Kinetic Description Of Martian Atmospheric Entry Plasma
1646IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 37, NO. 8, AUGUST 2009Kinetic Description of MartianAtmospheric Entry PlasmaD. Janette Drake, Svetozar Popović, Leposava Vušković, and Thao DinhAbstract—A kinetic model for Martian atmospheric entryplasma (MAEP) is presented. The model was calculated based ondata from Viking, Pathfinder, and the Mars Exploration RoversOpportunity and Spirit. Calculations of the density, temperature,and electron density across the shock front were made. We employed steady-state and nonsteady-state kinetic models to describethe rate of dissociation of CO2 in the Martian atmosphere as wellas the production of O2 . Water vapor was included in the modelssince it represents 0.03% of the surface atmospheric composition.With the addition of small amounts of water vapor, we found adecrease in the dissociation of CO2 in the Martian air as well asan increase in the production of O2 . This paper is to report ananalysis of MAEP and its interactions with the shock front whichforms in the front of the Martian Landers.Index Terms—Kinetic modeling, Martian atmospheric entry,oxygen production, plasma production.I. I NTRODUCTIONNASA’S MARS exploration program seeks to understandMars as a dynamic system, including measuring the structure of the upper atmosphere and ionosphere, understandingthe past and present climate, and its potential habitability. Inkeeping with these goals, many satellites and landers have beensent to Mars [1]. Each of these Martian probes and satellitesfaced numerous challenges on their long missions. For thesatellites, one of the most challenging phases is the aerobrakingphase. First used by the Magellan spacecraft while orbitingVenus in 1993 [2] and then by the Mars Global Surveyor in1997 [3], the satellites skimmed the atmosphere of the planetusing the friction between the atmosphere and the probes toslow their velocity. This subsequently led to a decrease in theorbital radius for the satellite. The major benefit of this processis that the naturally occurring forces were used to decrease thealtitude of the orbit as opposed to a deorbital burn which wouldinvolve the use of fuel to ignite the engines. Since less fuel wasneeded for the mission to get the satellite into orbit, the cost waslower. This friction also caused heating and ionization of thesurrounding atmosphere, generating Martian atmospheric entryplasma (MAEP), which is relatively poorly understood.The first report on hypersonic aerodynamic problems duringterrestrial reentry was apparently by Hermann [4]. He reportedhow the changes in the altitude and the strong heating at theManuscript received March 5, 2009. First published July 7, 2009; currentversion published August 12, 2009. This work was supported in part by theNASA’s Graduate student research’s program through Marshall Space FlightCenter and NASA Langley Research Center.D. J. Drake, S. Popović, and L. Vušković are with the Departmentof Physics, Old Dominion University, Norfolk, VA 23529 USA (e-mail:ddrak002@odu.edu).T. Dinh is with Berriehill Research Corporation, Dayton, OH 45459 USA.Digital Object Identifier 10.1109/TPS.2009.2023846stagnation point would affect the chemical composition of theair flowing around the probe and eventually cause ionization. Inaddition, he described the interactions of the ionized gas with ashock wave formed by a circular cylinder, sphere, and circularcone. Although this paper was concerned with reentry, it helpedto define the types of obstacles that the Mars exploration landerswould face during entry. A more comprehensive review of theeffects of planetary atmospheric entry at Mach numbers greaterthan 20 for Earth, Mars, and Jupiter was given by Gnoffo [5].Martian entry plasma is a complex mixture consisting ofnumerous atomic and molecular gases (CO2 , O2 , O, CO, NO, N2 , CN, C2 , N, C, and Ar), ions (C , O 2 , Ar , O , CN , O , CO , and NO ), and electrons. Modeling of these typesof plasmas is very computationally intensive. Gorelov et al. [6]showed, through a comparison of experiments and numericalsimulations at shock speeds of 4–9 km/s, that to model thistype of discharge, a weaker dissociation rate for CO2 moleculeshas to be included in the model. In addition, they showed that,for nonequilibrium ionization behind the shock fronts, a slowerionization rate for C and O atoms by electron impact and thenonequilibrium distribution of the free electron temperaturemust also be included. In an earlier work by Park et al. [7],a thermochemical model using the previously identified molecular, atomic, and ionic species was used to show that the vibrational temperature approaches the translational temperaturequickly behind the shock front.In recent years, there have been many models and experiments of CO2 /N2 hypersonic flows in a convergent–divergentnozzle for application to Martian atmospheric entry [8]–[10].By using a thermochemical nonequilibrium Navier–Stokessolver, these researchers have shown how the rotational temperature, vibrational temperature, number density, and molarfractions of the gases vary in a plasma arcjet under specificlaboratory conditions. The main problem with these models andexperiments is that they do not include argon as a key speciesin the discharges. In addition, they do not take into account thecorrect geometry for the Martian Landers.This paper is to report an analysis of MAEP and its interactions with the shock front which forms in the front ofthe Martian Landers. This paper is organized as follows. InSection II, we describe the free stream density measurementsfrom Viking Landers 1 and 2, and Pathfinder. In Section III,we describe the entry trajectory data and the calculations forthe shock-wave jump parameters for these probes as well as forthe Mars Exploration Rovers (MERs) Opportunity and Spirit.In Section IV, we describe the electron energy distributionfunctions (EEDFs) for CO2 under Martian entry conditions. Weshow the changes in the electron temperature and dissociation0093-3813/ 26.00 2009 IEEE
DRAKE et al.: KINETIC DESCRIPTION OF MARTIAN ATMOSPHERIC ENTRY PLASMA1647TABLE IM ARTIAN ATMOSPHERIC C OMPOSITION AT THE S URFACETABLE IIM ARTIAN U PPER ATMOSPHERIC C OMPOSITION FOR V IKING L ANDER 2Fig. 1. Atmospheric free stream density distribution for the Martian atmosphere. The Pathfinder Lander had three sensors in the upper atmosphereto measure the density as indicated by the data points about 140 km.rate coefficients for CO2 and O2 . Section V describes thecalculation of the electron density under jump conditions forthree Martian probes, given that the dominant neutral speciesin the MAEP will be CO2 , O2 , O, CO, NO, N2 , and Ar. InSection VI, we discuss steady-state and nonsteady-state chemical kinetic models used for the simulation of MAEP. Finally,in Section VII, we discuss the inclusion of water vapor in themodel and how it impacts the dissociation rate of CO2 in theMartian atmosphere.approximately 120 s. The Pathfinder probe had three diagnostictools, one set in each plane, during entry [14]. Each of thesetools collected data for the density, pressure, and temperaturefor the entire entry phase. The free stream density measurements for the Martian atmosphere are shown in Fig. 1.From this figure, we observe that the Viking measurementsare higher than the Pathfinder measurements at higher altitudes.This difference is due to seasonal density changes and diurnalvariations within the thermosphere [15]. Two models wereconstructed for the Pathfinder data using the altitudinal densitydistributionII. C HARACTERISTICS OF THE M ARTIAN ATMOSPHEREρ ρo e C1 hObservations made by several Martian probes and satellitessuggest that the atmosphere at the surface is primarily composed of CO2 with minor components of N2 and Ar (seeTable I) [11].The Viking Landers contained two diagnostic tools. The firstof these is the Viking Upper Atmospheric Mass Spectrometer(VUAMS), which took measurements of the free stream atmospheric density, pressure, and temperature between 160 and130 km [12]. The second is the Viking Atmospheric StructureInstrument, which provided data from 120 to 9 km. In the upperatmosphere, the VUAMS provided data about the compositionchanges from 200 to 110 km. Based on these results, Nier andMcElroy [13] created a simple model for the number density ofthe species for each of the Viking Landers. In Table II, we showthe composition changes for the upper atmosphere from VikingLander 2. From the table, we observe that composition changesgreatly with altitude. At 125 km, the composition is comparablewith the composition at the surface. Therefore, we will use thesurface atmospheric composition for altitudes below 100 km.Above this altitude, we will use the results from the VikingLanders.Beyond the basic composition of the atmosphere, there ismuch information from the Pathfinder and Viking Landersabout the free stream density and temperature during the entryphase into the atmosphere. This phase begins at approximately160 km above the surface and lasts until the parachutes aredeployed for landing, around 9 km, with a total elapsed time of(1)where ρ is the atmospheric free stream density, C1 is a constantdetermined by calculation, and h is the height in kilometers.Free stream temperature measurements were made duringentry by the Pathfinder and Viking (VL1 and VL2) Landers.From Fig. 2, we observe that the temperature is very high in theupper atmosphere which most likely is caused by solar radiation. The temperature then decreases, with decreasing altitudereaching a minimum at 80 km. According to Schofield et al.[14], this temperature minimum may be due to a superpositionof waves, such as thermal tides. These tides will propagatefrom the lower atmosphere to higher altitudes with increasingamplitude. After 80 km, the temperature once again beginsto increase until it reaches an average of about 220 K at thesurface.A comparison of the data from the three probes showsthat temperature measurements were less precise than the freestream density measurements in Fig. 1. Two models have beenemployed to estimate the temperature: the Glenn model and theLangley Atmospheric Upwind Relaxation Algorithm (LAURA)[16]. Neither of these was able to accurately describe the trendsin the data. All of the data were collected at relatively the sametime of year and same distance from the equator. The onlyknown difference in the data is from the fact that the Vikingdata were taken 20 years before the Pathfinder data. For ourresearch, we have estimated the average temperature over allsets of current Mars data; the resulting curve is labeled Present
1648IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 37, NO. 8, AUGUST 2009Fig. 2. Temperature distributions in the Martian atmosphere from VikingLanders 1 and 2, Pathfinder Lander, the Glenn model, LAURA model, and ourpresent model.Fig. 3. Velocity measurements for Pathfinder, Viking, MER Spirit, and MEROpportunity Landers.Model in Fig. 2. In addition, we have constructed an upperand lower limit for the temperatures, which can be seen asthe shading in the calculations which involve these temperaturemeasurements.III. M ARTIAN P ROBE E NTRY T RAJECTORYAND S HOCK PARAMETERSBy reconstructing the entry trajectory of the Martian probes,when we use the present model, we are able to develop an accurate portrait of all the atmospheric phenomena faced by them.In Fig. 3, we have reconstructed the velocity measurements foreach of the listed probes. We note that the velocity was constantdown to an altitude of 60 km. The velocity then dropped sharplyover the next 50 km. At an altitude of approximately 10 km, asmentioned earlier, the parachutes for each of the probes weredeployed. Due to the closeness of the MER Spirit and MEROpportunity data, we will use the MER Opportunity data forthe remainder of this paper. These data will have error barsto take into account the velocity measurements of the MERFig. 4. Calculated values of the Mach number for the Viking, Pathfinder, andMER Opportunity Landers.Spirit probe. It is important to note that, although there aresufficient velocity data for reconstruction of the trajectory of theMER Landers, there are no free stream temperature or densitymeasurements for these landers.The Mach number (M ) for each probe was determined byusing M v/cs , where v is the velocity of the probe and cs isthe speed of sound in the medium determined by γRTcs .(2)mmHere, γ is the specific heat ratio, R is the universal gasconstant, mm is the molecular mass, and T is the temperaturefrom Fig. 2, and we present these results in Fig. 4. Fromthe figure, we observe that the Mach number increases as theprobes enter the atmosphere with peak values of 42, 31, and25 for the Pathfinder, MER Opportunity, and Viking Landers,respectively. Then, the Mach number begins to decrease sharplydue to an increase in atmospheric density.In order to accurately estimate electron density and gascomposition for MAEP, we must construct a simple model forthe shock region in front of each probe during entry. For thismodel, certain assumptions must be made: 1) The gas mixturesgenerated during entry are thermodynamically perfect gases;2) ionization occurs instantly behind the shock front; 3) the gasmixtures are constant in the boundary layer behind the shockfront; and 4) gas parameters are defined by the free streamparameters and the relations across the shock. From theseassumptions, we are able to calculate the shock parameters(γ 1)AMAM (γ 1) 2 2(γ 1) AM 1T2 T 1 [γAM 1](γ 1)2 AMρ2 ρ(3)(4)where γ is the specific heat ratio, AM M (T )2 sin2 β, β isthe oblique shock angle, T is the atmospheric temperature fromFig. 2, and ρ is the atmospheric density from Fig. 1.The calculated density across the shock layer, or jump density, is given in Fig. 5. We calculated these values from (3) in
DRAKE et al.: KINETIC DESCRIPTION OF MARTIAN ATMOSPHERIC ENTRY PLASMA1649velocity distribution function f ( r, v , t), me is the mass of theelectron, and e is the charge of the electron. In the case of aweakly ionized gas, the right-hand side of (5) will take intoaccount the elastic and inelastic collisions between electronsand neutral atoms or molecules.Due to the complexity of (5), solutions can only be determined for selected cases. We have employed three approximations for our model: 1) the homogenous approximation;2) the steady-state approximation, where the electron collisionfrequency in the ionized gas is approximately two or threeorders of magnitude larger than the driving frequency; and3) symmetry along an axis of symmetry for the Martian probes.By applying these approximations, a steady-state isotropic solution can be obtained for a monatomic gas [18]Fig. 5.Jump density for the MAEP as a function of the altitude.13 2 eEε dfddf2me kB T d ε2 QmNdε Qm dεMdεdε 2me d 2(ε Qm f ) εf (ε)Qj (ε) M dεj (ε εj )f (ε εj )Qj (ε εj ) 0.(6) jFig. 6. Temperature across the shock layer in the MAEP. The shaded regionis due to the uncertainties of the measured free stream temperature data.which the specific heat ratio of the Martian atmosphere is 1.29and the oblique shock angle for all three probes was 78.9 .The jump temperature was calculated from (4) and is shownin Fig. 6. We observe that the temperature reaches a peakaverage value of about 36 000 K, 19 000 K, and 13 000 Kfor the Pathfinder, MER Opportunity, and Viking Landers,respectively. These maximum values occurred between 50 and60 km above the surface. The shaded regions give us a range ofvalues for the temperature. For example, the Pathfinder Landerhas a peak temperature of 36 000 K 2500 around 60 km. Athigher altitudes, all the distributions become wider since theatmosphere becomes less dense.Here, we have expressed, by convention, the solution in termsof the electron energy ε mv 2 /2 and have neglected superelastic collisions. In (6), Qm is the momentum transfer crosssection; Qj is the cross section of the jth inelastic collision; Eis the electric field; M , N , and T are the mass, density, andtemperature of the neutral gas molecules; kB is the Boltzmannconstant; and f is the isotropic EEDF.For a gas mixture, an appropriate modification to the crosssectionsof all gas species in (6) shouldQm be nmade: nnnnQGinthefirsttermandQ MQG/Mandmmmn nn nM n M G in the second and third terms. Here, Qnm isthe momentum transfer cross section, Gn is the mole fraction,and M n is the mass of the molecule of the nth species.We calculated the EEDF by employing a commercial numerical Boltzmann solver, Bolsig, for weakly ionized gasesunder steady-state conditions [19]. Bolsig provides numericalsolutions for the EEDF at different values of the reducedelectric field (E/N ). An upgraded version of Bolsig, known asBolsig [20], provides similar results as the old Bolsig programbut allows for more species to be included in the calculation ofthe discharge parameters and EEDF.From the EEDF values, we are able to calculate the averageelectron temperature (Te ) of the distribution and rate constants(kj ) for certain processesIV. E VALUATION OF THE EEDFIn the absence of a magnetic field, the flow of electronsthrough a unit velocity phase space is described by theBoltzmann transport equation [17] f ( r, v , t) ( r, v , t) eE · f ( r, v , t) f ( r, v , t) v · f. tm tcoll(5)Here, the distribution of electrons in their velocity space v atthe space coordinate r and at time t is given by the electron2Te 3 kj ε3/2 f (ε) dε02em 1/2(7) εQj (ε)f (ε) dε.(8)0To verify the accuracy of the numerical results, we present inFig. 7 a comparison of the results from Bolsig for pure carbondioxide with calculated values from Nighan [21]. We found that
1650IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 37, NO. 8, AUGUST 2009Fig. 7. EEDF values in a carbon dioxide plasma.Fig. 9. Calculated values of the dissociation rate coefficients for CO2 and O2in MAEP.Fig. 8. Calculated values of the average electron temperature in terrestrial andMartian air plasmas.the Bolsig data agreed fairly well with the calculated values, 20%.By applying (7), we calculated the average electron temperature at various values of E/N for terrestrial air and Martian airand present these results in Fig. 8. We observe that Te increasesalmost linearly with E/N values greater than 4 10 16 V ·cm2 . Below this point, the values for terrestrial air increase veryslowly and are almost constant, while the values for Martian aircontinue to increase linearly.The dissociation rate coefficients for CO2 and O2 were calculated by employing (8) and are shown in Fig. 9. We note that thecoefficients increased strongly with the reduced electric field.V. E VALUATION OF THE E LECTRON D ENSITYThe electron density was evaluated by using the Saha equation [22]logNe Nikεk5040 5040 1.5 logNk NikTeTe 26.9366 log2gkigk0(9)where Nik is the number density of ions from species k, Nkis the neutral species density, εk is the ionization potential ofspecies k, gki is the statistical weight of the ion species k,and gk0 is the statistical weight of the neutral species k. Inthis model, we have assumed that the electron temperature (Te )equilibrates to the gas temperature obtained in Fig. 6. We mustalso assume that the gas temperature is still high enough thatwe can neglect the effects of ion chemistry. In addition, we arenot including the free stream electron density in this calculation [23].Since Martian air is composed of many constituents, thiscalculation is very complex. Looking at Table I, we see that themain constituents of the Martian atmosphere are CO2 , N2 , andAr. During ionization, the major additional neutral species, dueto the dissociation of CO2 and N2 , will be O2 , O, CO, and NO.The number densities of the other neutral species mentionedin the introduction will be negligible in comparison with thedensities of the other seven species. As such, we have reducedthe number of equations needed to find the electron densit
Kinetic Description of Martian Atmospheric Entry Plasma D. Janette Drake, Svetozar Popovic, Leposava Vuškovi c, and Thao Dinh . ployed steady-state a
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