Attenuation of radio signals by the ionosphere of Mars

16 Pages·2011·0.69 MB·English

RADIO SCIENCE, VOL.46,RS2004, doi:10.1029/2010RS004450,2011
Attenuation of radio signals by the ionosphere of Mars:
Theoretical development and application
to MARSIS observations
Paul Withers1
Received10June2010;revised13December2010;accepted29December2010;published11March2011.
[1] We investigate the ionospheric conditions required to explain Mars Express Mars
Advanced Radar for Subsurface and Ionosphere Sounding topside radar sounder
observationsofionosphericattenuationinexcessof13dBat5MHzduringsolarenergetic
particle events. We develop theoretical expressions for the attenuation caused by a layer
of ionospheric plasma in cases of high, intermediate, and low radio frequency relative
to the electron‐neutral collision frequency at the ionospheric layer. We apply these
relationships to four layers: the M2 layer produced at 120 km by solar extreme ultraviolet
photons,theM1layer producedat100kmbysolarX‐rayphotonsandassociatedelectron
impact ionization, the meteoric layer produced at 85 km by meteoroid ablation, and a
putative plasma layer producedat 35 kmby cosmic rays. Attenuation is weaker in theM2
layer than in the M1 layer. Attenuation in the M1 and meteoric layers are comparable,
although their properties are quite variable. The greatest attenuation for radio frequencies
above 50 MHz occurs in the predicted plasma layer at 35 km, but its effects are relatively
smallatlowerfrequencies.Ifoptimallylocatedwithapeakaltitudeof50km,alayerwitha
peakplasma densityof109m−3issufficient toexplaintheobserved13dBattenuation.
Although the electron densities produced by solar energetic particle events at Mars have
not been directly simulated, the required electron densities are plausible. However, the
altitude at which solar energetic particles produce plasma is uncertain.
Citation: Withers, P.(2011), Attenuation ofradio signals bythe ionosphereof Mars: Theoretical development and
application to MARSISobservations, RadioSci.,46, RS2004,doi:10.1029/2010RS004450.
1. Introduction NASA’s Mars Reconnaissance Orbiter are conducting
orbital operations [Crisp et al., 2003; Saunders et al.,
[2] Ionospheres affect radio navigation and communi-
cation systems, as well as radio‐based scientific instru- 2004; Chicarro et al., 2004; Zurek and Smrekar, 2007].
Plannedfuturemissionsandtheiranticipatedlaunchdates
ments,bytheireffectsonradiowavepropagation[e.g.,
include NASA’s Mars Science Laboratory rover (2011),
Ratcliffe,1959;Budden,1985;Hargreaves,1992;Rawer,
1993; Reinisch et al., 2000; El‐Rabbany, 2002; Kliore Russia’s Phobos‐Grunt sample return mission (2011),
China’sYinghuo‐1orbiter(2011),andNASA’sMAVEN
et al., 2004; Blaunstein and Plohotniuc, 2008]. Here we
orbiter(2013).
investigate how the attenuation of radio signals by the
ionosphere of Mars affects topside radar sounders and [4] Typical missions to the surface of Mars have two
communicationspathways:adirect‐to‐Earthlink(∼5GHz)
existingandpotentialcommunicationssystems.
and a link between the surface and orbit (400 MHz)
[3] Mars is currently a major target for solar system
[RoncoliandLudwinski,2002;Crispetal.,2003].Typical
exploration. At the time of writing, NASA rovers Spirit
orbital missions also have these same two communica-
and Opportunity are conducting surface operations and
NASA’s Mars Odyssey, ESA’s Mars Express, and tions pathways [Edwards et al., 2003; Graf et al., 2005;
ZurekandSmrekar,2007].Thetransmissionofradiosig-
nalsthrough theionosphereofMarsistherefore common
1Center for Space Physics, Boston University, Boston, forbothsurfaceandorbitalmissionsatMars.Inaddition,
Massachusetts,USA. theoperationofradarinstrumentsonorbiters,suchasMars
AdvancedRadarforSubsurfaceandIonosphereSounding
Copyright2011bytheAmericanGeophysicalUnion. (MARSIS)(0.1–5.4MHz)onMarsExpress[Picardietal.,
0048‐6604/11/2010RS004450
RS2004 1of 16
RS2004 WITHERS:MARSIONOSPHERIC ATTENUATION RS2004
electron density profile from the ionosphere of Mars.
Figure 2 shows a schematic representation of the main
features and forcings associated with the ionosphere of
Mars. Solar photons are the main external forcing, but
thesolarwind’sfluxofenergeticparticlesandembedded
magnetic field affect the topside ionosphere and its
boundary with the magnetosheath above. The magnetic
environmentatMartianionosphericaltitudesisextremely
variable, and it is affected by crustal magnetic fields,
which vary with position, and the solar wind magnetic
field, which varies with position and time. This affects
both internal ionospheric processes, such as bulk plasma
motion and plasma instabilities, and external forcings,
Figure 1. A typical Mars Global Surveyor radio occul-
such as the flux of precipitating energetic particles. The
tation electron density profile, 0337M41A.EDS,with 1s
ionospheric state clearly depends on magnetic environ-
uncertainties. Note the M2 layer at 140 km and the M1
ment to some degree, but the detailed interactions and
layer at 110 km.
processesarenotcurrentlywellunderstood.
[8] Themainlayerintheionosphere,calledtheM2layer,
2004] and SHARAD (15–25 MHz) on Mars Reconnais- is produced by the photoionization of carbon dioxide
molecules,themostabundantatmosphericconstituent,by
sanceOrbiter[Seuetal.,2004],alsoinvolvesradiosignals
solar extreme ultraviolet (EUV) photons. The resultant
passingthroughtheionosphereofMars.
CO+ ions rapidly charge exchange with neutral oxygen
[5] Theaimofthisworkistoinvestigatehowtheiono- 2
atoms, the next most abundant atmospheric species at
sphereofMarsattenuatesthesignalsofsuchradiosystems
ionosphericaltitudes,toformlonger‐livedO+ions[Hanson
for a range of possible ionospheric states and a range of 2
etal.,1977;Chenetal.,1978;Fox,2004;FoxandYeager,
radiofrequencies.Therearetwomotivatingfactorsbehind
2006]. This is the dominant ion species. Since time con-
this work. First, we wish to determine what ionospheric
stantsintheM2layeraremuchshorterforphotochemical
properties are required to explain sporadic failures of the
processesthan for transportprocesses, thedominant loss
MARSIS instrument to detect the surface of Mars after
processhereisthedissociativerecombinationofO+ions.
solarenergeticparticleevents[Morganetal.,2006,2010; 2
As the photoionization cross section for CO is approxi-
Espleyetal.,2007;Nielsenetal.,2007].Second,wewish 2
matelyuniformacrossmuchoftheEUVspectrum[Schunk
todevelopthefoundationsnecessaryforfuturestudiesof
the effects of Martian ionospheric conditions on radio
navigation and communication systems at Mars during
extreme space weather events [Foullon et al., 2005;
Crosbyetal.,2008].
[6] The structure of this work is as follows. Section 2
introduces the ionosphere of Mars, the interaction of
radiowaveswithionosphericplasma,andpreviouswork
on theeffects of theMartian ionosphere onradio waves.
Section3developsgeneralexpressionsforthepowerlost
by a radio signal that propagates through a plasma layer
in an unmagnetized ionosphere. Section 4 applies these
expressionstofourMartianionosphericlayers.Section5
investigatestheionosphericconditionsnecessarytoexplain
the strong attenuation observed occasionally by the
MARSIS instrument. Section 6 summarizes the findings
of this work and Appendix A derives a mathematical
relationshipthatisimportantforsection3.
2. Background
2.1. The Ionosphere of Mars
[7] The dayside ionosphere of Mars was recently Figure 2. Schematic illustration of the main features
reviewed by Withers [2009]. Figure 1 shows a typical
and forcings associated with the ionosphere of Mars.
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RS2004 WITHERS:MARSIONOSPHERIC ATTENUATION RS2004
and Nagy, 2000], the M2 layer can be adequately during a particular solar energetic particle event, but did
represented as a Chapman layer for many purposes not describe ionization rates or resultant plasma densi-
[Chapman, 1931a, 1931b]. Several other processes can ties. Theoretical models of the production of plasma by
createadditionalplasmalayersatloweraltitudes. the precipitation of high‐energy particles are relatively
[9] First, a less dense layer, called the M1 layer, is unconstrained and immature.
foundat100–110kmaltitude,20kmbelowtheM2layer [12] The nightside and dayside ionospheres differ sig-
[Bougheretal.,2001;Fox,2004;Mendilloetal.,2006]. nificantly due to the absence of ionizing solar irradiance
Some plasma in the M1 layer is produced by photoioni- on the nightside. Nightside electron densities at the alti-
zation by solar soft X‐rays, but the majority is produced tudes of the M2 and M1 layers, where photochemical
by electron impact ionization. Photoionization by a soft lifetimes are on the order of minutes, are much smaller
X‐ray produces a very energetic photoelectron that ther- and more variable than on the dayside [Zhang et al.,
malizes (i.e., its speed reduces to the characteristic ther- 1990; Fox et al., 1993; Gurnett et al., 2008]. Electron
mal speeds of other particles) via collisions with neutral densities as high as 1010 m−3 at M1 and M2 layer alti-
molecules.Electron‐neutralcollisionsmayionizetheneu- tudes have been observed at solar zenith angles of 110°,
tral molecule if the impact energy is sufficiently large, althoughitisuncertainifthesedensitiesareproducedby
and each photoionization event in the M1 layer leads to the photoionization processes responsible for these day-
the production of ∼5–10 ion‐electron pairs [Fox, 2004; side layers or not. The nightside meteoric and EP layers
Nicholson et al., 2009]. After CO+ ions are produced in are probably more similar to their dayside counterparts.
2
theM1layer,theyfollowthesameseriesofphotochemical Atomic metal ions in the meteoric layer have photo-
processes as in the M2 layer. Since the solar soft X‐ray chemicallifetimesontheorderofhourstodays[Withers
spectrum is extremely variable, the properties of the M1 et al., 2008], so this layer is likely to persist onto the
layer are also extremely variable [e.g., Christou et al. nightside [Pätzold et al., 2005]. Plasma in the EP layers
2007]. This layer is significantly enhanced during solar isproducedbyparticles, notphotons,whichareincident
flares[Mendilloetal.,2006]. on the nightside of Mars. Solar energetic particles can
[10] Second, a layer attributed to meteoroid influx is precipitateonto all regionsofMars,notjustthedayside,
sporadicallypresentaround85kmaltitude[Pesnelland due to their large gyroradius, and the distribution of
Grebowsky,2000;Molina‐Cuberosetal.,2003;Pätzold galacticcosmicraysisrelativelyisotropic.Theiongyro-
et al., 2005; Withers et al., 2008]. The interaction of radius at Mars is comparable to the planetary radius
high‐speed meteoroids and atmospheric gases leads to [Leblancetal.,2002]duetoitsweakmagneticfield.For
the deposition in the atmosphere of species that would instance,a50keVO+ionina40nTmagneticfieldhasa
otherwise be absent, such as neutral Mg or Fe. Related gyroradiusofover3000km.Weexpectthemeteoricand
ions, such as Mg+ or Fe+, are also produced. These ions EPlayers,butnottheM1andM2layers,tobepresenton
may be produced directly during meteoroid ablation by thenightside.
the impact ionization of ablated neutral metal atoms in [13] The magnetic environment in an ionosphere, espe-
collisions with atmospheric molecules, indirectly by pho- cially the magnetic field strength, affects interactions
toionizationofneutralmetalatoms,orindirectlybycharge betweenaradiowaveandionosphericplasma.UnlikeEarth
exchange between neutral metal atoms and atmospheric or Jupiter, Mars does not possess a strong dipolar mag-
ions. The photochemical lifetimes of atomic metal ions netic field generated deep within the planetary interior.
aremeasuredinhours,muchlongerthanthefewminutes Instead, it possesses localized regions of magnetic fields
of O+ ions, so even a modest production rate can result generated by remanent magnetism in its crust [Acuña
2
in substantial plasma densities. etal.,1999,2001].Incertainlocations,themagneticfield
[11] Third, the precipitation of high‐energy particles, strength at ionospheric altitudes can exceed 1000 nT, as
suchascosmicraysandsolarenergeticparticles,canpro- shown in Figure 10 of Brain et al. [2003]. Elsewhere,
duce plasma at altitudes below the M2 layer [Molina‐ crustalfieldsareweakorabsent.Theselocalizedregions
Cuberosetal.,2002;Leblancetal.,2002;Haideretal., of strong field produce unusual magnetic topologies.
2009; Brain et al., 2009]. This is the only plasma layer Magneticfieldlinescanrotatefromverticaltohorizontal
discussed in this work where negative ion densities may over distances of a few hundred kilometers, a fraction
be significant. In this case, the electron density does not of the planetary radius. Minimagnetospheres are formed
necessarily equal the plasma density. We label the layer where closed and strong field lines isolate ionospheric
ofelectronsproducedbyenergeticparticleprecipitationas plasma from solar wind plasma. Cusp‐like conditions
the“EP”layer.Molina‐Cuberosetal.[2002]andHaider prevail on their boundaries, which permit solar wind
etal.[2009]eachsimulatedasinglepredictedionospheric plasmatoflowdownwardintotheionosphereeasily and
profile produced by cosmic rays with peak electron den- ionosphericplasmatoflowoutwardeasily[Brainetal.,
sitiesat35km.Leblancetal.[2002]reportedamodeled 2007]. In order to keep the scope and complexity of this
vertical profile of energy deposition rate above 80 km worktractable,wefocusontheunmagnetizedandweakly
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magnetized regions at Mars where the effects of mag- The received amplitude, E, is related to the transmitted
r
netic fields can be neglected. This simplification would amplitude,E,by
t
not be appropriate on Earth. The following introduction
ofrefractiveindexandattenuationisthereforelessgeneral E (cid:2)Z (cid:3) (cid:4) (cid:2)Z (cid:3)(cid:5)secðOZAÞ
and comprehensive than is typical for terrestrial applica- Er ¼exp kids ¼ exp kidz ; ð4Þ
tions [e.g., Rishbeth and Garriott, 1969; Budden, 1985; t
Schunk and Nagy, 2000; Gurnett and Bhattacharjee,
2005]. wherewehaveusedds=sec(OZA)dz,zasaltitude,OZA
astheanglebetweentheraypathandthevertical,andthe
2.2. Interaction of Radio Waves With Ionospheric assumption that the ionosphere can be treated as plane
Plasma parallelintheregionswhereitscontributiontotheatten-
[14] The complex refractive index of an unmagnetized uationissignificant.Inthecaseofanorbitingtransmitter
ionosphere,m ,inwhichionmotionisneglectedsatisfies andasurfacereceiver,OZAisthe“orbiterzenithangle”.
c
[Budden, 1985] Thepowerloss,P,isgivenindecibels(dB)by
!2
(cid:1)2c ¼1(cid:2)!ð!(cid:2)p i(cid:3)Þ; ð1Þ PðdBÞ¼(cid:2)20log10ðEr=EtÞ (cid:2)Z (cid:3)
¼(cid:2)20log10ðeÞsecðOZAÞ kidz ; ð5Þ
wherew istheplasmaangularfrequency,wistheangular
p
frequencyoftheelectromagneticwave,iisthesquareroot
of −1, and n is the electron‐neutral collision frequency. where e = exp (1) and the minus sign ensures that the
The plasma angular frequency satisfies wp2 = Nq2/me(cid:4)0 power loss is positive. Equations (3) and (5) show that
whereNiselectrondensity,qistheelementarycharge,me the effects of multiple plasma layers on the absorption
istheelectronmass,and(cid:4)0isthepermittivityoffreespace. coefficient, ki, and the power loss, P (dB), are additive
Equation(1)followsfromequation4.47ofBudden[1985] iftherealpartoftherefractiveindex,m,isclosetounity
wanidthUY==10−forina/nw.uTnmheasgeneextipzreedsspiloanssmfao,rXY=, XN,qa2n/mdeU(cid:4)0war2e, t(hwapt(cid:3)arewnofotrreaflceocltleisdiobnyleasnsyioionnoospsphhereer)ir.cFlaoyrerra,dtihoewatatvenes-
statedinequations3.22,3.5,and3.16ofBudden[1985]. uation caused by a multilayered ionosphere can be cal-
The plasma frequency, fp, equals wp/2p and the electro- culated by studying each individual layer in turn, then
magnetic wave frequency, f, satisfies f = w/2p. If we addingtheirrespectivevaluesofP(dB).
definetherealandimaginarypartsof thecomplexrefrac-
tiveindexbymc=mr+imi,thenitfollowsfromequation(1) 2.3. Previous Work on the Effects of the Martian
that Ionosphere on Radio Waves
!
!2 !4(cid:3)2 [16] Mostexistingworkonthistopichasbeenmotivated
(cid:1)4(cid:2) 1(cid:2) p (cid:1)2(cid:2) p ¼0: ð2Þ by theneed to understandthe functioning of potential or
r ð!2þ(cid:3)2Þ r 4!2ð!2þ(cid:3)2Þ2 actualscientificinstruments.Thefrequenciesusedbypast
and current communications systems have been high
In the collisionless limit where n (cid:3) w (electron‐neutral enoughthationosphericeffectsarerelativelysmall.Cur-
collision frequency much less than angular frequency of rent Mars missions use ∼5 GHz frequencies for commu-
the radio wave), m2 = (1 − w2/w2). Radio waves with nications with Earth [Tyler et al., 2001] and 400 MHz
r p
angular frequency w cannot propagate into regions where frequencies for short‐range communications [Edwards
m2 (w) < 0. If a radio wave encounters a surface where etal., 2003].However, futuremissions,particularlythose
r
m2 = 0 (equivalent to w = w in the collisionless limit), requiringcommunicationsbetweenlandedornear‐surface
r p
then it is reflected at that surface. explorers,mayuselowerfrequenciesthataremoreaffected
[15] Wedefinetherealandimaginarypartsofthecom- by the ionosphere. For certain applications, ionospheric
plex wave number k by k = k + ik, where k = wm /c effects arebeneficial; for example, thereflection oflow‐
c c r i c c
and c is the speed of light. The value of k controls how frequencyradiosignalscanextendrangeoverthehorizon
i
the radio wave amplitude changes due to absorption. If [e.g., Melnik and Parrot, 1999]. Terrestrial communica-
E is radio wave amplitude and s is distance, then k = tionssystemswithsub‐GHzfrequencies(http://www.ntia.
i
dE/(Eds).From equation (1), we have doc.gov/osmhome/osmhome.html)thatmightbeadapted
for Martian use include long‐range shortwave radios
ki ¼2(cid:2)c!(cid:1)2pr(cid:4)(cid:3)2þ(cid:3) !2: ð3Þ (a∼n1d0coMrdHlezs)s, tceilteizpehnosn’esba(nodne(CgeBn)eraratidoinosus(e∼s3046MMHHz)z,
frequencies).
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RS2004 WITHERS:MARSIONOSPHERIC ATTENUATION RS2004
[17] Melnik and Parrot [1999] investigated the propa- layers. Our results do not change greatly if a symmetric
gation of radio waves in the 10 Hz to 10 kHz range to Gaussian shape is assumed instead. The electron density
support radio wave experiments on the Russian Mars 96 N (z) satisfies
and Japanese Nozomi missions (both missions failed).
(cid:2) (cid:3)
They found that these low frequencies, which might be 1(cid:6) z(cid:2)z (cid:6) z(cid:2)z (cid:7)(cid:7)
naturally produced in the lower atmosphere by electrical N ¼N exp 1(cid:2) 0(cid:2)exp (cid:2) 0 ; ð6Þ
0 2 L L
discharges associated with airborne dust, could only
propagate upward to orbital detectors under nightside
ionospheric conditions at low solar activity and in the where N is the maximum electron density, z is the
0 0
presenceofastrongmagneticfield.Witasseetal.[2001] altitudeatwhichN=N ,andListhewidthofthelayer.
0
investigated the effects of meteoric layers on MHz radio Note that we do not require L to equal the neutral scale
signals to support radar experiments on Mars Express height, H, at this point. Although we will do so for the
and Nozomi. They found that the attenuation could Mars‐specific results in section 4, we increase the gen-
exceed tens of dB. erality of our theoretical results by permitting L ≠ H at
[18] Armandetal.[2003]investigatedthedistortionof present. Assuming a uniform composition and acceler-
1–5 MHz radar pulses by the ionosphere and its impact ation due to gravity, the electron‐neutral collision fre-
on subsurface radar sounding to support the MARSIS quency n (z) satisfies
instrument on Mars Express. They found that the iono- (cid:6) (cid:7)
z(cid:2)z
sphere creates significant pulse distortion at MHz fre- (cid:3) ¼(cid:3) exp (cid:2) 0 ; ð7Þ
quencies. Safaeinili et al. [2003] developed calibration 0 H
schemes for the MARSIS instrument in order to under-
standandcorrectforseveralionosphericeffects,including where n is the value of n at z = z . We define h as the
0 0
attenuation,Faradayrotation,andpulsedispersion.They ratio of the width of the ionospheric layer (L) to the
predicted attenuation on the order of 1 dB at MHz fre- neutralscaleheight(H)sothath=L/H.Wealsodefinex,
quenciesundernormaldaysideionosphericconditions. a dimensionless measure of altitude, as x = (z − z )/L,
0
[19] Mendillo et al. [2004] investigated ionospheric whichgives dz=Ldx =hHdx.Therefore,
effects on a hypothetical global positioning system at (cid:2) (cid:3)
1
Marsandfoundthatrangeerrorsareontheorderof1m N ¼N exp ð1(cid:2)x(cid:2)expð(cid:2)xÞÞ ð8Þ
at1GHz.Theydidnotconsidertheeffectsofpowerloss. 0 2
The range error is proportional to the line‐of‐sight total
electroncontent(TEC)abovethereceiver,wherethesub- and
solarverticalTECisapproximately1016m−2.Evenduring
a large solar flare or solar energetic particle event, the (cid:3) ¼(cid:3) expð(cid:2)(cid:5)xÞ: ð9Þ
0
verticalTECisunlikelytoexceedtwicethevaluefoundat
thesubsolarpointundersolarmaximumconditions.
3.2. High‐Frequency Limit (w ≫ n)
3. General Expressions for Power Loss [21] In this limit, where the angular frequency of the
radio wave, w, is much greater than the electron‐neutral
3.1. Basic Assumptions collision frequency, n, equation (3) becomes
[20] Inthissection,wederivesomegeneralexpressions (cid:2)!2(cid:3)
for the power loss experienced by a radio signal during k ¼ p
one‐way propagation through an ionosphere. These will i;hi 2c!2 (cid:2) (cid:3)
be applied to the specific case of Mars in section 4. It is (cid:2)N q2(cid:3) 1
clearfromsection2.2thatthestudyofpowerlosshastwo ¼ 2m(cid:4)0 c!20exp 2½1(cid:2)ð1þ2(cid:5)Þx(cid:2)expð(cid:2)xÞ(cid:6) :
end‐member cases (w (cid:5) n and w (cid:3) n) separated by an 0
ð10Þ
intermediate case. We make one significant assumption:
thattherealpartoftherefractiveindexisunityorm =1.
r The power loss becomes
The validity of this assumption, which is clearly reason-
ableforwp(cid:5)w(cid:5)n,isexaminedforMarsinsection4. (cid:5)HN q2(cid:3)
Weconsiderasingleionosphericlayerwiththeshapeof P ðdBÞ¼20log ðeÞsecðOZAÞ 0 0
hi 10 2m(cid:4) c!2
a Chapman layer, which is a reasonable approximation Z (cid:2) 0 (cid:3)
∞
fortheshapeofmanyionosphericlayers,especiallythose (cid:7) exp 1ð1(cid:2)ð1þ2(cid:5)Þx(cid:2)expð(cid:2)xÞÞ dx:
found on Mars [Withers, 2009]. A Chapman layer shape (cid:2)∞ 2
is reasonable for a single layer, not a sum of multiple ð11Þ
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As shown in Appendix A, the value of the dimen- Truncatingtheintegralinthismannerpreventsthefailure
sionlessintegralinequation(11)isJ(1;h),whichequals of equation (13) at high altitudes from contributing any
exp(1/2)2(h+1/2)G(h+1/2)whereGisthemathempaffitffiffiifficffiffial erroneousattenuation.Thisleadsto
gammafunction.Ifh=1,thenJ(1;h)=exp(1/2) 2(cid:6)=
4.133.Withthissubstitution,wehave Z Z
z¼zL (cid:2)(cid:5)HN q2 x¼xL
k dz¼ 0
(cid:5)HN q2(cid:3) z¼(cid:2)∞ i;lo 2mec(cid:2)(cid:4)0(cid:3)0 x¼(cid:2)∞ (cid:3)
P ðdBÞ¼20log ðeÞsecðOZAÞ 0 0Jð1;(cid:5)Þ: 1
hi 10 2m(cid:4) c!2 (cid:4)exp ½1(cid:2)ð1(cid:2)2(cid:5)Þx(cid:2)expð(cid:2)xÞ(cid:6) dx;
0 2
ð12Þ ð16Þ
3.3. Low‐Frequency Limit (w ≪ n) wherex ,whichequals(z −z )/hH,ispositive.Wehave
L L 0
[22] In this limit, where the angular frequency of the not found a general solution to this integral even for the
radio wave, w, is much less than the electron‐neutral special case where h = 1 (the width of the ionospheric
collision frequency, n, equation (3) becomes layer,L,equalstheneutralscaleheight,H).However,an
approximate solution can be obtained. The integrand in
(cid:2)!2 equation (16) can be separated into the product of two
k ¼ p terms,exp(0.5+(h−0.5)x)andexp(−0.5exp(−x)).The
i;lo 2c(cid:3) (cid:2) (cid:3) secondtermisclosetounityforx>0andcanbeneglected.
(cid:2)N q2 1 Therefore,
¼ 0 exp ½1(cid:2)ð1(cid:2)2(cid:5)Þx(cid:2)expð(cid:2)xÞ(cid:6) :
2m(cid:4) c(cid:3) 2
0 0 Z
ð13Þ z¼∞ (cid:2)(cid:5)HN q2
k dz(cid:8) 0
z¼(cid:2)∞(cid:4)Zi;lo 2(cid:2)mec(cid:4)0(cid:3)0 (cid:3)
Formally, ki,lo tends to infinity at high altitudes for h > x¼0 1
1/2(thatis,theratioofthewidthoftheionosphericlayer, (cid:7) exp ð1(cid:2)ð1(cid:2)2(cid:5)Þx(cid:2)expð(cid:2)xÞÞ dx
L, to the neutral scale height, H, exceeds 1/2). However, Z x¼(cid:2)∞ (cid:2) 2 (cid:3) (cid:5)
x¼x
real ionospheres do not have infinite absorption coeffi- L 1
þ exp ð1(cid:2)ð1(cid:2)2(cid:5)ÞxÞ dx : ð17Þ
cients, ki, at altitudes high above their ionospheric peak. x¼0 2
Since the angular frequency of the radio wave, w, is
constant, but the electron‐neutral collision frequency, n,
decreases exponentially with increasing altitude, approx- The term in square brackets is simply a dimensionless
imationofequation(3)byequation(13)isformallyinvalid scalingfactorthatrelatesthepowerlosstothekeyphysical
athighaltitudes.Sincetheattenuationcontributedatalti- parameters (h, H, N0, n0). The first dimensionless term
tudes many scale heights above the ionospheric layer is inside the square brackets has no obvious analytical
negligible, we can continue to use equation (13) at all solution.Itsvaluedependsonhonly,sowerepresentitby
altitudesifwemakeadditionalapproximationsthatelim- the function g (h). The function g (h) decreases mono-
inateerroneousattenuationfromhighaltitudes. tonically with increasing h. At h = 1 (the width of the
[23] Wefirstlabelthealtitudewherew=naszL,which ionospheric layer, L, equals the neutral scale height, H),
satisfies g = 0.689. The value of g is on the order of unity for
h < 6, which means that it can be replaced by 1 for all
but the broadest of layers without explicit calculation.
zL ¼Hlnð(cid:3)s=!Þ; ð14Þ Theseconddimensionlessterminsidethesquarebrackets
hasasimplesolution.Accordingly,equation(17)becomes
wheren istheelectron‐neutralcollisionfrequencyatthe
s
surface (z = 0). In order for w (cid:3) n to be satisfied at the Z
z¼∞ (cid:2)(cid:5)HN q2
ionospheric peak, plasma densities must be negligible at k dz(cid:8) 0
z > zL. Accordingly, we approximate the integral of ki,lo z¼(cid:2)∞ i;lo 2m(cid:4)ec(cid:4)0(cid:3)0
with respect to altitude as follows: 2
(cid:7) (cid:7)ð(cid:5)Þ(cid:2) expð0:5Þ
Z Z ð1(cid:2)2(cid:5)Þ
z¼∞ z¼z (cid:2) (cid:2) (cid:3) (cid:3)(cid:5)
ki;lo dz¼ Lki;lo dz: ð15Þ (cid:4) exp (cid:2)ð1(cid:2)2(cid:5)ÞxL (cid:2)1 : ð18Þ
z¼(cid:2)∞ z¼(cid:2)∞ 2
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RS2004 WITHERS:MARSIONOSPHERIC ATTENUATION RS2004
Thus, Thisisequivalenttotheattenuationproducedbyauniform
layer of electron density N and width 2H at an altitude
0
(cid:5)HN q2 where n/(n2 + w2) equals its maximum value of 1/2w
P ðdBÞ¼20log ðeÞsecðOZAÞ 0
lo (cid:4) 10 2mec(cid:4)0(cid:3)0 (equation(3)).Thus,attenuationinexcessof1dBcanbe
2 producedatMHzfrequenciesbyevenaweakplasmalayer
(cid:7) (cid:7)ð(cid:5)Þ(cid:2) expð0:5Þ
ð1(cid:2)2(cid:5)Þ whose peak density is less than 1% of the subsolar peak
(cid:2) (cid:2) (cid:3) (cid:3)(cid:5) electron density in the M2 layer (2 × 1011 m−3) if the
(cid:4) exp (cid:2)ð1(cid:2)2(cid:5)ÞxL (cid:2)1 : ð19Þ plasmalayerislocatedattheoptimalaltitude.Measurable
2
attenuation, even on the nightside, does not necessarily
require large electron densities, which has implications
3.4. Intermediate Frequencies that will be explored in section 5. Having developed
[24] Sometimesneitherw(cid:5)n (section3.2)norw(cid:3)n approximate analytical formulae for thepower lossatall
frequencies, albeit with the assumption that m = 1, we
(section 3.3) is satisfied at all altitudes with appreciable r
plasmadensities.Insuchcases,neitherequation(12)nor now show that they are reasonable for Mars.
equation (19) will provide an accurate estimate of the
powerloss.Herewestatewithoutproofanexpressionfor 4. Application to Mars
the power loss at intermediate frequencies. We show in
section 4 that this expression is appropriate for Mars. 4.1. Model of the Atmosphere of Mars
[25] RecallthatzListhealtitudewherew=n.Thepeak [27] Theneutralatmosphereof Marscanberepresented
altitude (z ) at which (z − z ) = −hH = −L is significant asanidealgasofpureCO withuniformtemperature,T,
0 0 L 2
here and we label the value of P (equation (19)) at this of 150 K and uniform gravitational acceleration, g, of
lo
altitude as P*. If the peak altitude (z ) is more than one 3.7 m s−2 [Owen, 1992; Zurek et al., 1992]. This leads
lo 0
layerwidthbelowz ,thenthelow‐frequencylimitforthe to a uniform neutral scale height, H, of 7.6 km. Spec-
L
powerlosscanbeused.Ifz issmallenoughthat(z −z )/ ifying the surface pressure, p, to be 600 Pa determines
0 0 L s
hH ≤ − 1 then P = P from equation (19). If z is larger the neutral density at all altitudes [Zurek et al., 1992].
lo 0
thanthisand(z −z )/hH>−1thenPequalsthesmaller The electron‐neutral collision frequency, n, equals (cid:8)n
0 L
ofP* andP (equation (12)).Therefore, where n is neutral number density and (cid:8), a momentum
lo hi
transfer coefficient, is 10−13 m3 s−1 for CO [Hake and
2
P¼Plo if ðz0(cid:2)zLÞ=(cid:5)H (cid:9)(cid:2)1 Phelps, 1967; Nielsen et al., 2007]. The value of (cid:8)
(cid:9) (cid:10) ð20Þ equals the product of the electron thermal speed and the
P¼minimum P*lo;Phi if ðz0(cid:2)zLÞ=(cid:5)H >(cid:2)1: electron‐neutral collision cross section [Gurnett and
Bhattacharjee, 2005]. Both of these factors depend on
temperature. However, Figure 13 of Hake and Phelps
As a rough rule of thumb, P equals Pl*o for the altitude [1967] shows that the temperature dependence of (cid:8) is
range1>(z −z )/hH>−1andP forthealtituderange
0 L hi relatively small, so we assume a constant value for this
(z − z )/hH>1.
0 L exploratory survey. Note that a wide range of values of
[26] The value of Pl*o can be estimated as follows. At (cid:8) have been used in previous work [Melnik and Parrot,
(z − z ) = −hH, n = ew and, for h = 1, the value of the
0 L 0 1999;Witasseetal.,2001;Safaeinilietal.,2003].Some
term in square brackets in equation (19) is close to e.
previous workers adopted the terrestrial (N /O ) value
Hence, for (cid:8), which is 2 orders of magnitude small2er 2than the
(cid:2) (cid:3)
HN q2 value in the CO2 atmosphere of Mars [Nielsen et al.,
P*ðdBÞ¼20log ðeÞsecðOZAÞ 0 : ð21Þ 2007].Thisshouldbeconsideredwhencomparingresults.
lo 10 2mec(cid:4)0!
4.2. Example of Power Loss Using 5 MHz
A sense of the numerical value of P* can be obtained
lo Radio Waves
by replacing H, N , and f with representative values of
0
10 km, 109 m−3, and 100 MHz, respectively. Hence, [28] We assumed that the layer width equalled the
neutralscaleheight(h=1)andcalculatedthepowerloss
(cid:2) (cid:3)(cid:2) (cid:3)
H N as a function of N and z for vertical radio wave
P*loðdBÞ¼0:73secðOZAÞ 10 km 109m0(cid:2)3 propagation (that is,0an OZA0 of 0°) and f = 5 MHz,
(cid:2) (cid:3) where this frequency was chosen in order to support
f (cid:2)1
interpretation of MARSIS observations in section 5.
(cid:4) : ð22Þ
100 MHz Results are shown in Figure 3. Two separate results are
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RS2004 WITHERS:MARSIONOSPHERIC ATTENUATION RS2004
Table 1. TypicalHeightsandElectronDensitiesofIonospheric
LayersConsideredinThisWork
Layer Height(km) ElectronDensity(m−3)
M2 120 2×1011
M1 100 1×1011
Meteoric 85 2×1010
EP 35 1×108
4.3. Power Losses due to Four Ionospheric Layers
[29] Weconsiderfourionosphericlayers,theM2layer
at120km[Withers,2009],theM1layerat100km[Fox,
2004; Pätzold et al., 2005; Mendillo et al., 2006; Fox
andYeager,2006],themeteoriclayerat85km[Pätzold
et al., 2005; Withers et al., 2008] and the putative EP
Figure 3. Powerloss(dB)inanionosphericlayerasa
layer at 35 km [Molina‐Cuberos et al., 2002; Haider
function of peak electron density, N , and altitude, z ,
0 0 etal.,2009],andverticalradiowavepropagation(OZA=0).
forf=5MHzandlayerwidthequaltotheneutralscale
LayerheightsandelectrondensitiesarelistedinTable1.
height. Black contours show the results of a numerical
Theassumptionthatthelayerwidthisequaltotheneutral
integration.Greycontoursshowtheanalyticalapproxima-
scaleheight(h=1)isreasonableforalltheselayers.For
tionsderivedinsection3.Greycontoursmaybeinvisible
eachlayer,weconsiderradiofrequenciesbetween1MHz
beneathblackcontourswhentheapproximationsarevery
and 100 GHz (section 2.3) and peak electron densities
accurate.Insuchinstances,thegreycontourscanbeseen between106m−3and1012m−3.
beneath the numbers that label the black contours. Con-
4.3.1. Power Loss due to the M2 Layer
tourswithverysmallpowerlossesareshowntoillustrate [30] The high‐frequency limit for power loss can be
thefunctionaldependenciesoftheresultsplottedhere,not
used here because w > n at 120 km for all frequencies
because a power loss of 10−4 dB has great practical sig-
greaterthan660Hz.ThepowerlossduetotheM2layeris
nificance. The dashed black curve indicates the 13 dB
shown in Figure 4. Two separate results were calculated,
contour(section5).Thehorizontalgreylineindicatesz ,
L
thevalueofzwheren=w,whichistheboundarybetween
thehigh‐frequencyandlow‐frequencylimits.Thevertical
greylineindicatesthevalueofN atwhichthemaximum
0
plasmafrequency,f ,equalstheradiofrequency,f.
p,max
shown:blackcontoursrepresentnumericalintegrationof
equation(5)usingthecomprehensiveexpressionsforthe
absorption coefficient, k, and the real part of the
i
refractiveindex,m,fromequations(2)and(3),andgrey
r
contours represent the analytic approximations devel-
oped in section 3. The approximations are reasonably
accurate for all frequencies. At low peak plasma fre-
quencies (low N ), the black and grey contours are dis-
0
tinguishablefromeachotheronlyatpeakaltitudeswhere
the intermediate expression of section 3 applies (that is,
Figure4. Powerloss(dB)intheM2layer(120km)asa
wherez ≈z ).Asthepeakplasmafrequencyapproaches
0 L functionofpeakelectrondensity,N ,andradiofrequency.
the radio frequency (that is, as N increases), differences 0
0 Results of a numerical integration, not the analytical
between the black and grey contours become more
approximations derived in section 3, are shown here.
apparent as the assumption that m = 1 weakens. Yet the
r Notethatsomecontoursdonotextendtolowfrequencies.
weakening of this assumption does not have terrible
Radio signals at these frequencies are reflected by the
consequences. For regions of Figure 3 where the radio
ionospherebeforetheycanreachthesurface.Radiowaves
frequencyisgreaterthantwotimesthemaximumplasma
cannot propagate through the ionosphere if the angular
frequency, the value of the real part of the refractive
frequencyoftheradiowaveissmallerthanthemaximum
index, m, differs from unity by less than 0.13.
r valueoftheionosphericplasmafrequency.
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RS2004 WITHERS:MARSIONOSPHERIC ATTENUATION RS2004
flares,whichmayleadtoN >1011m−3forshortintervals.
0
We defer the study of the rise and fall of attenuation
during an intense solar flare to future work.
4.3.3. Power Loss due to the Meteoric Layer
[32] The high‐frequency limit for power loss can be
used here because w > n at 85 km for all frequencies
greater than 66 kHz. The power loss due to themeteoric
layer is shown in Figure 6. Two separate results were
calculated, one for numerical integration and one for the
analyticalapproximationsofsection3,butonlythosefor
numerical integration are shown here. However, the
results are indistinguishable, except where the radio fre-
quency is close to the maximum plasma frequency. For
regionswheretheradiofrequencyisgreaterthantwotimes
the maximum plasma frequency, the value of m differs
r
Figure5. AsinFigure4butfortheM1layer(100km). from unity by less than 0.13, confirming that the key
assumption of section 3 is valid here. Note that Figure 6
isconsistentwiththeresultsofWitasseetal.[2001].The
one for numerical integration and one for the analytical largest electron densities observed in the meteoric layer
approximationsofsection3,but only thosefor numerical to date are 2 × 1010 m−3 [Pätzold et al., 2005; Withers
integrationareshownhere.However,theresultsareindis- et al., 2008]. Using this value for N means the power
0
tinguishable, exceptwhere the radio frequency is close to loss exceeds 1 dB only for frequencies below 18 MHz.
the maximum plasma frequency. For regions where the Theattenuationcausedbythemeteoriclayerexceedsthat
radio frequency is greater than two times the maximum caused by the subsolar M2 layer if N is greater than
0
plasmafrequency,thevalueofm differsfromunitybyless about3×109m−3,whichissmallerthanallpeakmeteoric
r
than0.13,confirmingthatthekeyassumptionofsection3 densitiesyetreported[Pätzoldetal.,2005;Withersetal.,
isvalidhere.Thetypicalsubsolarpeakelectrondensityof 2008].Ifthemeteoric layerispresent, thenitseffects on
the M2 layer is 2 × 1011 m−3 [Withers, 2009], so the ionospheric attenuation of radio signals are comparable
subsolar power loss exceeds 1 dB only for frequencies to those of the larger M1 layer and greater than those of
below 6 MHz. the M2 layer.
4.3.2. Power Loss due to the M1 Layer [33] Nielsen et al. [2007] found identical attenuation
[31] The high‐frequency limit for power loss can be for the M1 layer at 110 km with peak density of 6 ×
used here because w > n at 100 km for all frequencies 1010 m−3 and the meteoric layer at 90 km with peak
greaterthan9kHz.ThepowerlossduetotheM1layeris density of 8 × 109 m−3. Equation (12) predicts that the
showninFigure5.Twoseparateresultswerecalculated, attenuation of these two layers will be identical if the
one for numerical integration and one for the analytical productoftheneutralandplasmadensitiesattheirpeaks
approximationsofsection3,butonlythosefornumerical
integration are shown here. However, the results are
indistinguishable, except where the radio frequency is
close to the maximum plasma frequency. For regions
where the radio frequency is greater than two times the
maximumplasmafrequency,thevalueofm differsfrom
r
unity by less than 0.13, confirming that the key assump-
tion of section 3 is valid here. The typical peak electron
densityintheM1layerisone‐thirdtoone‐halfofthepeak
electron density in the M2layer [Fox and Yeager, 2006;
Withers, 2009]. If we assume N = 1011 m−3, then the
0
subsolar power loss exceeds 1 dB only for frequencies
below 16 MHz. The attenuation caused by the M1 layer
exceeds that caused by the subsolar M2 layer if N in
0
the M1 layer is greater than about 2 × 1010 m−3, or one‐
fifth of our adopted value of 1011 m−3. Thus, the iono-
sphericattenuationexperiencedduringtypicalconditions
isdominatedbytheM1layer,nottheM2layer.Electron Figure 6. As in Figure 4 but for the meteoric layer
densities inthislayeraresignificantly enhancedbysolar (85 km).
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RS2004 WITHERS:MARSIONOSPHERIC ATTENUATION RS2004
areidentical. Fromtheequivalenceofthesetwoattenua-
tions, equation (12) predicts that the neutral scale height
is 10 km, which is exactly the value that was used by
Nielsen et al. [2007]. This successful prediction shows
that the relationships derived in section 3 are consistent
with the results of Nielsen et al. [2007].
4.3.4. Power Loss due to the EP Layer
[34] AtthealtitudeoftheEPlayer(z0=35km),w=n
at 46 MHz. Thus, the high‐, intermediate‐, and low‐
frequencycasesmustbeconsideredhere,ratherthanjust
asinglecase.ThepowerlossduetotheEPlayerisshown
inFigure7.Twoseparateresultswerecalculated,onefor
numerical integration and one for the analytical approx-
imations of section 3, but only those for numerical inte-
grationareshownhere.ThecurvaturevisibleinFigure7,
Figure 8. Power loss (dB) in theEP layer (35 km) asa
which is not present in Figures 4–6, appears because the
function of peak electron density, N , and radio fre-
frequency range of 1 MHz to 100 GHz spans the high‐, 0
quency. Black contours show the results of a numerical
intermediate‐, and low‐frequency cases. Only the high‐
integration. Grey contours show the analytical approxi-
frequencycaseisrelevantforFigures4–6.Thetransition
mationsderivedinsection3.Greycontoursmaybeinvis-
betweenhigh‐,intermediate‐,andlow‐frequencycasesis
iblebeneathblackcontourswhentheapproximationsare
more clearly visible in Figure 8, where results for both
veryaccurate.Insuchinstances,thegreycontourscanbe
numerical integration and the analytical approximations
seen beneath the numbers that label the black contours.
ofsection 3areshown.Thetwosets ofresults areindis-
The dashed black curve indicates where the maximum
tinguishable,exceptwheretheradiofrequencyiscloseto
plasma frequency equals the radio frequency, f. In the
themaximumplasmafrequencyandbetween f=100MHz
regiontotheleftofbothverticalgreydashedlines,thegrey
andf=400MHz(w=2×to8×n0).Forregionswhere contoursusethelow‐frequencyexpressionofsection3.In
theradiofrequencyisgreaterthantwotimesthemaximum
theregionbetweenthetwoverticalgreydashedlines,the
plasmafrequency,thevalueofmrdiffersfromunitybyless grey contours use the intermediate‐frequency expression
than0.06,confirmingthatthekeyassumptionofsection3
ofsection3.Intheregiontotherightofbothverticalgrey
isvalidhere.Therearenodirectobservationsofthislayer dashed lines, the grey contours use the high‐frequency
or its maximum electron density, but two independent
expressionofsection3.
modelspredictthatelectrondensitiesof108m−3arepro-
duced at 35 km by typical galactic cosmic ray fluxes
[Molina‐Cuberosetal.,2002;Haideretal.,2009].Using caused by this plasma layer exceeds that caused by the
thisvalueforN meansthepowerlossexceeds1dBonly M2 layer, the M1 layer, and the meteoric layer for all
0
forfrequenciesbelow4MHz.Evenataslowafrequency frequencies greater than 50 MHz (Figure 9).
as 1 MHz, the power loss is only 3 dB. The attenuation
5. Application to Attenuation Observed
by MARSIS
5.1. MARSIS Observations and Blackouts
[35] TheMARSIStopsideradarsounderinstrumenton
Mars Express has provided direct measurements of the
attenuation of radio signals by the ionosphere of Mars
[Gurnettetal.,2008].Thisradar,whichoperatesat0.1to
5.4 MHz, usually detects surface reflections at frequen-
cies greater than the ionosphere’s plasma frequency, but
sometimes itdoesnot.The M2andM1 layerscannot be
responsible for the MARSIS blackouts, which persist
deepintothenightsidewheretheselayersareabsent.The
spatial patchiness of observed meteoric layers [Pätzold
etal.,2005;Withersetal.,2008]isinconsistentwiththe
global scale of the MARSIS blackouts. Blackouts, while
they last, are continuous, whereas meteoric layers occur
Figure 7. As in Figure 4 but for the EP layer (35 km).
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