Gravitational Lensing of Rays through the Levitating Atmospheres of Compact Objects

1.6 MB·

by  Adam Rogers

universe
Article
Gravitational Lensing of Rays through the Levitating
Atmospheres of Compact Objects
AdamRogers
DepartmentofPhysicsandAstronomy,UniversityofManitoba,Winnipeg,MBR3T2N2,Canada;
[email protected]
AcademicEditors:ValerioBozza,FrancescoDePaolisandAchilleA.Nucita
Received:24November2016;Accepted:22December2016;Published:date
Abstract: Electromagnetic rays travel on curved paths under the influence of gravity.
7
1 Whena dispersiveopticalmediumisincluded, these trajectoriesarefrequency-dependent. Inthis
0 work we consider the behaviour of rays when a spherically symmetric, luminous compact
2
object described by the Schwarzschild metric is surrounded by an optically thin shell of plasma

n
supported by radiation pressure. Such levitating atmospheres occupy a position of stable radial
a
J equilibrium, where radiative flux and gravitational effects are balanced. Using general relativity

0 andaninhomogeneousplasmawefindtheexistenceofastablecircularorbitwithintheatmospheric
2
shellforlow-frequencyrays.Weexplorefamiliesofboundorbitsthatexistbetweentheshellandthe

] compactobject,andidentifysetsofnovelperiodicorbits. Finally,weexamineconditionsnecessary
c
forthetrappingandescapeoflow-frequencyradiation.
q
-
r
g Keywords: gravitation-plasmas-pulsars;general-stars;neutron
[

1
v
3 1. Introduction
9
6 Novelbehavioursforelectromagneticraysresultfromthecombinationofgravitationallensing
5
and the optical effects from a dispersive plasma medium [1,2]. The joint effects of gravitation,
0
. refraction and dispersion have been studied using homogenous [3,4] and inhomogenous plasma
1
distributions [5] on a variety of scales [6,7] and geometries [8–10]. Bound orbits of rays under the
0
7 effectof gravityandplasma havealsobeendiscussed [5,11]. The effectof apower-lawdistribution
1
of transparent plasma on the pulse profiles of a compact object (CO) have been shown to produce
:
v frequency-dependentshifts[12],andtrappingofrays[13].
i
X Intheenvironmentofahighlyluminousneutronstar(NS),adistributionofplasmaisstrongly
r affectedbybothradiationpressureandgravity[14–16]. IntheNewtonianpicture,theradiativeforce
a
decreases as r 2, and therefore if radiation overpowers gravity at a particular radius it is true for
−
all space. However, the situation described by general relativity is more subtle. In Schwarzschild
space-timetheradiativeforcechangesatafasterratethangravitationaleffects[17,18]. Thus,thereis
only a single radius at which the effects of gravity and radiation on a test particle are balanced.
ThispositionofradialequilibriumdefinesasurfacecalledtheEddingtoncapturesphere(ECS[17,19]).
Atthisradius,astableshellofplasmacancollect,detachedfromtheNSsurface[20].Theselevitating
atmosphericshellshavebeenstudiedinopticallythinandopticallythickconditions[21,22].
Inthiswork,weassumethattheopticallythinlevitatingatmospherearoundaCOcanbetreated
asatransparentplasmashellandthatradiationisfreetopropagatebetweenthestellarsurfaceandthe
shell. Wecalculatetheresultingfrequency-dependentraytrajectoriesundertheeffectsofgravitation
fromtheCOaswellasrefractionanddispersionfromtheplasmashell. InSection2we discussthe
theoreticalaspectsoflevitatingatmospheresandtheray-tracingprocedure. InSection3we present
ournumericalresults.Section4providesdiscussionofourcalculationsandthegeneralresultsthatwe
haveobtained,aswellasexampledensityfunctions. Finally,wesummarizeourfindingsinSection5.
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2. Theory
LetususethesetofunitsinwhichG =c = h¯ =1. Weassumesphericalsymmetry,andwithout
lossofgeneralityworkintheequatorialplaneθ =π/2. TheSchwarzschildlineelementis
dr2
ds2 = A(r)dt2+ +r2dφ2 (1)
− A(r)
with
2M
A(r) =1 . (2)
− r
As a first approach, let us treat the plasma shell as a non-magnetized, inhomogenous
optically-thinmediumwithindexofrefractionoftheform
ω (r)2
n2(r) =1 p (3)
− ω2
withrayfrequencyωandlocalplasmafrequency
4πq2N
ω2 = 0N(r)= kN(r) (4)
p m
where the plasma particles have charge q and mass m, and N(r) describes the number density
distributionoftheplasma,scaledbythemaximumN . Ontherighthandsidewehaveconsolidated
0
the coefficients into the constant k. For the remainder of our numerical work we use a simple
exponentialfunctiontodescribetheshelldensity
(r r )2
N(r)=exp − 0 . (5)
− σ2
(cid:20) (cid:21)
with the maximum located at a distance r > R, where R is the radius of the CO. Despite these
0
assumptions,ourconclusionswillbegeneral,andwillnotdependontheexactdetailsofthedensity
function. The numerical examples in the body of the text use Equation (5), however we consider
otherdensityfunctionsinSection4. Infact,anyshell-like densityprofilewillgeneratequalitatively
similarresults. ThroughoutthetextwewillrefertotheplasmadensityasN,andsuppresstheradial
dependenceforsimplicity.
TheHamiltonian foranelectromagneticrayunderthe effectof gravityandanopticalmedium
was first considered by Synge [23]. This result was subsequently specialized to non-homogenous
plasma[2,3],whichwestateusingEquations(3)–(5),
1
H = gijp p +kN . (6)
i j
2
(cid:16) (cid:17)
Theequationsofmotionare
dxi ∂H
= = gijp (7)
j
dλ ∂p
i
and
dpi = ∂H = 1gjkp p k (N) . (8)
dλ −∂xi −2 ,i j k− 2 ,i
Thederivativesofthetandφmomentavanish,providingtheconstantsofmotion
pt = E = ω∞ (9)
− −
whereω∞ istherayfrequencyanobserveratinfinitywoulddetectprovidedtherayescapes,and
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p = L (10)
φ
where L is the angular momentum. Since we restrict the trajectories to the equatorial plane,
themomentum p vanishes. Withthisrestrictiontheindicesi, jandkeachtakethevaluest,randφ.
θ
Forastaticmediumtheeffectiveredshiftrelationshipisgivenby
ω∞
ω(r)= . (11)
1
A(r)2
Thecorrespondingcoordinatederivativesare
dt E
= (12)
dλ A(r)
dφ L
= . (13)
dλ r2
For spherical symmetry, there is no orbital motion out of the θ = π/2 plane. The radial
momentumisgivenbyEquation(6)vanishing,
1
L E2 1 kN 2
p = A(r) + (14)
r ±A(r) L2 − r2 L2
(cid:20) (cid:18) (cid:19)(cid:21)
with the sign of p denoting a radially infalling ray as p < 0 and an outgoing ray with p > 0.
r r r
Thederivativeoftheradialmomentumis
dp M M L2 kdN
r = E2 p2+ , (15)
dλ −r2A2(r) − r2 r r3 − 2 dr
givingtheradialcoordinatederivative
dr
= A(r)p . (16)
r
dλ
Finally,wefindthechangeinangleasafunctionofradiususingEquations(13),(14)and(16),
dφ 1
= . (17)
dr ± 1
r2 E2n(r)2 A(r) 2
L2 − r2
(cid:16) (cid:17)
Theeffectivepotentialthatincludestheeffectoftheplasmais
2M L2
V(r) = 1 +kN , (18)
− r r2
(cid:18) (cid:19)(cid:18) (cid:19)
withfirstandsecondderivatives
dV = 2L2 1 3M
dr − r3 − r (19)
+k (cid:16)1 2M (cid:17)dN + 2MN
− r dr r2
h(cid:16) (cid:17) i
d2V(r) = 6L2 1 4M
dr2 r4 − r (20)
+k (cid:16) 1 2M(cid:17) d2N + 4MdN 4MN .
− r dr2 r2 dr − r3
h(cid:16) (cid:17) i
Thesegeneralexpressionswithpower-lawdensityreproducethecorrespondingequationsfrom[12].
Circularorbits arefound atradialdistances where the derivative of the effectivepotentialvanishes
[24,25].Wecalltheseextremalradiir ,whereetakesthelabelssforastableorbitanduforanunstable
e
orbit. Stableminimafurtherrequire
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d2V(r)
>0. (21)
dr2 (cid:12)
(cid:12)rs
(cid:12)
Solvingforthecorresponding LwithdV/dr = 0g(cid:12)ivestheangularmomentumrequiredtoproduce
(cid:12)
acircularorbit
1
L2 = re4k21 1 2M dN + 2MN 2 (22)
e 2(re−3M) "(cid:18) − re (cid:19) dr (cid:12)re re2 e#
(cid:12)
(cid:12)
with Ne = N(re). (cid:12)
Levitating spherical shell-type atmospheres around COs are physically motivated and have
anestablishedtheoreticalfoundationintheliterature[14,22]. Toexplorethebasicsofthesesolutions,
let us follow the approach in [21] and write the observed stellar luminosity as ℓ∞. Then the local
stellarluminosityasafunctionofradialdistancerfromtheCOisgivenby
ℓ
ℓ(r)= ∞ . (23)
A(r)
TheobservedEddingtonluminosityis
4πMm
ℓ = (24)
Edd
σ
T
where m is the plasma particle mass and σ is the cross-section for Thomson scattering. However,
T
the Eddington luminosity can also be used to define a local critical luminosity required to produce
aradiativeforcewhichbalancesgravityatadistancer[15]. Thelocalcriticalluminosityis
ℓ
ℓ (r) = Edd . (25)
c 1
A2(r)
The radius atwhich the stellar luminosity ℓ is equal to the critical luminosity ℓ defines a spherical
c
surfacearoundtheCOcalledtheEddingtoncapturesphere[15,21]withradius
2M
r = , (26)
ECS 1 λ2
−
wheretheratiooftheobservedluminosityandtheEddingtonluminosityis
ℓ
∞
λ = . (27)
ℓ
Edd
PlasmaparticlesneartheCOaresubjecttoradiationdragandhavetheirangularmomentumreduced
[26]. TheseparticlesgatherontheECSsinceitisanequilibriumpositionintheradialdirection[17,19]
andisstableagainstradialoscillations[20].
The stability of the ECS allows for the formation of a levitating shell of plasma. For larger
values of the luminosity ratio, 0 < λ < 1, these levitating atmospheres are separated from the
NS surface with no significant density between. Analytical density profiles have been constructed
for isothermal and polytropic levitating atmospheres in the optically thin case [21]. The polytropic
atmospheres have thickness that depends on the temperature T, with higher T producing more
distended atmospheres. The optically thick case has also been investigated numerically [22]. The
radiativeequilibriumconditionhasbeenusedintheanalysisofphotosphericexpansionX-raybursts
fromNSs[27–30],andtheeffectofalevitatingatmospheremayalsocontributetothenatureofthese
bursts[21,26].
Our assumed density N (Equation (5)) approximatesthe rapid drop-off from the peak density
r = r ofthelevitatingpolytropicfluidshells[21]whileremaininganalyticallysimple. However,
0 ECS
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our conclusions are qualitatively insensitive to the particulars of N, provided a maximum external
to the stellar surface. Thus, we are not concerned with the exact details that lead to the levitating
atmosphere solution and rather seek to capture the general character of these plasma shells in our
numericalexamples.
We will examine the trajectories of electromagnetic rays by ray-tracing. We assume a ray of
energyEandangularmomentum Lbeginsataninitialposition(t,r,φ),directedinwardoroutward
depending on the choice of sign for p . We then solve Equations (9)–(16) numerically to find the
r
pathoftherayundertheinfluenceofgravityandtheeffectoftheplasmagivenbythedensity N in
Equation(5). Theintegrationoftheequationsofmotion isceasedwhenthetrajectoryintersectsthe
stellarsurfaceorescapestoasignificantdistance(r 100R).Forboundrays,wefollowtheorbitsfor
≥
anarbitrarynumberoftime-steps,chosentogiveafewperiastron-apastronpairs.
3. NumericalResults:AVarietyofNovelOrbits
To initialize our numerical ray-tracingroutine, we select the physical parametersthat describe
our CO and the levitating atmospheric shell. We choose the mass of the CO as M = 1.4M . To
⊙
maximizethegravitationaleffects,weuseanextremelyrelativisticcompactnessratioof R/M = 1.6,
at the stiff end of the nuclear EoS [13,31]. We choose a relatively high ratio of stellar to Eddington
luminosity of λ = 0.9,which givesthe centerof the shellatr = 10.5M. Finally, we estimate the
ECS
thicknessoftheshellusingtheapproximateformulafrom[21],
2M
H 2 10−3 (28)
≈ × (1 λ)2
−
whichgives H 1km for our choicesof parameters. We thenset σ = H andr = r in our test
0 ECS
≈
function,giveninEquation(5).
Theequationsofmotionshowusthattheconstantk setsthefrequencyatwhichplasmaeffects
become relevant to the ray trajectories, but the dynamics of the rays are unchanged provided we
redefinethemomenta p = p /√k,aswellastheaffineparameterλ = √kλ. Thus,inournumerical
′i i ′
exampleswesimplysetk =1anddiscusstherelevantfrequencyscaleinSection4.
3.1.AStableCircularOrbitforElectromagneticRays
Whenanopticallythinlevitatingatmospherewitharefractiveindexisconsidered,theeffective
potentialissignificantlymorecomplicatedthanthevacuumcaseandallowsforavarietyofcircular
orbits. Asanumericalexample,letusconsideraraythathasafrequencyω∞ = 1. Sucharaywith
angular momentum L = 15.1689 has a stable circular orbit at r = 13.6468. This stable circular
s s
orbit is well within the radius of the ECS, r = 14.7368. The unstable circular orbit radius for
ECS
electromagneticraysatthepotentialmaximum,r =14.7194,isshiftedinwardslightlyfromtheECS
u
radiusduetothecontributionfromtheangularmomentumtermintheeffectivepotential.
Generally, any shell-like plasma atmosphere with a density profile N that has a maximum at
r > R and which is physically realistic (decreases as a function of r in both directions r > r
0 ECS
and r < r and which vanishes at infinity) will produce a minimum in the effective potential,
ECS
resulting in a stable circular orbit. The existence of this stable circular orbit does not require any
additionalsymmetry argumentsbe imposed on the shell density, provided the previousconditions
are satisfied. For example, an asymmetrical density profile was found for an isothermal levitating
atmosphere[21],thoughthisisnotaphysicallyrealisticsolutionsincethedensitydoesnotvanishat
infinity. However,suchadensityprofilewouldstillproduceastablecircularorbitwithinr .
ECS
3.2.PeriodicOrbits
Bound raytrajectoriesadmitperiodicsolutions withthe appropriateangular momentum L for
agivenenergyE. Considerthetimeelapsedbetweentheclosestdistanceoftheorbitfromthecentral
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CO,theperiastronradiusr andthefurthestdistance,theapastronradiusr . Letusdefinetheangle
p a
accumulatedastheintegralofEquation(17)foraboundorbit,
∆φ =2 t(ra) dφdt=2 ra dφdr. (29)
r
Zt(rp) dt Zrp dr
UsingEquation(17)therighthandsidegives
∆φ =2 ra dr . (30)
r 1
Zrp r2 E2n(r)2 A(r) 2
L2 − r2
h i
Thequantity∆φ istheratiooftheradialandangularfrequenciesfoundfromtheHamiltonianusing
r
action-anglevariables[32],
∆φr = ωφ =Λ. (31)
2π ω
r
PeriodicorbitshavearationalvalueofΛ. IntheupperpanelofFigure1wecalculateΛasafunction
ofL,holdingE = ω∞ =1,viaEquation(30). WethennumericallyfindthecorrespondingvaluesofL
forwhichΛequals1,1/2,1/3,1/4,1/5,1/6and1/7. ThestablecircularorbithasΛ =0. ForΛ =1
(dashed line) and Λ = 1/7 (dash-dotted line) we plot the corresponding effective potentials in the
lowerpanel,andshowthepotentialexperiencedbytheboundorbitasathickblackline. TheratioΛ
isboundedinLsinceaboundorbitrequiresapotentialminimum. WhenListoolow,noinner(stable)
potentialboundaryexistsbetweenRandr .ForLtoohigh,orbitsarenotboundbythecontribution
ECS
totheeffectivepotentialfromtheshell. InFigure2weplottheperiodicorbitscorrespondingtothese
Λvalues.
FortheperiodicorbitsthedenominatoroftherationalΛvaluesdenotethenumberofdeflections
from the effective potential barrier provided by the plasma in a complete orbit. The numerator
describesthecomplexityoftheorbit. Forexample,letΛ = α/β bearationalnumber. Theresulting
orbithasβreflectionsoffoftheplasmashellandthepathintersectsitself(α 1)βtimes. Afamilyof
−
periodic orbits with Λ = 1/5, 2/5, 3/5 and 4/5 is plotted in Figure 3 to demonstrate. These orbits
eachcontain5reflectionsfromtheplasmashellandfeature0,5,10and15intersectionsoftheorbital
path.
In terms of angular momentum, the periodic configurations are preceded and followed by
orbits with similar morphology but which precess in the opposite sense. For example, the Λ = 1
periodicorbitisshowninthecenterpanelofFigure4. Weintegratetheequationsofmotionuntilthe
coordinatetime componentreachesanarbitraryvaluesufficienttoprovideseveralorbitsoftheCO.
Theinitialpositionismarkedasablackdotandthefinalpositionisasquare. Givenaperiodicorbit
with Λ = q, alower angularmomentum orbitwith Λ > qprecessesinacounter-clockwisemanner
(toppanel),andanorbitwithΛ <qprecessesintheoppositesense(bottompanel).
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Λ vs. Angular Momentum
1.2
1
0.8
Λ 0.6
0.4
0.2
0
8 10 12 14 16
L
Effective Potential
2
1.5
V(r) 1
0.5
0
5 10 15
r
Figure1. Toppanel: ΛasafunctionofangularmomentumLisplottedforE = ω∞ = 1asthethick
blackline. OnthispanelweplotthevalueofΛ =1asadashedline,1/2,1/3,1/4,1/5,1/6asthin
solidlinesandΛ=1/7asadash-dottedline. TheintersectionsoftheΛ=1andΛ=1/7lineswith
Λ(L) are marked as white discs. The effective potential corresponding to the angular momentum
specifiedbythesediscsareplottedinthelowerpanel. Theshadedregionrepresentstheinteriorof
theCOandtheheavyblacklinesaretheeffectivepotentialsexperiencedbytheray. TheΛ = 1and
Λ=1/7casesarerepresentedbydashedanddash-dottedlinesrespectively.
Λ=1 Λ=1/2 Λ=1/3 Λ=1/4
10 10 10 10
y 0 y 0 y 0 y 0
−10 −10 −10 −10
−10 0 10 −10 0 10 −10 0 10 −10 0 10
x x x x
Λ=1/5 Λ=1/6 Λ=1/7 Λ=0 (Circular)
10 10 10 10
y 0 y 0 y 0 y 0
−10 −10 −10 −10
−10 0 10 −10 0 10 −10 0 10 −10 0 10
x x x x
Figure2. PeriodicorbitswithangularmomentumLfoundfromΛ =1,1/2,1/3,1/4,1/5,1/6,1/7
andthestablecircularorbit.TheCOistheshadeddisc,andtheunstablecircularorbitismarkedwith
adashedline.
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Λ=1/5 Λ=2/5 Λ=3/5 Λ=4/5
10 10 10 10
y 0 y 0 y 0 y 0
−10 −10 −10 −10
−10 0 10 −10 0 10 −10 0 10 −10 0 10
x x x x
Figure3.AfamilyofperiodicorbitswithLfoundfromΛ=1/5,2/5,3/5and4/5.Theorbitsreflect
offoftheplasmashell5times,andtheorbitalpathintersectsitself0,5,10and15times,respectively.
TheCOistheshadeddisc,andtheunstablecircularorbitismarkedwithadashedline.
L=7.35000
20
10
y 0
−10
−20
−20 −10 0 10 20
x
L=7.35835
20
10
y 0
−10
−20
−20 −10 0 10 20
x
L=7.37000
20
10
y 0
−10
−20
−20 −10 0 10 20
x
Figure4. AnexampleofprecessionasafunctionofangularmomentumforΛ = 1(middlepanel).
ForΛ > 1wefindprecessioninthecounter-clockwisedirection(toppanel)andforΛ < 1theorbit
precessesinthe opposite sense (bottompanel). Initial positionsare plotted as black dots and final
positionsaresquares.
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3.3.FrequencyWindows
We define two particularly significant frequency ranges for rays that interact with the plasma
atmosphere. Theescapewindow(EW)isdefinedbythefrequencyrangeω∞0 <ω∞ ω∞+. Raysin
≤
this frequency window can escape from the CO surface to reach an observer at infinity, but the
trajectoriesarestronglyinfluencedbythepresenceoftheplasmaatmosphere.BelowtheEWasecond
frequencybandexists,ω∞ < ω∞ ω∞0,whichwerefertoastheanomalouspropagationwindow
− ≤
(APW). In this frequency range, rays are trapped by the plasma. Rays in the APW that leave the
surfaceoftheCOtraveltoamaximumaltitudeandthenturnbacktotheCOsurface. Thistrapping
is analogous to the anomalous propagation of low-frequency radio wavesin the atmosphere of the
Earth. Inaddition to a family of trappedraysemitted fromthe CO surface, the APW also includes
asetofraysthatapproachthetheplasmaatmospherefromtheoutsideandarescatteredoffofit. The
EW and APW were explored in [13] for a CO surrounded by a cloud of plasma with a power-law
density.
We define the asymptotic plasma frequency ω∞p in analogy with the ray frequency at infinity
(Equation(9))usingtheredshiftdefinition(Equation(11)),
1
ω∞p = A(r)2ωp(r), (32)
such that the propagation of radiation requires ω∞ to exceed ω∞p along the path of a ray. In fact,
the quantity on the right hand side is simply the square root of the effective potential with L = 0,
forwhichr =r .
u ECS
ThelimitingfrequencyforaradiallydirectedraytoescapeisgivenbyEquation(32)atr ,
ECS
ω∞0 = V(ru) =λ kN(rECS)= λ√k. (33)
q q
The condition on radially directed rays defines the floor of the EW and the APW ceiling. We will
focusoneachofthefrequencyrangesseparatelyinthefollowingsections.
3.3.1.RayTrapping
Outgoing radially-directed rays with ω∞ ω∞0 cannot propagate through the plasma
≤
atmosphere. Since the effective potential contains a contribution from the angular momentum,
turningpointsexistforallrayswithfiniteL,
ω∞ = V(r) >ω∞p. (34)
q
Therefore,providedL =0,allrayswithω∞ ω∞0arescattered.Rayswithfiniteangularmomentum
6 ≤
that are emitted from the CO surface reach a maximum height and return to the CO surface. This
trappingrequiresapotentialmaximumoutsidethestellarsurface,V(R) V(r ),andthuswedefine
u
≤
thelowerAPWlimitas
ω∞ = V(R). (35)
−
q
For incoming rays that approach the CO atmospheric shell externally, we write the angular
momentum as L = ω∞b where b is the impact parameter. For non-vanishing b, the approaching
raysreachaminimumdistancebeforescatteringfromtheplamaatmosphereduetothepresenceof
aturningpointandreturntoinfinity.
We plot examples of rays in the APW in Figure 5 and use ω∞ = √1/2, well below the APW
ceilingω∞0 = λk1/2 = 0.9forthechoiceofexampleparametersinourunits. Weplotthetrajectories
oftenrayslaunchedfrom R atposition φ = π/2. Wevarytheangularmomenta ofthe raysfrom0
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(dashedray)to0.99L (dashed-dottedray),wherethemaximumangularmomentumisfoundby
max
settingbyrelatingthesquareofthefrequencyandtheturningpointatthesurface,
R
Lmax = ω∞ 1. (36)
A(R)2
AtandaboveL ,therayisinaboundorbitandonlyskimsthesurfacetangentiallyatR. TheECS
max
radiusisplottedasadottedcircle.Externalraysthatapproachfromthe+xdirectionarescatteredby
the plasma atmosphere. For these rayswe used the same frequencyand anidenticalset of angular
momenta,givingtheimpactparameters
L
b= . (37)
ω∞
Thefiducialraydoesnotpropagateintheplasmasinceω∞ = ω∞p forthe L = 0case. Inthebottom
panelweplottheeffectivepotentialforthe L =0ray(dashedcurve)andthepotentialforaraywith
0.99Lmax(solidcurve). Thehorizontallineisω∞2. TheinterioroftheCOistheshadedregionandthe
ECSradiusisplottedastheverticaldottedline.
Rays in the APW
20
10
y 0
−10
−20
−20 −10 0 10 20
x
Potentials at the APW limits
1
0.8
V(r) 0.6
0.4
0.2
0
5 10 15
r
Figure5. TrappedandscatteredraysintheAPW.Theupperpanelplotsthepathsof10raysemitted
fromtheCOsurface,R,atφ=π/2.TheseraysareevenlyspacedbetweenL=0(dashed),andLmax
(dashed-dotted). Alsoplottedarethepathsof10raysthatapproachtheECSexternallyfromthe+x
direction, and are reflected by the potential boundary. These external rays are plotted as the solid
blackcurves.TheECSradiusisthedottedcircle.Thelowerpanelshowsthecorrespondingeffective
potential for the bound rays L = 0 (dashed) and 0.99Lmax (solid). The square of the asymptotic
frequencyω∞2 isthehorizontalline.TheshadedregionistheCOinterior,andtheverticaldottedline
istheECSradius.Bothinteriorandexteriorradiallydirectedrays(L=0)havefrequenciesω∞ =ω∞p.
Theserays,andallrayswithlowerfrequency,cannotpropagatethroughtheplasma.