:51) = ( m R R Sun + b R ) (cid:18) + . e − R Sun − . . (cid:19) , (30)with m R = − . b R = . . (31)where m R and b R are fitting parameters, so that (cid:51) is expressed inm s − . The column density η is integrated numerically.In order to obtain an average e ff ect, we chose here (cid:51) =
600 m s − , f d = . × − (1AU / R Sun ) s − , correspond-ing to low solar activity conditions (Huebner & Mukherjee 2015,including all photodissociation and photoionization channels),and a constant solar wind bulk speed of U sw =
400 km s − .Figure 3 shows the beam fractionation for helium (left) andhydrogen (right) as a function of cometocentric distance, and forthree heliocentric distances: 1 . Q = . × s − ), 2 AU( Q = . × s − ), and 3 AU ( Q = . × s − ). We notethat the 1 . ff ects(Simon Wedlund et al. 2017). That said, the validity of our modeldependd on several parameters: the outgassing rate, solar EUVintensity, and solar wind parameters. It also depends on the po-sition of the spacecraft in a highly asymmetric plasma environ-ment with respect to the Sun-comet line. All of these parametersmay significantly fluctuate in a real-case Rosetta -like scenario.This implies that the validity range of our model with respectto the cometocentric distance may extend or shrink dependingon these parameters, and should thus be carefully evaluated inspecific case studies. At 1 . + become graduallyconverted into an equal mixture of He + ions and He energeticneutral atoms, which become predominant below 100 km come-tocentric distance. Correspondingly, all curves in Fig. 3 (initiallyin black for 1 . Rosetta was outside the SWIC region, that is, for R Sun (cid:38) < r <
100 km.For a constant solar wind speed of 400 km s − , this results inHe + being the most important helium species for most of thetime during the solar wind ion measurements, with a proportionof about / each for (He + , He + , He ) at the limit at 10 km come-tocentric distance.The hydrogen system presents a much simpler picture, withH − accounting for less than 7% of the total hydrogen beam atany cometocentric and heliocentric distances. At 1 . Rosetta ,that is, for 30 km distance and below. All scales considered, theseconclusions are in qualitative agreement with those of Ekenbäcket al. (2008), who used an MHD model to image hydrogen ENAsaround the coma of comet 1P / Halley.For reference, Appendix C shows how the collision depth τ cx i = η ( r ) σ i that is due to charge-changing processes ina H O atmosphere varies with cometocentric and heliocentricdistances. It shows that the atmosphere is almost transparentto H and He ENAs, whereas H + and He + ions will becomemuch more e ffi ciently charge-exchanged on their way to the in-ner coma.As with the equilibrium charge states, charge distributionswith a solar wind Maxwellian temperature of T = . × K anda solar wind bulk speed U sw =
400 km s − were also computed.Temperature e ff ects are mostly seen for the hydrogen case, withan increase in loss cross sections from H , which are more ef-ficiently converted back into H + and H − ions. Hence protonsare not any more totally converted into their lower charge stateswhen the solar wind becomes significantly heated. For our Sun, the solar wind speed varies typically between 300and 800 km s − and is not modified with increasing heliocentricdistance (Slavin & Holzer 1981). The main variations are due tothe regular (corotating interaction regions due to the Sun’s rota-tion) and transient (coronal mass ejections) nature of the solaractivity, and its subsequent dynamics in interplanetary space. Inextreme cases, the solar wind speed, and thus the impact speedof the protons and α particles, may increase up to several thou-sand km s − in a matter of hours (Ebert et al. 2009). Adding solarwind temperature e ff ects and heating at shock-like structures tothese variations in bulk speed, large combined e ff ects may arisein the charge distribution of solar wind particles.Figure 4 shows the monochromatic charge distributions as afunction of cometocentric distance for helium species (left) andfor hydrogen species (right) for solar wind bulk speeds rangingbetween 100 and 2000 km s − . The calculations were made herefor a distance of 2 AU, hence at the limit when Rosetta enteredthe SWIC; they are comparable to the gray curves in Fig. 3. Aheliocentric distance of 2 AU corresponds to a water outgassingrate of Q = . × s − and a neutral speed (cid:51) ≈
600 m s − ,chosen at inbound conditions (Hansen et al. 2016). Article number, page 9 of 17 & A proofs: manuscript no. ICA_solar_wind_charge_exchange_PaperII Cometocentric distance r comet [km]00.10.20.30.40.50.60.70.80.91 C ha r ge d i s t r i bu t i on Helium He + He Cometocentric distance r comet [km]
Hydrogen + H H - Fig. 3.
Normalized charge distributions of helium (left) and hydrogen (right) species in a H O 67P-like atmosphere for di ff erent outgassing rates(or heliocentric distances). The solar wind speed is assumed to be constant and equal to 400 km s − . (X i + , X ( i − + , X ( i − + ) components of projectilespecies X of initial positive charge i are plotted as solid, dashed, and dotted lines, respectively. At this level of cometary activity for 67P, no full-fledged bowshock structure is expected to have formed yet, although indi-cations of a bow shock in the process of formation have beenreported in Gunell et al. (2018) already at around 2 . + ions move further downstream before being a ff ected by the heat-ing due to the presence of the shock-like structure. This impliesthat in this case, our model would be valid at lower cometocen-tric distances for helium particles than for hydrogen particles.Moreover, during these events, Rosetta likely explored di ff er-ent locations in the comet-Sun plane containing the solar windconvection electric field because of the asymmetry of the bow-shock-like structure in this plane, hence modifying the validityrange of the model depending on the o ff - x -axis position of thespacecraft. Therefore, our model is expected to be valid at a 67P-like comet down to typically a few tens of kilometers from thenucleus. Therefore, in this section, no thermal velocity distribu-tion for the solar wind particles is assumed.Owing to the velocity dependence of charge-changing reac-tions, a change in velocity in Fig. 4 results for helium speciesin a complex behavior where the proportion of He + ions (solidlines) first increases slightly from 100 to 400 km s − , decreasesby about 10% from 400 to 800 km s − , and finally increasesagain toward 2000 km s − at any cometocentric distance to levelssimilar to those for 100 km s − . In parallel, the proportion of He + (dashed lines) dramatically increases until about 800 km s − ,where it settles at a maximum around 45% ( ∼
10 km cometocen-tric distance), a value that does not change much above this so-lar wind speed. This tendency can be more clearly seen in Fig. 5(left, black curves), where we calculate the charge distributionsas a function of solar wind speed at 50 km from the nucleus.Regarding He , it is interesting to note that the lower the solarwind speed, the larger the fraction of He atoms. This is linkedto the high double charge capture cross section of He + at theseenergies as compared to the single charge capture, as discussedlater in Section 3.3 (see also Bodewits et al. 2004b). At high im-pact speeds, this e ff ect becomes reversed, and He + ions becomerelatively more abundant than He atoms, and are the main losschannel of He + ions. For hydrogen species, the proportion of protons H + first di-minishes (10% decline between 100 and 400 km s − on average)and then increases with solar wind speed in the 800 − − range ( +
30% on average). The proportion of neutral atomsH peaks below 400 km s − solar wind speed; they may be-come dominant over H + at cometocentric distances below about30 km. These two e ff ects are also shown in Fig. 5 (right, blackcurves). Similar to what we observed for the heliocentric dis-tance study (Section 3.2.3), H − ions make up only 10% or lessof the solar wind, with a small increase seen below 10 km come-tocentric distance, where the atmosphere becomes increasinglydenser; the maximum e ff ect is reached when the solar wind bulkspeeds are about 800 km s − . We investigate now the e ff ect of individual processes on the com-position of the beam at a heliocentric distance of 2 AU (justoutside of the SWIC, see Behar et al. 2017, and previous sec-tion), and a cometocentric distance of 50 km. The latter distancewas chosen as a typical orbital distance of Rosetta at 2 AU. Weused the recommended fitted monochromatic charge-changingcross sections of Paper I, with solar wind speeds ranging from100 km s − to 5000 km s − .Figure 5 shows the charge distribution of helium (left) andhydrogen (right) species as a function of solar wind speed at50 km cometocentric distance. From Fig. 9 of Paper I, doublecharge exchange (DCX) He + → He is expected to be the mainloss of α particles at solar wind speeds below 300 km s − , lead-ing to the creation of He atoms, whereas single charge exchangeHe + → He + starts to play a more important role at higher so-lar wind speeds. When we set the DCX cross sections to zero( σ = σ =
0) in our simulations (Fig. 5, blue curves),the solar wind contains less than 2% He atoms at any impactspeed, whereas their proportion climbs up to almost 20% at100 km s − when DCX is taken into account. As expected fromthe shapes of the cross sections and the relative abundance ofHe + and He , the most important e ff ect is for the 2 → σ , Article number, page 10 of 17yril Simon Wedlund et al.: Solar wind charge exchange in cometary atmospheres Cometocentric distance r comet [km]00.10.20.30.40.50.60.70.80.91 C ha r ge d i s t r i bu t i on Helium
100 km s -1
400 km s -1
800 km s -1 -1 He He + He Cometocentric distance r comet [km]
Hydrogen
100 km s -1
400 km s -1
800 km s -1 -1 H + H H - Fig. 4.
Normalized charge distributions of helium (left) and hydrogen (right) species in a H O 67P-like atmosphere for di ff erent solar wind speeds( U sw = − − ) and at a heliocentric distance of 2 AU. σ , σ ) impact the charge distributions with respect to solarwind speed. This is shown as gray curves in Fig. 5. No drasticchange is seen when the EL processes are turned on or o ff in oursimulations (black and gray curves are almost superimposed inthis figure).For hydrogen species, neither EL nor DCX processes seem toplay any significant role in the composition of the beam at 2 AU,implying that the main processes populating all three species atthe comet are single-electron capture. This analysis is furthervindicated by the behavior of the hydrogen system with respectto heliocentric and cometocentric distances (see Sections 3.2.2and 3.2.3). That said, EL processes may start playing a role atsolar wind speeds above 800 km s − and for cometocentric dis-tances below about 10 km, where the neutral column density be-comes comparatively much higher.Maxwellian-averaged cross sections can also be used here;because DCX usually peaks at low impact velocities, He + ionswill be less e ffi ciently converted into He atoms with increasingsolar wind temperature. Di ff erences in the charge compositionof the solar wind, especially below 300km s − , will start to ap-pear (figure not shown) for temperatures T (cid:38) + and He + over He ), and for T (cid:38) + overH ).Because EL processes are expected to play a minor role at Rosetta ’s position around comet 67P, flux charge distributionsand arguably simpler expressions for the reduced EL-free systemcan be derived. These equations are presented in Appendix B forclarity.
Rosetta mission
To finalize our theoretical study of charge-changing processes ata 67P-like comet, we now first turn to evaluating the normalizedcomposition of the solar wind helium and hydrogen charge dis-tributions in the vicinity of 67P throughout the
Rosetta mission(2014-2016). Using the analytical model inversions presentedin Section 2.3, we then show how the outgassing rate and so-lar wind upstream fluxes can be reconstructed from the in situknowledge of the He + -to-He + ratio and proton flux, and we ap- ply this technique to the complex trajectory of Rosetta aroundcomet 67P. Validations of these inversions are presented in Sec-tions 3.4.2 and 3.4.3 and follow the following scheme: ( i ) gen-erate virtual measurements from basic upstream solar wind andcometary outgassing rate parameters, ( ii ) use forward analyticalmodel to produce the expected solar wind charge distributionslocally at the geometric position of Rosetta , and ( iii ) performinversions from the locally generated fluxes to retrieve the up-stream solar wind conditions or the outgassing rate.
This paragraph aims at simulating what an electron-ion-ENAspectrometer would observe at the location of
Rosetta aroundcomet 67P. Because of the large dataset that we attempt to sim-ulate, we derived here the column density η following the sim-ple 2D integration of Beth et al. (2016), and set the exponen-tial term in equation (29) to one. At the position of Rosetta , thedi ff erence between including or excluding the exponential lossterm that is due to photodestruction is negligible, as discussedin Section 2.3. The column density is given by equation (22)with the neutral outgassing rate and speed (cid:51) parameterized byequations (28) and (30) (see Hansen et al. 2016). The geome-try of Rosetta in di ff erent coordinate systems, including CSEq,is accessible via the European Space Agency Planetary ScienceArchive (PSA). For simplicity, an average solar wind speed of400 km s − (Slavin & Holzer 1981) was chosen to calculate thetotal charge-changing cross sections during the mission. Solarwind propagation from point measurements at Mars (with theMars Express, MEX, spacecraft) and at Earth (with the ACEsatellite) would provide a more physically accurate upstream so-lar wind, although at the expense of simplicity in our theoreticalinterpretation. For the comparison with Rosetta observations, werefer to Paper III.Figure 6 shows the simulated normalized charge distribu-tions at the position of
Rosetta for comet 67P between 2014-2016 for helium (left) and hydrogen (right) species. The SWIC,where the solar wind was mostly prevented from entering theinner coma, is shown as a gray-graded region and spans almosteight months between late April and early December 2015 (Be-
Article number, page 11 of 17 & A proofs: manuscript no. ICA_solar_wind_charge_exchange_PaperII
500 1000 1500 2000 2500 3000 3500 4000 4500 5000Solar wind speed [km s -1 ]00.10.20.30.40.50.60.70.80.91 C ha r ge d i s t r i bu t i on Helium He He + He AllNo ELNo DCXHe He + He
500 1000 1500 2000 2500 3000 3500 4000 4500 5000Solar wind speed [km s -1 ] Hydrogen H + H H - AllNo ELNo DCXH + H H - Fig. 5.
Normalized charge distributions of helium (left) and hydrogen (right) and e ff ect of individual processes for a cometocentric distanceof 50 km and a heliocentric distance of 2 AU. Double charge exchange (labeled "DCX") and electron loss (labeled "EL") cross sections aresequentially set to zero and compared to the full set (labeled ’All’). For hydrogen species, di ff erences between all three runs are minimal. Solarwind speeds range from 100 km s − to 5000 km s − . har et al. 2017). It corresponds to times where the column densitytraversed by the solar wind beams becomes comparatively high.In this ion cavity, both He + and H + beams are strongly de-pleted in favor of lower charge states, which coincides with thelack of in situ observations during this period (Behar et al. 2017).Whether this cavity has a well-defined surface, or how dynami-cal it is (with regard to the spacecraft position), are questions thatare unanswered as of now because they challenge the ion sen-sors at the limit of their capacity (field-of-view limitations andsensitivity). Our analytical model does not take into account thecomplex trajectories of solar wind particles in the inner coma (aspointed out in Behar et al. 2018; Saillenfest et al. 2018), which islikely to increase the e ffi ciency of the CX because of the curvi-linear path of projectiles, which also depends on their chargeand mass. Consequently, the correct origin of the SWIC may bebetter investigated by a self-consistent modeling that includesthe physico-chemistry of the coma, such as a quasi-neutral hy-brid plasma model (Koenders et al. 2015; Simon Wedlund et al.2017; Lindkvist et al. 2018). In our results, the analytical calcu-lations should in this region only be seen as an indication of thecharge distribution of the solar wind for rectilinear trajectoriesof the incoming solar wind.For the helium system, He + constitutes the bulk of thecharged states, reaching percentages of at least 70% outside ofthe SWIC. He + ions and He atoms have a similar behavior andare each about 15% of the total helium solar wind. Because ofthe changing geometry and outgassing rate, the compositionalfractions are asymmetric with respect to perihelion. Because noENA detector was on board Rosetta , the full charge distributionof the solar wind cannot be determined; a new mission to anothercomet could thus usefully include such an instrument (Ekenbäcket al. 2008). In two instances before perihelion, in February 2015and at the end of March 2015 ( R Sun ∼ and He + increased dramatically after the spacecraft orbitedwithin 10 km from the nucleus. As shown in Fig. 3, the pro-portion of these two charge states increases dramatically in andaround this cometocentric distance and closely matches that ofHe + ions. For the hydrogen system, in a way similar to the helium sys-tem, the solar wind contains mostly protons, with an averagepercentage of 70% outside of the SWIC. In the two instances de-scribed above, H atoms also become more abundant than H + .In agreement with the previous sections, hydrogen negative ionsH − only seem to be of note around perihelion, where it reachesabout 3 −
4% of the total (outside of the SWIC, the abundancelevels are closer to 0 . Rosetta . Burch et al.(2015) ascribed the observed H − to the two-step charge-transferprocess H + → H → H − from solar wind protons around 1 keVenergy ( ∼
437 km s − ). At this bulk speed, our values of σ (1 . × − m ) and σ − (6 . × − m ) are similar within afactor 2 to those used by Burch et al. (2015), whereas our valueof σ − (4 . × − m ) is a factor 3 . for this reaction). Consequently, their mainconclusions remain unchanged: the two-step process is in ournew calculations about 23 times more e ffi cient than DCX reac-tions to produce H − anions. When making the numerical appli-cation and correcting their two-step process to 1 × − F sw , andof double capture to 3 × − F sw , Burch et al. (2015) shouldhave found a ratio of about 3 . − component in our simulation is very faintand therefore points to the presence of favorable neutral-plasmaconditions (increased outgassing, small cometocentric distance,increased solar wind flux, or combinations thereof) in order to bedetectable. This conclusion is contained in the account of Burchet al. (2015). This section aims at validating our inversion procedure for theoutgassing rate, using the He + / He + ratio as a proxy of the neu-tral outgassing at the comet. We follow three steps: ( i ) calcu-lation of the He + / He + ratio at Rosetta during the mission, us-ing the forward analytical model with the neutral atmosphere of
Article number, page 12 of 17yril Simon Wedlund et al.: Solar wind charge exchange in cometary atmospheres
Sep Nov Jan Mar May Jul Sep Nov Jan Mar May Jul SepTime [UT]00.10.20.30.40.50.60.70.80.91 C ha r ge d i s t r i bu t i on Helium He + He Sep Nov Jan Mar May Jul Sep Nov Jan Mar May Jul SepTime [UT]
Hydrogen + H H - Fig. 6.
Expected normalized charge distributions of helium (left) and hydrogen (right) during the
Rosetta mission (2014-2016) at the location ofthe spacecraft. The SWIC encountered at comet 67P is marked as a gray region, with a smooth gradient to indicate its dynamic nature.
Hansen et al. (2016) as inputs (as in Fig. 6), ( ii ) computationof the geometric factor (cid:15) ( r , χ ) entering in the expression of thecolumn density (see equation (22)), which depends on Rosetta ’sposition around comet 67P during the mission, ( iii ) final recon-struction of the outgassing rate from the local He + / He + ratio,using equation (24).Figure 7 presents the results of this approach and comparesour reconstructed outgassing rate (black) with the productionrate fit of Hansen et al. (2016), which was used in the first placeto generate the charge distributions in Fig. 6. Very good agree-ment within 15% on average is found throughout the mission,except for occasional events, such as the cometary tail excur-sion around April 2016 or at the end of the mission. This stemsfrom the approximation made in the inversion procedure detailedin Section 2.3, with the condition F ∞ i / F i (cid:28) , for i = , + but not for He + during thetail excursion (because of the large cometocentric distance andcomparatively low column density), and during the early andlater parts of the mission (very low column density for a com-paratively small cometocentric distance). The sweet spot of theretrieval method with the fulfilled condition for He + is conse-quently achieved in the region where the He + charge fraction ispeaking with respect to the cometocentric distance (see this re-gion in Fig. 3, left, dashed lines). Outside of these regions, and ifthe solar wind speed is about 400 km s − or above, a much sim-pler approach, as detailed in Simon Wedlund et al. (2016) andepitomized by equation (25), may prove better suited. This isdemonstrated by the yellow line in Fig. 7, which at this constantsolar wind speed agrees with the input outgassing rate of Hansenet al. (2016) to within 5%. However, during the Rosetta mission,this solar wind speed value is only encountered episodically, ascan be seen in the solar wind velocities measured by the RPC-ICA ion spectrometer (Behar et al. 2017) and the more com-plex approach developed in the present study, with six charge-changing cross sections, is warranted.
The second inversion introduced in Section 2.3 enables recon-structing the upstream solar wind flux or density from local mea- surements made deep into the coma. To test our inversion, wefirst created synthetic upstream solar wind conditions, which wepropagated with the forward analytical model at the position of
Rosetta .According to the parameterization of Slavin & Holzer (1981)with respect to heliocentric distance, the undisturbed proton den-sity is n p = × R − m − , with R Sun expressed in AU. For aconstant 400 km s − solar wind bulk speed, this is equivalent toan upstream solar wind proton flux F p = . × R − m − s − ,which is commensurable to the flux levels measured by the RPC-ICA instrument on board Rosetta (Nilsson et al. 2017a,b). Onaverage, the solar wind is composed of about 4% He + ions(e.g., Simon Wedlund et al. 2017). We first applied the analyt-ical model to the inputs above and calculated the resulting localproton and helium ion fluxes at the position of Rosetta during themission; this is equivalent to multiplying the normalized chargedistributions in Fig. 6 by the upstream solar flux F p for protons,and by F α = / F p for α particles. Using equation (27), we thenreconstructed the upstream solar wind flux from the syntheticfluxes.The results are presented in Fig. 8, where the solar wind in-put flux and the reconstructed upstream flux match perfectly. Inconformity with Fig. 6, the solar wind fluxes are expected to de-crease by almost one order of magnitude around perihelion at theposition of Rosetta as a result of CX processes.For comparison purposes, we also calculated the e ff ect ofvery high Maxwellian temperatures for the solar wind, with T = × K reminiscent of a strong heating at a full-fledgedbow shock structure in the upstream solar wind. These temper-atures correspond to thermal velocities of 1000 km s − for pro-tons and 500 km s − for α particles. This is shown in Fig. 8 asdotted lines. Increasing the temperatures leads to a similar trend,albeit reinforced, to the trend that we previously described inSection 3.1: proton fluxes are reinforced (factor × . + fluxes undergo a decrease by a factor of about 1 . Rosetta ’s position (as in Behar 2018), and how they connect
Article number, page 13 of 17 & A proofs: manuscript no. ICA_solar_wind_charge_exchange_PaperII O u t ga ss i ng r a t e Q [ s - ] P e r i he li on E xc u r s i on E xc u r s i on (a) Input (Hansen et al., 2016)Reconstruction (inversion)Simon Wedlund et al. (2016)Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep OctTime [UT]10 r c o m e t [ k m ] R S un [ A U ] (b) Fig. 7. (a) Outgassing rate Q reconstructed from the inversion of the analytical model (black line), compared to the original input outgassing rate(Hansen et al. 2016) (blue stars) during the Rosetta mission (2014-2016). The orange line is the result of the simplified approach of equation (25)where only He + → He + reactions were considered (Simon Wedlund et al. 2016); this method works well at the constant solar wind speedof 400 km s − chosen here. (b) Geometry parameters of the spacecraft Rosetta at comet 67P during its two-year mission: black, cometocentricdistance; red, heliocentric distance. Two large cometocentric distance excursions are indicated in blue: the dayside excursion (September-October2015), and the cometary tail excursion (March-April 2016). Gray-shaded regions mark years. with the local flux measurements made with RPC-ICA, will bediscussed in our next study (Paper III).
4. Conclusions
We have developed a 1D analytical model of charge-changingreactions at comets based on the fluid continuity equation andwithin the assumptions of stationarity and of particle motionalong solar wind streamlines at the same bulk speed. A sensi-tivity study on several cometary parameters was then conductedfor helium and hydrogen three-component systems. The resultsare listed below. – Double charge transfer is important for helium, especially atsolar wind velocities below about 500 km s − . – Electron loss (stripping) plays only a minor role in the com-position of the solar wind at any solar wind impact speedand at the typical cometocentric and heliocentric distancesencountered by the
Rosetta spacecraft. For high solar windspeeds ( >
800 km s − ) and much higher column densities,stripping e ff ects may start to appear, especially for hydrogenprojectiles. – Solar wind temperature e ff ects start to play a role at tempera-tures T > × K, in accordance with Simon Wedlund et al.(2018b). At comet 67P at the position of
Rosetta , this resultsin an increase in proton fluxes by a factor 3 − α particles are further depleted compared toa monochromatic (monoenergetic) solar wind.We have also shown that with this analytical model, thecharge-state distribution of helium and hydrogen species in cometary atmospheres can be predicted, with the use of a to-tal of 12 charge-changing reactions in a water atmosphere (seeSimon Wedlund et al. 2018b, for recommended cross sections).We predict at a 67P-like comet the formation of a region be-low 2 AU where the incoming solar wind ions are e ffi ciently lostto lower charge states and ENAs through CX reactions alone.In combination with kinetic plasma e ff ects and the formation ofshock-like structure upstream of the nucleus (Gunell et al. 2018),CX may thus play an additional role in the creation of the so-lar wind ion cavity characterized with Rosetta by Behar et al.(2017). From the knowledge of in situ ion composition appliedto He + and He + , we also demonstrated that it is possible to re-trieve the outgassing rate of neutrals and solar wind upstreamconditions purely from geometrical considerations and from lo-cal measurements made deep into the coma, assuming a spheri-cally symmetric 1 / r expansion for the neutral atmosphere.This article is the second part of a triptych on charge-transfere ffi ciency around comets. The first part gives recommendationson low-energy charge-changing and ionization cross sections ofhelium and hydrogen projectiles in a water gas. The third part,presented in Simon Wedlund et al. (2018a), aims at applying thisanalytical model and its inversions to the Rosetta Plasma Con-sortium (RPC) datasets, and in doing so, at quantifying charge-transfer reactions and comparing them to other processes duringthe Rosetta mission to comet 67P.
Appendix A: Hydrogen system, forward model
We present here the explicit analytical model for the system of(H + , H , H − ) and its six charge-changing reactions with a neutral Article number, page 14 of 17yril Simon Wedlund et al.: Solar wind charge exchange in cometary atmospheres
Fig. 8.
Solar wind flux reconstructed from the inversion of the analytical model compared to the original input solar wind upstream flux (Slavin &Holzer 1981) during the
Rosetta mission (2014-2016). Proton and α particle fluxes (4% of the total solar wind ions) are considered. Dotted linesare results of the forward analytical model with a solar wind Maxwellian temperature T = × K. Otherwise, the caption is the same as forFig. 7. atmosphere. The solution is identical to that presented in Sec-tion 2.2 for the helium system and is given here for completenessin the manner of Allison (1958). The relevant cross sections arein this case σ : H + −→ H single capture σ − : H + −→ H − double capture σ : H −→ H + single stripping σ − : H −→ H − single capture σ − : H − −→ H + double stripping σ − : H − −→ H single strippingWe may correspondingly pose σ = σ + σ − for H + σ = σ + σ − for H σ − = σ − + σ − for H − (cid:88) σ i j = σ + σ + σ − , with (cid:80) σ i j the sum of all six cross sections.For an upstream solar wind flux F sw , and with F , F and F − representing H + , H and H − fluxes, the matrix system (11)of equations for the reduced (H + , H ) system with N = η (cid:34) F F (cid:35) = (cid:34) a a a a (cid:35) (cid:34) F F (cid:35) + F sw (cid:34) σ − σ − (cid:35) , (A.1)with a = − ( σ + σ − ) , a = σ − σ − , a = σ − σ − , a = − ( σ + σ − ) . Solution (19) for the helium system can be made to apply tothe hydrogen system by subtracting every finite index of flux and cross section by 1, so that F = F sw (cid:32) F ∞ + q (cid:0) P e q η − N e − q η (cid:1) e − (cid:80) σ ij η (cid:33) (A.2)with F ∞ = F ∞ F ∞ F ∞− = D − a σ − + a σ − ( a σ − − a σ − ) σ − ( a − a ) + a ( a + σ − ) − a ( a + σ − ) , P = P P P − = ( t + q ) (cid:16) − F ∞ (cid:17) − a F ∞ a (cid:16) − F ∞ (cid:17) + ( t − q ) F ∞ − ( t + q + a ) (cid:16) − F ∞ (cid:17) − ( t − q − a ) F ∞ , N = N N N − = ( t − q ) (cid:16) − F ∞ (cid:17) − a F ∞ a (cid:16) − F ∞ (cid:17) + ( t + q ) F ∞ − ( t − q + a ) (cid:16) − F ∞ (cid:17) − ( t + q − a ) F ∞ , recalling t =
12 ( a − a ) , q = (cid:112) ( a − a ) + a a , (cid:88) σ i j = − ( a + a ) , and D = a a − a a . Appendix B: Electron loss-free helium system andsimplified formula for outgassing rate
As shown in Section 3, electron loss reactions do not play a sig-nificant role at typical solar wind speeds and for the heliocentricdistances encountered during the orbiting phase of
Rosetta . Ig-noring the three electron loss reactions σ , σ , and σ , that is,with only CX reactions considered, the flux continuity equationfor the (He + , He + , He ) helium system with fluxes ( F , F , F ) Article number, page 15 of 17 & A proofs: manuscript no. ICA_solar_wind_charge_exchange_PaperII reduces to d F d η = − ( σ + σ ) F d F d η = σ F − σ F , d F d η = σ F + σ F (B.1)where, by definition, F = F sw − ( F + F ). Solving this systemof single di ff erential equations, we find the expression of fluxesdepending on column density η : F = F sw e − ( σ + σ ) η F = σ σ − ( σ + σ ) (cid:0) F − F sw e − σ η (cid:1) , F = F sw − ( F + F ) (B.2)which is considerably simpler than the full six-reaction solu-tion (19). This expression yields results that are almost identi-cal to the full six-reaction model in the conditions probed by Rosetta . Di ff erences between the two approaches are negligi-ble at solar wind speeds of 400 km s − and below, but may be-come noticeable for higher values, when the maxima of strippingcross sections are approached. An illustration of the di ff erenceexpected at comet 67P between the six-reaction model and thepresent electron loss-free solution is shown in Fig. B.1 at 2 AUfor a solar wind speed of 2000 km s − . Such high speeds can beencountered in extreme solar transient events such as coronalmass ejections (Meyer-Vernet 2012). In this case, electron lossreactions start to play a role below 10 km cometocentric distancefor helium and below about 30 km for hydrogen.We may extract the column density η , which depends oncometocentric distance r and solar zenith angle χ , by calculat-ing the flux ratio R = F / F as in Section 2.3: η ( r , χ ) = ln (cid:16) + σ + σ − σ σ R (cid:17) σ + σ − σ . (B.3)Taking the definition of the approximate cometaryneutral column density from equation (22), that is, η ( r , χ ) = Q π (cid:51) r χ sin χ = Q (cid:51) (cid:15) ( r , χ ), the cometary neutraloutgassing rate is under these assumptions Q = (cid:51) (cid:15) ( r , χ ) ln (cid:16) + σ + σ − σ σ R (cid:17) σ + σ − σ . (B.4)This expression reduces further to equation (25) when wepose σ = σ = Appendix C: Note on collision depth
Figure C.1 displays the charge-changing collision depth, definedas τ cx i = η ( r ) σ i , where σ i is the sum of loss cross sectionsfor each charge state i (equation [12]) of helium and hydro-gen. In analogy with the Beer-Lambert optical depth, τ cx ≥ ff ectively"opaque" to charge-changing reactions: particles experience sig-nificant charge-changing collisions. It depends on the projectilestate, its energy, and on the neutral atmosphere, parameterized bya Haser-like model (see equation [29], with (cid:51) =
600 m s − ). Fora solar wind bulk speed of 400 km s − , the atmosphere is almost transparent to He and H ENAs over the full range of cometo-centric distances and for all heliocentric distances. This tendencyis enhanced even further when decreasing the solar wind speedto 100 km s − , with τ cx i = ffi ciently charge-exchanged intolower charge states by the time they reach the typical cometo-centric distances probed by the Rosetta spacecraft.
Acknowledgements.
The work at University of Oslo was funded by the Norwe-gian Research Council "
Rosetta " grant No. 240000. Work at the Royal BelgianInstitute for Space Aeronomy was supported by the Belgian Science Policy Of-fice through the Solar-Terrestrial Centre of Excellence. Work at Umeå Universitywas funded by SNSB grant 201 /
15. Work at Imperial College London was sup-ported by STFC of UK under grant ST / K001051 / / N000692 /
1, ESA,under contract No.4000119035 / / ES / JD. The work at NASA / SSAI was sup-ported by NASA Astrobiology Institute grant NNX15AE05G and by the NASAHIDEE Program. C.S.W. would like to thank S. Barabash (IRF Kiruna, Sweden)for useful impetus on the work leading to the present study and for suggestingto investigate electron stripping processes at a comet. The authors thank the ISSIInternational Team "Plasma Environment of comet 67P after
Rosetta " for fruit-ful discussions and collaborations. C.S.W. thanks M.S.W. and L.S.W. for help instructuring this immense workload and for unwavering encouragements through-out these two years of work. Datasets of the
Rosetta mission can be freely ac-cessed from ESA’s Planetary Science Archive ( http://archives.esac.esa.int/psa ). References
Allison, S. K. 1958, Rev. Mod. Phys., 30, 1137Allison, S. K. 1959, Rev. Mod. Phys., 31, 839Balsiger, H., Altwegg, K., Bochsler, P., et al. 2007, Space Sci. Rev., 128, 745Banks, P. M. & Kockarts, G. 1973, Aeronomy, Part A (Academic Press, NewYork and London)Behar, E. 2018, PhD thesis, Luleå University of Technology, Space TechnologyBehar, E., Lindkvist, J., Nilsson, H., et al. 2016a, A&A, 596, A42Behar, E., Nilsson, H., Alho, M., Goetz, C., & Tsurutani, B. 2017, Month. Not.Roy. Astron. Soc., 469, S396Behar, E., Nilsson, H., Wieser, G. S., et al. 2016b, Geophys. Res. Lett., 43, 1411Behar, E., Tabone, B., Saillenfest, M., et al. 2018, A&ABeth, A., Altwegg, K., Balsiger, H., et al. 2016, Month. Not. Roy. Astron. Soc.,462, S562Bodewits, D., Christian, D. J., Torney, M., et al. 2007, A&A, 469, 1183Bodewits, D., Hoekstra, R., Seredyuk, B., et al. 2006, Astrophys. J., 642, 593Bodewits, D., Juhász, Z., Hoekstra, R., & Tielens, A. G. G. M. 2004a, Astrophys.J., 606, L81Bodewits, D., Lara, L. M., A’Hearn, M. F., et al. 2016, The Astronomical Journal,152, 130Bodewits, D., McCullough, R. W., Tielens, A. G. G. M., & Hoekstra, R. 2004b,Phys. Script., 70, C17Burch, J. L., Cravens, T. E., Llera, K., et al. 2015, Geophys. Res. Lett., 42, 5125Combi, M. R., Harris, W. M., & Smyth, W. H. 2004, in Comets II, ed. M. C.Festou, H. U. Keller, & H. A. Weaver (1510 E. University Blvd., P.O. Box210055, Tucson, AZ 85721-0055: The University of Arizona Press), 523–552Cravens, T. E. 1997, Geophys. Res. Lett., 24, 105Cravens, T. E., Robertson, I. P., Snowden, S., et al. 2009, in American Instituteof Physics Conference Series, Vol. 1156, Am. Inst. Phys. Conf. Ser., ed. R. K.Smith, S. L. Snowden, & K. D. Kuntz, 37–51Dennerl, K. 2010, Space Sci. Rev., 157, 57Ebert, R. W., McComas, D. J., Elliott, H. A., Forsyth, R. J., & Gosling, J. T.2009, J. Geophys. Res. (Space Physics), 114, A01109Ekenbäck, A., Holmström, M., Barabash, S., & Gunell, H. 2008, Geo-phys. Res. Lett., 35, L05103Festou, M. C. 1981, A&A, 95, 69Fougere, N., Altwegg, K., Berthelier, J.-J., et al. 2016, Month. Not. Roy. Astron.Soc., 462, S156Fuselier, S. A., Shelley, E. G., Goldstein, B. E., et al. 1991, ApJ, 379, 734Galand, M., Héritier, K. L., Odelstad, E., et al. 2016, Month. Not. Roy. Astron.Soc., 462, S331Glassmeier, K.-H. 2017, Phil. Trans. Roy. Soc. Lond. Ser. A, 375, 20160256Greenwood, J. B., Chutjian, A., & Smith, S. J. 2000, Astrophys. J., 529, 605Gunell, H., Goetz, C., Simon Wedlund, C., et al. 2018, A&A, 619, L2Hansen, K. C., Altwegg, K., & et al. 2016, Month. Not. Roy. Astr. Soc., 1, 1Haser, L. 1957, Bull. Soc. Roy. Scie. Liège, 43, 740Hässig, M., Altwegg, K., Balsiger, H., et al. 2015, Science, 347, aaa0276Heritier, K., Galand, M., Henri, P., et al. 2018, A&A, 617, 1
Article number, page 16 of 17yril Simon Wedlund et al.: Solar wind charge exchange in cometary atmospheres Cometocentric distance r comet [km]00.10.20.30.40.50.60.70.80.91 C ha r ge d i s t r i bu t i on Helium, R sun = 2 AU, U sw = 2000 km s -1 He He + He He no ELHe + no ELHe no EL 10 Cometocentric distance r comet [km]
Hydrogen, R sun = 2 AU, U sw = 2000 km s -1 H + H H - H + no ELH no ELH - no EL Fig. B.1.
Normalized charge distributions of helium (left) and hydrogen (right) with respect to cometocentric distance for a solar wind speed of2000 km s − and a heliocentric distance of 2 AU. The distributions are analytically calculated with and without electron loss reactions (EL). Cometocentric distance r comet [km]00.10.20.30.40.50.60.70.80.91 cx [ ] Helium He + He Cometocentric distance r comet [km]
Hydrogen + H H - Fig. C.1.
Charge-changing collision depth τ cx i = η ( r ) σ i for helium (left) and hydrogen species (right) in a comet 67P-like H O atmosphere forthree typical heliocentric distances. The solar wind speed is assumed to be constant and equal to 400 km s − . Heritier, K. L., Altwegg, K., Balsiger, H., et al. 2017, Month. Not. Roy. Astron.Soc., 469, S427Hoekstra, R., Anderson, H., Bliek, F. W., et al. 1998, Plasma Phys. Control. Fus.,40, 1541Huang, Z., Tóth, G., Gombosi, T. I., et al. 2016, J. Geophys. Res. (SpacePhysics), 121, 4247Huebner, W. F. & Mukherjee, J. 2015, Plan. Space Sci., 106, 11Ip, W.-H. 1989, Astrophys. J., 343, 946Isler, R. C. 1977, Phys. Rev. Lett., 38, 1359Khabibrakhmanov, I. K. & Summers, D. 1997, J. Geophys. Res., 102, 2193Kharchenko, V., Rigazio, M., Dalgarno, A., & Krasnopolsky, V. A. 2003, Astro-phys. J., 585, L73Koenders, C., Glassmeier, K.-H., Richter, I., Ranocha, H., & Motschmann, U.2015, Plan. Space Sci., 105, 101Läuter, M., Kramer, T., Rubin, M., & Altwegg, K. 2019, Month. Not. Roy. As-tron. Soc., 483, 852Lindkvist, J., Hamrin, M., Gunell, H., et al. 2018, A&A, 616, A81Lisse, C. M., Dennerl, K., Englhauser, J., et al. 1996, Science, 274, 205Meyer-Vernet, N. 2012, Basics of the Solar Wind (Cambridge, UK: CambridgeUniversity Press)Mullen, P. D., Cumbee, R. S., Lyons, D., et al. 2017, ApJ, 844, 7 Nilsson, H., Lundin, R., Lundin, K., et al. 2007, Space Science Reviews, 128,671Nilsson, H., Stenberg Wieser, G., Behar, E., et al. 2015a, Science, 347, 571Nilsson, H., Stenberg Wieser, G., Behar, E., et al. 2015b, A&A, 583, A20Nilsson, H., Wieser, G. S., Behar, E., et al. 2017a, Month. Not. Roy. Astron.Soc., 469, S804Nilsson, H., Wieser, G. S., Behar, E., et al. 2017b, Month. Not. Roy. Astron.Soc., 469, S252Saillenfest, M., Tabone, B., & Behar, E. 2018, A&ASchmidt, H. T., Reinhed, P., Orbán, A., et al. 2012, in Journal of Physics Confer-ence Series, Vol. 388, J. Phys. Conf. Ser., 012006Schwadron, N. A. & Cravens, T. E. 2000, ApJ, 544, 558Simon Wedlund, C., Alho, M., Grono ff , G., et al. 2017, A&A, 604, A73Simon Wedlund, C., Behar, E., Nilsson, H., et al. 2018a, submitted to A&A, 1Simon Wedlund, C., Bodewits, D., Alho, M., et al. 2018b, submitted to A&A, 1Simon Wedlund, C., Kallio, E., Alho, M., et al. 2016, A&A, 587, A154Slavin, J. A. & Holzer, R. E. 1981, J. Geophys. Res., 86, 11401Tawara, H. 1978, At. Data Nucl. Data Tab., 22, 491Tawara, H., Kato, T., & Nakai, Y. 1985, At. Data Nucl. Data Tab., 32, 235Tawara, H. & Russek, A. 1973, Rev. Mod. Phys., 45, 178Wegmann, R. & Dennerl, K. 2005, A&A, 430, L33, G., et al. 2017, A&A, 604, A73Simon Wedlund, C., Behar, E., Nilsson, H., et al. 2018a, submitted to A&A, 1Simon Wedlund, C., Bodewits, D., Alho, M., et al. 2018b, submitted to A&A, 1Simon Wedlund, C., Kallio, E., Alho, M., et al. 2016, A&A, 587, A154Slavin, J. A. & Holzer, R. E. 1981, J. Geophys. Res., 86, 11401Tawara, H. 1978, At. Data Nucl. Data Tab., 22, 491Tawara, H., Kato, T., & Nakai, Y. 1985, At. Data Nucl. Data Tab., 32, 235Tawara, H. & Russek, A. 1973, Rev. Mod. Phys., 45, 178Wegmann, R. & Dennerl, K. 2005, A&A, 430, L33