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P og ess in ae ospace sciences
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A icle publica / Published pape :
Li adio i, S. [e al.]. A e iew o gas-su ace in e ac ion models o
o bi al ae odynamics applica ions. "P og ess in ae ospace sciences",
No emb e 2020, ol. 119, p. 100675/1-100675/16. DOI:
<10.1016/j.pae osci.2020.100675>.
©2021. This manusc ip e sion is made a ailable unde he CC-BY-NC-
ND 4.0 license h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/
A Re iew o Gas-Su ace In e ac ion Models o O bi al
Ae odynamics Applica ions
Sab ina Li adio i*a, Nicholas H. C ispa, Pe e C.E. Robe sa, S ephen D. Wo alla, Vi o T.A.
Oikoa, S e e Edmondsona, Sa ah J. Haigha, Clai e Huy ona, Ka ha ine L. Smi ha, Luciana A.
Sinpe ua, B andon E. A. Holmesa, Jona han Becedasb, Rosa Ma ´
ıa Dom´
ınguezb, Valen ´
ın
Ca˜
nasb, Simon Ch is ensenc, Ande s Mølgaa dc, Jens Nielsenc, Mo en Bisgaa dc, Yung-An
Chand, Geo g H. He d ichd, F ancesco Romanod, S e anos Fasoulasd, Cons an in T aubd,
Daniel Ga cia-Almi˜
nanae, Sil ia Rod iguez-Donai ee, Miquel Su edae, Dhi en Ka a ia , Badia
Belkouchig, Alexis Con eg, Jose San iago Pe ezg, Rachel Villaingand Ron Ou lawh
aThe Uni e si y o Manches e , Ox o d Road, Manches e , M13 9PL, Uni ed Kingdom
bElecno Deimos Sa elli e Sys ems, Calle F ancia 9, 13500 Pue ollano, Spain
cGomSpace AS, Langage ej 6, 9220 Aalbo g Eas , Denma k
dUni e si y o S u ga , P a enwald ing 29, 70569 S u ga , Ge many
eUPC-Ba celonaTECH, Ca e de Colom 11, 08222 Te assa, Ba celona, Spain
Mulla d Space Science Labo a o y (UCL), Holmbu y S . Ma y, Do king, RH5 6NT, Uni ed Kingdom
gEu oconsul , 86 Boule a d de S´ebas opol, 75003 Pa is, F ance
hCh is ophe Newpo Uni e si y Enginee ing, Newpo News, Vi ginia 23606, Uni ed S a es
Abs ac
Renewed in e es in Ve y Low Ea h O bi s (VLEO) - i.e. al i udes below 450 km - has led o
an inc eased demand o accu a e en i onmen cha ac e isa ion and ae odynamic o ce p edic-
ion. While he o me equi es knowledge o he mechanisms ha d i e densi y a ia ions in
he he mosphe e, he la e also depends on he in e ac ions be ween he gas-pa icles in he
esidual a mosphe e and he su aces exposed o he low. The de e mina ion o he ae odynamic
coe icien s is hinde ed by he nume ous unce ain ies ha cha ac e ise he physical p ocesses oc-
cu ing a he exposed su aces. Se e al models ha e been p oduced o e he las 60 yea s wi h
he in en o combining accu acy wi h ela i ely simple implemen a ions. In his pape he mos
popula models ha e been selec ed and e iewed using as disc imina ing ac o s ele ance wi h
ega ds o o bi al ae odynamics applica ions and heo e ical ag eemen wi h gas-beam expe i-
men al da a. Mo e sophis ica ed models we e neglec ed, since hei inc eased accu acy is gene -
ally accompanied by a subs an ial inc ease in compu a ion imes which is likely o be unsui able
o mos space enginee ing applica ions. Fo he sake o cla i y, a dis inc ion was in oduced
be ween physical and sca e ing ke nel heo y based gas-su ace in e ac ion models. The phys-
ical model ca ego y comp ises he Ha d Cube model, he So Cube model and he Washboa d
model, while he sca e ing ke nel amily consis s o he Maxwell model, he Nocilla-Hu lbu -
She man model and he Ce cignani-Lampis-Lo d model. Limi s and asse s o each model ha e
been discussed wi h ega ds o he con ex o his pape . Whe e e possible, commen s ha e
been p o ided o help he eade o iden i y possible u u e challenges o gas-su ace in e ac ion
science wi h ega ds o o bi al ae odynamic applica ions.
Keywo ds: Gas-Su ace In e ac ion, Ve y Low Ea h O bi , O bi al Ae odynamics
P ep in submi ed o P og ess in Ae ospace Sciences No embe 24, 2020
a Xi :2010.00489 3 [physics.space-ph] 22 No 2020
Di icul ies in modelling he in e ac ion o he nea -Ea h ae odynamic en i onmen wi h
sa elli es in Low Ea h O bi (LEO) a e due o a lack o knowledge on he mechanisms ha de-
e mine he he mosphe e o al densi y a ia ion, he magni ude and he di ec ion o he he mo-
sphe ic wind ec o and he dynamics o he Gas-Su ace In e ac ion (GSI). These unce ain ies
a ec he compu a ion o he accele a ion ha d ag - he main sou ce o pe u ba ion o al i udes
below 600 km [1] - exe s on sa elli es:
aD=−1
2ρV2
el
CDS e
m
V el
|V el|(1)
whe e S e is he e e ence su ace adop ed o pe o m he compu a ion and mis he sa elli e’s
mass, o en he only pa ame e known wi h subs an ial accu acy unless any p opellan consump-
ion needs o be acknowledged. In Eq. 1 unce ain ies a e ound in he assessmen o he o al
densi y (ρ), he ae odynamic d ag coe icien (CD)and he sa elli e eloci y wi h ega ds o he
o a ing a mosphe e V el. Since hese sou ces o unce ain ies a e mu ually linked, any a emp
o discuss hem sepa a ely is imp ope . Howe e , he complexi y o he p oblem demands some
o m o simpli ica ion. The e o e, challenges encoun e ed in es ima ing ρand V el, whose a i-
a ions a e gene ally associa ed wi h luc ua ions in he he mosphe e en i onmen , will no be
ea ed in his e iew. Consequen ly, he key ac o s ha o a gi en eloci y o he low con-
ibu e o de e mina ion o he dynamic p essu e q(Eq. 2) will be dis ega ded o de o e mo e
a en ion o hose enginee ing a iables ha can be modi ied h ough p ope design and ma e ials
selec ion:
q( )=1
2ρ( )V el( )2(2)
Fo he eade ’s knowledge, comp ehensi e wo ks co e ing he mechanisms a ec ing he es i-
ma e o dynamic p essu e can be ound in [2–8].
In he ollowing sec ions o his pape , he e ec o GSI dynamics on d ag e alua ion and
compu a ion o he ae odynamic coe icien s will be discussed. Some in o ma ion ega ding he
ae odynamic egime expe ienced by sa elli es in LEO - and especially in Ve y Low Ea h O bi
(VLEO) - will be p o ided in an a emp o c ea e an adequa e backg ound o he discussion.
A en ion will be ocused on s a is ical and physical GSI models which ha e ob ained conside -
able success bo h in enginee ing applica ions and su ace science o hei capabili y o desc ibe
he complex p ocesses occu ing a he su ace wi h ela i e simplici y. In his ega d, he in en-
ion is o c ea e con inui y wi h a ecen e iew pape on he opic by Mos aza P ie o e al. [9],
which ocuses on classical analy ical models used o ae odynamic compu a ion in LEO. Gas-
beam expe imen al esul s conduc ed in he physical egimes o in e es o his pape will inally
be p esen ed, and he beha iou o he models desc ibed will be discussed, whe e e possible,
agains he iden i ied ends in sca e ing. The objec i e is hus o highligh he poin s o s eng h
and he limi s o he heo e ical models agains he a ailable expe imen al da a. In his way pos-
sible ea u e de elopmen s can be discussed and hope ully a easonable pic u e o he di icul ies
encoun e ed in app oaching he GSI p oblem o o bi al ae odynamics can be p o ided.
Inc eased knowledge o he in e ac ion mechanisms occu ing in he gas-solid phase sys em
is c ucial no only o scien i ic achie emen , bu also o he possibili y o imp o ing ae ody-
namic pe o mance o spacec a ope a ing in VLEO [10, 11]. This would e lec in inc eased
con idence in assessing he ad an ages and he d awbacks o employing ae odynamic o ques
o o bi [12–18] and a i ude con ol pu poses [19–28]. O e es ima ing o unde es ima ing he
ae odynamic o ques induced by he ac ua ion o ae odynamic con ol su aces has an impac on
2
he al i ude ange o which ae odynamic manoeu ing is expec ed o be easible. This seems el-
e an especially o missions ope a ing in pe iods o low sola ac i i y, when he he mosphe ic
densi y alues a al i udes abo e 200 km a e signi ican ly lowe han hose expec ed du ing high
sola ac i i y [9]. In e ms o a i ude con ol implemen a ion, he achie able ae odynamic con-
ol au ho i y abou he oll, pi ch and yaw body axes d i es he design o he con olle selec ed.
This is especially ue i con en ional ac ua o s (e.g. eac ion wheels, magne o que s) a e mean
o be used in syne gy wi h he designa ed con ol su aces. Undesi ed ae odynamic o ques may
coun e ac he con ol ac ion o he wheels, dis u bing he a i ude ask and e en ually leading o
sa u a ion. Simila ly, ae odynamic o bi con ol [29], o ma ion lying [30–33] and endez ous
manoeu es [34, 35] would signi ican ly bene i om any imp o emen in GSI models, espe-
cially wi h ega ds o he possibili y o p oducing con ol o ques in he di ec ion pe pendicula
o he o bi plane. A eliable es ima ion o he ae odynamic coe icien s is also undamen al in
he e alua ion o he impac ha ae odynamic based manoeu es may ha e on he a e o decay
o sa elli es in o bi . This knowledge could po en ially be use ul o a be e p edic ion o he
sa elli e e-en y ajec o y [16, 18] o o achie e a be e knowledge o he expec ed ae ody-
namic o ces and o ques induced on he am su aces du ing con olled e-en y in a mosphe e.
This knowledge ep esen s a conside able ad an age e en o spacec a ha a e al eady in o bi ,
assuming ha he ma e ials employed o he ex e nal su aces a e known and ha a good p e-
dic ion o he en i onmen al condi ions is achia able. Ne e heless, a be e knowledge o he
phenomena occu ing a he su ace could po en ially d i e a mo e a ional design o he sa elli e
geome y acco ding o he desi ed ae odynamic pe o mances. The same geome y is indeed
expec ed o p o ide a di e en ae odynamic beha iou wi h a ying sca e ing e-emission pa -
e ns. Compa ably, he design o A mosphe ic B ea hing Elec ic P opulsion (ABEP) sys ems
is d i en by he ae odynamic pe o mance expec ed o he ma e ials employed [36]. Any im-
p o emen s in he eliabili y o he GSI models may ansla e in a mo e con iden de ini ion o
op imal pe o mance anges and i can possibly pa e he way o new design c i e ia.
1. The Ae odynamic En i onmen
A he al i udes whe e VLEO sa elli es o bi , i.e. below 450 km [37], he a mosphe e is so
enuous ha he low can no longe be conside ed a con inuum. In his scena io, he p inciples
ha ule he in e ac ion o gas cons i uen s wi h each o he and wi h a body imme sed in he
low change, as do he assump ions used o in es iga e he ae odynamic en i onmen . The dis-
c imina ing ac o employed o dis inguish be ween he di e en egimes is a dimensionless a io
known as he Knudsen numbe (Kn), which compa es he o de o magni ude o he molecula
mean ee pa h (λ)wi h a cha ac e is ic dimension o he low ield (L):
Kn=λ
L(3)
As e idenced by he de ini ion, he Knudsen numbe is no s ic ly a low p ope y since
i s alue is pa ially con olled by he e e ence leng h adop ed o desc ibe he ield o mo ion.
Lis gene ally assumed o coincide wi h a signi ican dimension o a body in he low, bu his
associa ion is no unique. Be ween λand ρ he e is an in e se p opo ional co ela ion, acco ding
o which, high alues o he mean ee pa h (and hus Kn) a e usually associa ed wi h low densi y
le els o gas-su ace in e ac ions occu ing a nano-scale leng h.
3
Th ee undamen als egimes a e usually iden i ied acco ding o he Knudsen numbe . Small
Kn(0<Kn<0.1)a e ypical o he amilia con inuum dynamics, whe e collisions be ween pa -
icles is he p e alen mechanism o in e ac ion. When Kn→ ∞, he leng h a elled by he
pa icles be o e impingemen wi h o he pa icles in he gas mix u e is conside able compa ed o
he cha ac e is ic low- ield dimension. In hese condi ions, he low is cha ac e ised by a s uc-
u e in which he in e ac ion o he gas-pa icles wi h he su ace domina es o e in e -pa icle
collisions. The dis ance a e lec ed pa icle a els be o e colliding wi h he ee s eam is o he
same o de o magni ude o λand consequen ly, he low i sel can be conside ed collisionless. In
hese condi ions, i can be assumed ha he p esence o a body in he low- ield does no pe u b
he dis ibu ion o he inciden s eam o pa icles in he icini y o he body i sel , and conse-
quen ly, no shock wa es a e expec ed o a ise. This beha iou is ypical o a highly a e ied low
egime, mo e commonly known as ee molecula low (FMF). The majo i y o au ho s ag ee
o se Kn>10 as he lowe limi o iden i y his egion [9, 27, 38–42], wi h some a ia ions
[43]. In e media e Kn alues iden i y he nea - ee molecula egime (Kn1)and he complex
ansi ional egime (Kn∼1), whe e con inuum and a e ied dynamics phenomena a e o simila
ele ance [44].
The de e mina ion o he Knudsen numbe o he o bi al ae odynamics p oblem is especially
dependen on a judicious choice o he mean ee pa h o he coo dina e sys em used. Fo he
ee molecula assump ion o be alid, 1) he in e ac ions o he inciden pa icles wi h each o he
and 2) he in e ac ions o he inciden pa icles wi h he e lec ed pa icles should be negligible
compa ed o he p obabili y o collision o he inciden pa icles wi h he su ace. Fo condi ions
1) and 2) o be me simul aneously, i is necessa y ha he o de o magni ude o he associa ed
mean ee pa hs is such ha :
Kn,λii =λii
L>10 Kn,λi =λi
L>10 (4)
whe e λii is he mean ee pa h e e ing o he in e ac ion wi h o he inciden pa icles and λi
is he mean ee pa h ela ed o he in e ac ion wi h he e lec ed pa icles. Acco ding o i s
de ini ion in kine ic heo y, he mean ee pa h is in e sely p opo ional o he e ec i e collision
c oss-sec ional a ea (πd2) and he numbe densi y o he pa icles (n) [40]:
λ∝1
πd2n(5)
whe e dis he adius o he sphe e o in luence. Fo he p oblem examined, he eloci y o he
sa elli e h ough he gas is conside able and he e-emission o he pa icles om he su ace
is ypically he mal. In hese condi ions, he numbe densi y o he e lec ed molecules can
signi ican ly inc ease, especially in p oximi y o he su ace, possibly changing he na u e o he
low locally. I he coo dina e sys em is assumed o be ixed wi h he body imme sed in he low,
λi ends o be an o de o magni ude smalle han λii [40] and should hus be p e e ed o a
conse a i e es ima ion o Kn. Fo many applica ions, howe e , he ee s eam mean ee pa h
(λ∞), de ined wi h ega ds o a coo dina e sys em ixed wi h he gas, is adop ed:
λ∞=1
√2πd2ni
(6)
whe e niis he numbe densi y o he ee s eam inciden pa icles. The ela ion be ween λi and
λ∞is p o ided by Sen man [40]:
4
λi = π
2
1
s T
Ti
λ∞(7)
whe e sis he molecula speed a io de ined la e in his sec ion, T is he empe a u e o he
e lec ed pa icles and Tiis he empe a u e o he inciden pa icles. Gene ally o sa elli es in
VLEO, s>5 and T /Ti<1. As a consequence, λ∞can be conside ably bigge ha λi . Resul s
shown by Walke e al. [45] sugges ha he Knnumbe e e ed o ees eam should be in he
o de o 103 o assu e o e all ee molecula condi ions. De ailed analysis o he unce ain ies
ela ed o he Knnumbe compu a ion o o bi al ae odynamics applica ions is p o ided in [40].
Because o he ex emely low densi y o he uppe a mosphe e, VLEO is gene ally cha ac-
e ised as a FMF en i onmen . While he Knudsen numbe de ines he deg ee o a e ac ion
o he gas, ano he pa ame e is needed o indica e how he ela i e magni ude o he sa elli e’s
eloci y and he mos p obable gas eloci y a ec s he ae odynamics. Jus as he Mach numbe
exp esses he ela ionship be ween he body’s mac oscopic eloci y and he speed o sound, he
molecula speed a io indica es he a io be ween he gas mac oscopic eloci y ( m)and he mos
p obable molecula he mal eloci y ( )acco ding o a Maxwell-Bol zmann dis ibu ion:
s= m
=V∞
q2RT∞
mm
(8)
In Eq. 8, Ris he gas cons an , mmis he gas molecula mass and T∞is he gas kine ic empe a u e
o he ee s eam. Fo he o bi al ae odynamics p oblem, he eloci y o he body imme sed in
he FMF is so high ha he in es iga ion conce ns he mo ion o a body a elling a high speeds
h ough a gas a es . Consequen ly, in Eq. 8 he gas mac oscopic eloci y e ec i ely co esponds
o he sa elli e’s eloci y (V∞)and he wo can be used in e changeably.
A a ce ain al i ude, a ia ions in he mal eloci y - and hus in e nal ene gy - occu wi h
al e a ions in he amoun o ene gy abso bed by he a mosphe e [4, 46, 47]. In hese condi ions,
he andom he mal eloci y may play a ole in he de e mina ion o he induced ae odynamic
o ces. This beha iou is gene ally associa ed wi h small alues o sand he FMFs a e acco d-
ingly e e ed as hypo he mal lows.
On he con a y, when s→ ∞, he e ec o he bulk eloci y o he pa icles on he ae ody-
namic o ces es ima ion is conside ed p edominan . Unde hese condi ions, he low is said o be
hype he mal and app oxima e kine ic heo ies igno ing he d i caused by he andom he mal
mo ion o he pa icles a e usually p e e ed. Typical alues o sa VLEO al i udes a e g ea e
han 5. Fo gene al applica ions, he hype he mal assump ion is conside ed alid o s>5
[9, 27] and, o his eason, i is equen ly employed.
The pa icula e low impinges on he ex e nal su aces o a spacec a , gene a ing induced
ae odynamic o ces whose magni ude is only dependen on he na u e o he in e ac ion. The
ae odynamic o ces expe ienced by he sa elli e in he body-axes e e ence sys em con en ionally
used o ligh -mechanics applica ions can be modelled e e ing o he amilia exp ession:
Fae o =ma=1
2ρV2
elS e CF(9)
whe e, simila ly o Eq.1, ρis he he mosphe ic densi y, V el is he sa elli e eloci y wi h ega ds
o he oncoming low, S e is he selec ed e e ence su ace and CF=[CA,CS,CN]Tis he
ec o o he h ee ae odynamic componen s along he axial, he side and he no mal di ec ion,
5
espec i ely. Simila ly, he esul ing ae odynamic o ques e e enced o he cen e o mass a e
gi en by:
Tae o = PO ×ma=1
2ρV2
elS e l e CM(10)
whe e CM=[Cl,Cm,Cn]Tis he ec o o he oll, pi ch and yaw momen um coe icien s and
PO is he posi ion ec o de ining he dis ance be ween he ae odynamic cen e o p essu e and
he cen e o mass. The magni ude o he ae odynamic o ces and coe icien s has been es ima ed
in li e a u e o bodies o di e en shapes making use o bo h analy ical [41, 48–51] and nume -
ical echniques [52, 53]. Bo h app oaches ha e bene i s and d awbacks and, ideally, he mos
ad an ageous s a egy would be o adop hem in syne gy, when pe mi ed.
Rega dless o he simula ion echnique, he es ima ion o he ae odynamic coe icien s elies
on he models employed o physically desc ibe he unde lying mechanism o GSI. CFand CM
a e gene ally compu ed ex ending he in eg als o he local s ess coe icien s (cF) o he su ace
exposed o he inciden low:
CF=ZS
cFdS (11)
CM=ZS
PO ×cFdS =ZS P− O×cFdS (12)
The a ie y o p oposed GSI models ansla es in o a a ie y in cF o mula ions. In pa icula , he
exp essions ound in li e a u e sugges ha he ae odynamic coe icien s a e a complex unc ion
o a numbe o pa ame e s which, once again, a y wi h he model adop ed (Fig. 1 and 2). A
ce ain ag eemen is howe e ound in he use o he so-called accommoda ion coe icien s, he
wall empe a u e which is usually assumed cons an , he inciden gas kine ic empe a u e and
he molecula speed a io s. These pa ame e s a e likely o depa om hei ini ial alue wi h
a ia ions in su ace con amina ion, composi ion and s uc u e, su ace he mal p ope ies and
inciden pa icle ene gy and eloci y. Fu he dependencies on geome y and eloci y ec o
di ec ion a e inco po a ed by he componen s o s ess ac ing pe pendicula ly and angen ially o
he su ace.
The inciden ee s eam, assumed o be in local Maxwellian equilib ium, in e ac s wi h he
su ace ans e ing bo h ene gy and momen um o i . Kine ic heo y-based GSI models y o de-
sc ibe he physics unde lying his phenomenon acco ding o he con ibu ion coming om bo h
he inciden and e-emi ed s eam o pa icles, wi h majo di icul ies ound in es ablishing a sa -
is ac o y ma hema ical model o he la e . The amoun o ene gy and momen um exchanged is
a measu e o he equilib ium he impinging pa icles achie e wi h he su ace be o e e-emission.
Bo h phenomena a e desc ibed by means o a se o a e age phenomenological coe icien s. The
he mal o ene gy accommoda ion coe icien , i s in oduced by Knudsen [54]:
αT=Ei−E
Ei−Ew
=Tk,i−Tk,
Tk,i−Tw
(13)
desc ibes he ene gy exchange, assuming ha he ansla ional, o a ional and ib a ional ene -
gies o he pa icles a e all a ec ed o he same deg ee by he in e ac ion wi h he wall [43]. In
Eq. 13, Eiand E a e he kine ic ene gies ca ied by he inciden and he sca e ed luxes, while
Ewdeno es he ene gy ha would be ca ied away om he su ace by he sca e ed lux i com-
ple e he mal equilib ium was achie ed and pa icles we e e-emi ed acco ding o a Maxwellian
6
dis ibu ion co esponding o he su ace empe a u e (Tw). Simila ly, Tk,iand Tk, indica e he ki-
ne ic empe a u es o he inciden and e lec ed s eams. In acco dance o wha will be discussed
in he ollowing sec ions, i is also app op ia e o in oduce a pa ial he mal accommoda ion
coe icien [55], whose alue depends on he speci ic inciden (θi) and sca e ing (θ ) di ec ions
selec ed wi h ega ds o he no mal o he su ace:
αT,Pθi, θ =Eiθi−E θ
Eiθi−Ew
(14)
To desc ibe he momen um exchange, i is common p ac ice o e e o he momen um coe -
icien (σ)[56]. Be e physical co ela ion is usually achie ed by adop ing wo sepa a e accom-
moda ion coe icien s o desc ibe he no mal (σn) and he angen ial (σ ) momen um exchange
[43]:
σn=pi−p
pi−pw
(15)
σ =τi−τ
τi
τw=0 (16)
The quan i ies in Eq. 15 and 16 a e analogous o hose al eady desc ibed o αT, he only di -
e ence being ha in his case hey e e o he momen um a he han he ene gy o he luxes.
The in o ma ion o mos signi ican alue p o ided by αT,σnand σ is ha he dis ibu ion o
he e-emi ed pa icles and eloci y is deeply in luenced by he deg ee o accommoda ion o he
inciden molecules wi h he su ace. By e e ing o hese quan i ies, wo classical and ex eme
mechanisms o in e ac ion a e iden i ied, namely specula e lec ion and di use e-emission. I
specula e lec ion occu s wi hou any he mal accommoda ion, he molecules a e elas ically e-
lec ed, no he mal ene gy is ans e ed o he body and momen um exchange occu s only along
he no mal o he su ace (αT=σn=σ =0). The angle ha he eloci y ec o o he e lec ed
pa icles o m wi h he su ace is equal o he one o he inciden s eam and i lies in he same
plane o he inciden eloci y ec o and he no mal o he su ace (Fig.3, le ).
I iso he mal di use e-emission wi h comple e he mal accommoda ion occu s (αT=σn=σ =
1), pa icles ha e ime o each equilib ium wi h he su ace and hey a e e-emi ed acco ding o a
p obabilis ic eloci y and di ec ion dis ibu ion de e mined by he wall empe a u e, ega dless o
he inciden s eam’s his o y (Fig.3, igh ). Expe imen al esul s discussed in Sec ion 4 howe e
sugges ha mo e complex sca e ing in e ac ions a e expec ed o occu a he su ace.
In VLEO, sola emissions in he Ex emely Ul a Viole (EUV) wa eleng h ha e su icien
ene gy o gene a e A omic Oxygen (AO) om he diassocia ion o O2. The chances o he high
eac i e AO o eassocia e o o m O2o O3a e qui e low because o he e y la ge mean ee
pa h cha ac e ising ex emely a e ied low egimes. Because o his, AO ep esen s he main
a mosphe ic cons i uen a VLEO al i udes and he main sou ce o con amina ion and deg ada-
ion o su aces exposed o he low. E ec s o AO in e ac ion wi h polyme s and me als include
oxygen e osion and inclusion, as well as o ma ion o ola ile and non- ola ile eac ion p od-
uc s [57]. Because o he high deg ee o con amina ion o he su aces in VLEO, mos wo ks
assume di use e lec ion wi h comple e he mal accommoda ion. Howe e , i highly accom-
moda ed pa icles a e likely o be p edominan below 300 km [58], he same canno be said a
highe al i udes, whe e he a mosphe e g adually becomes less and less dense, hus limi ing he
con aminan adso p ion o he spacec a su aces. Acco ding o his, Moe and Moe [58] p o-
posed a Maxwellian-like model o compu e he d ag coe icien . The la e uses a modi ied o m
7
Figu e 1: GSI models compa ed: pa ame e s o dependence o sca e ing dis ibu ions and ae odynamic coe icien s.
Black do s a e used o iden i y pa ame e s explici ly decla ed in he o mula ions, while whi e hombus indica e implici
pa ame e s o dependence. Pa ame e s a e g ouped acco ding o he ollowing amilies: [1] low angles; [2] ene gy
pa ame e s; [3] accommoda ion coe icien s; [4] beam shape pa ame e s; [5] eloci ies; [6] in e ac ion pa ame e s; [7]
mass pa ame e s.
8
The in e ac ion phenomenon in he angen ial di ec ions is desc ibed by he same accommoda ion
coe icien so ha , ollowing su ace iso opy, he exp essions o he sca e ing ke nels o he wo
angen ial componen s o eloci y a e in he o m o :
KCLLξi, →ξ , =1
pπσ 2−σ ×exp−ξ , −1−σ ξi, 2
σ 2−σ (32)
which sa is ies:
exp−ξi, 2KCLLξi, →ξ , =exp−ξ , 2KCLL−ξ , → −ξi, (33)
Z∞
−∞
KCLLξi, →ξ , dξ , =1 (34)
As he equa ions abo e sugges , he dynamic o he in e ac ion is egula ed by wo adjus able
pa ame e s: he no mal ene gy accommoda ion coe icien (αn) and he angen ial momen um
accommoda ion coe icien (σ ). Lo d con ibu ed signi ican ly o he success o he CL model
whils adap ing i o DSMC implemen a ion [74] and u he ex ended i s alidi y o cases ex-
cluded by he o iginal model, among which di use e-emission wi h incomple e accommoda ion
[73, 75] . Because o his, i is gene ally p e e ed o e e o he model as he Ce cignani-Lampis-
Lo d model (CLL).
A possible implemen a ion o he CLL model in closed- o m solu ions was p oposed by
Walke e al. [45]. The au ho s sugges ed modi ied exp essions o he Schaa and Chamb e
[41, 43] (S&C) closed o m equa ions wi h he CLL model. The a emp is a he di icul since,
among o he pa ame e s (Fig. 1), he S&C closed- o m solu ions a e w i en as a unc ion o σn
and σ . While an immedia e ela ion can be ound be ween he angen ial momen um accommo-
da ion coe icien and he angen ial ene gy accommoda ion coe icien (α ):
α =σ 2−σ (35)
he same canno eally be said o αnand σn. App oxima e analy ic σn−αn ela ions, w i en
as a unc ion o ou bes - i pa ame e s, we e ound adop ing a leas squa es e o app oach and
sensi i i y analysis. Ranges o a ia ion we e selec ed o some meaning ul pa ame e s abou
hei nominal alues. In his way he ag eemen be ween he σn−αn ela ion and he expec ed
CD alues could be e alua ed o he nominal condi ions and o e he ange o a ia ion o he
selec ed pa ame e s. These las we e iden i ied wi h he bulk eloci y o he pa icles, he ee
s eam empe a u e, he su ace empe a u e, he no mal he mal accommoda ion coe icien and
he angen ial momen um accommoda ion coe icien . Good co ela ion be ween he compu ed
CDand he alues p o ided by he CLL model implemen ed in DSMC Analysis Code was seen
o he modi ied S&C closed- o m solu ions w i en as a unc ion o he de i ed σn−αnlaws.
Howe e , he se o alues o be chosen o he bes - i pa ame e s is no cons an bu a ies
wi h he gas species conside ed, he ype o body impinged and, in he case o ligh e molecula
species, he αn ange assumed. Values sugges ed by Walke e al. o ep esen e i e molecula
species and body shapes could be ound in he o iginal pape om he au ho s [45]. Bigge
unce ain ies a e ound in he case o He and H o alue o αnclose o uni y.
15
3. Physical GSI Models
Physical GSI models ake ad an age o expe imen al esul s o desc ibe how he he mal mo-
ion o he su ace in luences he sca e ing dynamics o he impinging gas-beam. These models
a e hus based on assump ions ega ding he su ace in e ac ion po en ials, he su ace mo pho-
logical s uc u e and he su ace elas ici y/s i ness cha ac e is ics. Among he as numbe o
models p esen in li e a u e, special a en ion will be de o ed o he simple quasi one-dimensional
Ha d Cube model and i s mos success ul expansions: he So Cube model and he Washboa d
model. Two and h ee-dimensional la ice models [76–79] will be neglec ed in his e iew. These
las a e gene ally cha ac e ised by a mo e complex implemen a ion which leads o highe accu-
acy and also compu a ional ime; ac o s po en ially limi ing he ange o applicabili y o hese
models o he con ex o his pape . Mo eo e , i he inc ease in complexi y is jus i ied in
he ame o pu e gas-su ace in e ac ion, he same migh no be ue o o bi al ae odynamics
enginee ing applica ions. The numbe and ange o unce ain ies a ec ing he p oblem o ae o-
dynamic o ces and o ques es ima ion in VLEO is qui e high. Because o his, he inc eased
le el o complexi y is likely o be unjus i iable agains he nume ous sou ces o e o s obse ed.
3.1. The Ha d Cube Model (HC)
The Ha d Cube Model, as p oposed in i s ea lie o m by Goodman [80], has ound success
wi h Logan’s and S ickney’s [81] o mula ion. Despi e i s inhe en simplici y, esul ing om he
assump ions adop ed in i s de elopmen , he model is able o quali a i ely ep oduce he expe -
imen al lobal sca e ing ypical o clean and polished su aces. The model assumes he su ace
o be pe ec ly smoo h and he gas pa icles and su ace a oms in ol ed in he in e ac ion o be
ideally elas ic and igid. The dynamics o he collision is simpli ied by assuming ha each gas
pa icle in e ac s solely wi h a su ace a om ep esen ed as an isola ed cube in he la ice, so ha
any impac o he su ace s uc u e on he sca e ing p ope ies is neglec ed. Du ing he colli-
sion, he gas pa icle and he su ace a om in e ac as ee pa icles so ha a one-dimensional
impulsi e- epulsi e po en ial, wi h no a ac i e well, can be conjec u ed. In his way, he impac
o in e ac ion imes on he collision mechanism can be igno ed wi h bene i s in e ms o sim-
plici y and wi h only a pa ial loss o accu acy. The cubes comp ising he su ace a e o ien ed
so ha one o hei ou aces lies in he di ec ion pa allel o he su ace con ou and hey a e
cha ac e ised by an ini ial Maxwellian no mal eloci y dis ibu ion de e mined by he su ace
empe a u e. The momen um exchange is assumed o be due uniquely o he no mal compo-
nen o he gas pa icle eloci y ( n)as he angen ial componen ( )is p ese ed by he su ace
p ope ies (Figu e 6). The model p edic s a quasi-specula e-emission bo h abo e and below
he specula ange wi h each sca e ing angle θ being de e mined by a unique alue o ,n o a
gi en i, = , .
Along wi h he nume ical o mula ion, an analy ical app oach in which mean eloci y alues
a e adop ed ins ead o eloci y dis ibu ions o bo h he gas pa icles and he su ace a oms was
p oposed by Logan and S ickney [81]. Acco ding o his app oach, closed o m solu ions can be
de i ed and he pa ame e s on which he in e ac ion depends can be mo e easily iden i ied. The
la su ace assump ion allows he es ic ion o he analysis o he plane iden i ied by he su ace
no mal and he inciden eloci y ec o , so ha :
θ =co −1"co (θi) 1−µ
1+µ+16µ
9π(1+µ)
Tw
Tgcos2(θi)!# (36)
whe e µ, he gas pa icle-su ace a om mass a io, is es ic ed o a y in he ollowing ange:
16
Figu e 6: Ha d Cube Model, ep oduced om [81].
0< µ =mg
ms
<1
3(37)
and Tgis he gas-beam sou ce empe a u e. Cons ain s on he possible alues o µ esul om
u he assuming ha he gas pa icle expe iences a single in e ac ion wi h he su ace a om
conside ed. Acco ding o equa ion (36), he sca e ing di ec ion depends on he mass a io µ, he
incidence angle and he su ace- o-gas empe a u e a io. Simplici y and ease in implemen a ion
a e howe e ob ained a he p ice o a gene al loss in accu acy in desc ibing he expe imen al
esul s compa ed o he ex ended model desc ibed in [81], o which nume ical simula ion is
needed.
Some expansions o he HC model ha e been p oposed, he mos success ul one discussed
la e in his sec ion. Hu s e al. [82] and Nichols e al. [83, 84] modi ied he model o cap u e
o a ional dynamics o ellip ically-shaped dia omic molecules sca e ed om he su ace. Doll
[85] add essed he o a ional dynamics modelling dia omic homonuclea molecules as igid o-
o s wi h mo ion es ic ed o a single plane. The impo ance o his model o mul iple collisions
wi h he su ace a om a ising om he o a ional s a e we e also discussed. T apping phenomena
we e add essed by Weinbe g and Me ill [86]. T illing and Hu kmans [87] in oduced an a ac-
i e long- ange Coulomb po en ial, an exponen ial sho - ange epulsi e po en ial and modi ied
he su ace geome y ea ing a oms as ”sphe ical caps” a he hen cubes. Si z e al. [88] ex-
panded he HC model o quali a i ely desc ibe momen um o ien a ion in he sca e ing o N2
om smoo h Ag(111), in oducing a ic ional o ce along he su ace angen ial di ec ion. The
addi ional le el o complexi y cha ac e ising hese models seems inapp op ia e o he applica-
ions o which his e iew pape is in ended and, because o his, hey will no be discussed mo e
ho oughly in he ollowing sec ions.
17
Figu e 7: The So Cube Model, ep oduced om [89].
3.2. The So Cube Model
The So Cube Model p oposed by Logan and Keck in 1968 [89] owes i s name o he in o-
duc ion o a mo e ealis ic ”so ” po en ial o cap u e he physics o he gas-su ace in e ac ion.
The a oms ha comp ise he la su ace and ake pa in he collision a e assumed o beha e
like independen cubes linked o he unde lying la ice by means o single linea sp ings (Fig-
u e 7). Su ace a oms a e hus ega ded as oscilla o s cha ac e ised by a na u al equency ω
and a Maxwellian ene gy dis ibu ion co esponding o he su ace empe a u e Tw. Simila ly
o he HC model, he in e ac ion wi h a gas pa icle in ol es a single cube in he la ice and
he ene gy exchange, esponsible o he accommoda ion coe icien alue, is due solely o he
a ia ion o he no mal componen o eloci y a e he collision ( i, = , ). The in e ac ion
is howe e mo e ealis ically cap u ed assuming non-negligible collision imes, desc ibed by a
non-impulsi e in e ac ion po en ial consis ing o wo componen s. In addi ion o a epulsi e
exponen ial po en ial, which subs i u es he impulsi e epulsi e in e ac ion assumed in he HC
model, an a ac i e long- ange squa e-well po en ial componen is in oduced. The model can
be e en ually employed o ob ain an es ima e o he ac ion o pa icles ha emain apped on
he su ace a e he collision and depa om i a e achie ing su icien ene gy.
Compa ison wi h expe imen al esul s is ob ained by p ope ly selec ing he alue o h ee
modi iable pa ame e s: 1) he po en ial well-dep h W, 2) he in e ac ion ange band 3) he oscil-
la o equency ω, whose alue is assumed o be gi en by he Debye empe a u e (ΘD). Combi-
na ions o band W ha ep oduce, wi h sa is ac o y ag eemen , he expe imen al da a e e ed o
a selec ion o gas-su ace sys ems can be ound in he o iginal pape by Logan and Keck [89].
3.3. The Washboa d Model
Like he So Cube Model, he Washboa d model [90] can be ega ded as an a emp o
imp o e he Ha d Cube model ag eemen wi h he expe imen al esul s. Compa ed o he o he
GSI models discussed so a , he Washboa d model has he ad an age o add essing he e ec o
su ace co uga ion on he sca e ing p ope ies while p ese ing ela i e cla i y and simplici y.
18
Figu e 8: The washboa d model, ep oduced om [90].
The su ace con ou is simpli ied assuming a sinusoidal p o ile applied exclusi ely in one di-
ec ion, hus making he model app op ia e o bidimensional bu no o ou -o -plane sca e ing
e alua ion. Simila ly o he HC model, he cube wi h which he colliding gas pa icle in e ac s
is o ien ed along he su ace con ou and i s eloci y is de e mined by a Maxwellian dis ibu ion
a he su ace empe a u e. Because o he su ace co uga ion, he cubes a e il ed wi h ega ds
o he no mal o he la su ace so ha , o any impac poin , a local no mal and angen ial di-
ec ions can be iden i ied (Figu e 8). The maximum de ia ion o he local no mal om he la
su ace no mal di ec ion is measu ed by he co uga ion s eng h pa ame e (ΩC). The in oduc-
ion o his pa ame e allows he model o adap o di e en le els o su ace co uga ion, hus
p o iding good quali a i e ag eemen wi h expe imen al esul s anging om smoo h o ough
su aces. The a ac i e po en ial well Wp oduces e ac ion in he gas pa icle ini ial ajec o y,
hus a ying i s no mal and angen ial componen o eloci y. E en in his case he na u e o
he in e ac ion is impulsi e so ha in he local no mal- angen ial e e ence sys em, he angen ial
momen um is unchanged and he ene gy exchange is de e mined solely by he no mal momen-
um a ia ion. As a consquence, in he xz la su ace e e ence sys em he angen ial momen um
is no conse ed and he s ic assump ion cha ac e ising bo h he Ha d and he So Cube model
is acco dingly emo ed.
Analy ic exp essions o he angula sca e ing eloci y and kine ic ene gy dis ibu ions we e
p o ided o small su ace co uga ions. The pa ame e s on which hese depend a e he co uga-
ion s eng h coe icien ΩC, he inciden angle θi, he inciden kine ic ene gy Ei, he mass a io
µ, he po en ial well Wand he su ace empe a u e Tw. E en ually, he apping p obabili y can
also be add essed. Fu he ex ensions o he washboa d model include he wo ks o Yan e al.
[91] and Liang e al. [92].
4. Compa ison o GSI Models wi h Gas-Beam Expe imen s
Molecula beam expe imen s a e a use ul means o gaining in o ma ion conce ning he en-
e gy exchange a he su ace and he wall-gas sys em cha ac e is ics a a omic scale. Fo app o-
p ia e selec ion o inciden ene gy dis ibu ions, he sca e ing o neu al molecules on a a ge
19
is ep esen a i e o o bi al condi ions and can hus be used o in es iga e he ae odynamic be-
ha iou o speci ic ma e ials in he con ex o space applica ions. The li e a u e co e ing he
opic is as and his sec ion does no claim o be exhaus i e, since such an ac i i y would e-
qui e a speci ic e iew e o . The objec i e is hus o p o ide an o e all pic u e o he subjec ,
ocusing he a en ion on he physical egimes ha a e signi ican o he pu pose o his pape ,
hus add essing, whe e e possible, he poin s o s eng h and he limi s o he models p e i-
ously discussed in he con ex o VLEO ae odynamics. A VLEO al i udes, AO is he dominan
cons i uen o he esidual a mosphe e, wi h a oms impac ing on he exposed su aces wi h an
a e age eloci y o ∼7.8km s−1co esponding o a Maxwellian mean inciden ene gy dis i-
bu ion o ∼5eV. Acco ding o his, a en ion will be de o ed especially o esul s e e ing o
molecula beam sca e ing o monoa omic species om a ge s in a a ie y o condi ions. S udies
analysing dia omic and polya omic beams sca e ing a e nume ous bu hei esul s a e ypically
mo e di icul o in e p e : he in e ac ion depends on bo h he ansla ional and in e nal ene -
gies o he molecule conside ed and on he local aspec o he in e ac ion po en ial. Howe e ,
esul s o N2and O2sca e ing om Ag(111) epo ed by Asada e al. [93] show mean eloc-
i y and mean ene gy dis ibu ions wi h sca e ing angle which a e simila o hose ob ained o
monoa omic molecules. Simila ly, analysis o adso p ion and deso p ion a es, which a e im-
po an especially o hea ie molecula species, equi es co ela ion be ween he cha ac e is ics
o he sys em in e ac ion po en ial and he s icking p obabili y. These, in u n, a y wi h ini ial
o a ional and ansla ional ene gies, binding ene gies, o ien a ion o he molecules, incidence
angle and su ace empe a u e [94]. Because o his, he spa ial dis ibu ions ob ained a e he
esul o a complex mechanism o in e ac ion [95, 96] and mo e ad anced nume ical ecnhiques
in ol ing molecula dynamics o bina y collision app oxima ion a e equi ed. Ex ensi e e iews
on he opic a e howe e a ailable in [97–102] wi h some mo e da ed esul s epo ed in [103].
The sca e ing beha iou s obse ed a e no cons an as hey end o change subs an ially wi h
he sys em conside ed and, in pa icula , wi h he a io be ween he mass o he gas pa icles and
he su ace a oms, he ange o in e ac ion, he ene gy/ empe a u e o he inciden beam wi h
ega ds o he su ace empe a u e, he molecula o a omic species in ol ed in he expe imen ,
he p esence o adso ben s on he a ge su ace, he mo phology o he su ace conside ed despi e
he le el o oughness and he ela i e posi ion and o ien a ion o he gas pa icles and su ace
a oms [97]. A he ime o w i ing, a comp ehensi e heo y capable o cap u ing each possible
scena io is no a ailable such ha mul iple models, sui able o speci ic physical egimes, a e
adop ed ins ead. Fo ligh pa icles and low incidence ene gies [98] compa ed o solid maximum
phonon ene gy, he in e ac ion is p edomina ly elas ic. In hese condi ions, no ene gy ans e
occu s in he sys em and quan um mechanical phenomena, such as di ac ion, a e expec ed o
be p edominan [97].
As gas pa icles mass and inciden ene gy inc eases, he collisions become mo e inelas ic
in na u e and classic heo y is adequa e enough o ep oduce he sca e ing beha iou . Fo his
physical ange, whe he pa icles gain o ans e ene gy o he su ace depends on he ela i e
magni ude o Eiand Tw. Gene ally h ee mechanisms o in e ac ion a e iden i ied: 1) Single
gas-su ace collision wi h mode a e ne ene gy exchange; 2) Mul iple gas-su ace collisions wi h
no adso p ion and delayed sca e ing; 3) Mul iple gas-su ace collisions wi h adso p ion o he
su ace and e en ual deso p ion in he sca e ed gas. The la e in e ac ion mode is ypical o
highly con amina ed su aces [104–106]: in hese condi ions, he adso bed pa icles ha e ime o
each equilib ium wi h he su ace and hey a e sca e ed acco ding o a cosine dis ibu ion wi h
θ and a Maxwellian ansla ional eloci y dis ibu ion co esponding o he wall empe a u e.
The majo i y o wo ks published in li e a u e, howe e , e e o he i s wo in e ac ion scena ios,
20
ypically obse ed in sca e ing om clean la su aces in Ul a High Vacuum (UHV) condi ions.
In his case, lobal e-emission dis ibu ions cha ac e ised by a p edominan asymme ical quasi-
specula componen a e obse ed o a ying su ace p ope ies, inciden pa icles ene gies and
wall empe a u e [93, 106–110, 110, 111]. The gene ally small amoun o pa icles ha unde go
mul iple collisions be o e being sca e ed in he gas-phase de e mines he wid h o he lobe o
dis ibu ion and seems o be suscep ible o he mo phology o he su ace conside ed, despi e he
le el o oughness. Wide angula dis ibu ions we e ob ained o smoo he su aces e en when
highe inciden ene gies we e employed [107]. This seems o sugges ha he in e ac ions wi hin
a oms in he same su ace laye o adjacen laye s may play a ole in de e mining he sca e ing
cha ac e is ic.
In he inelas ic sca e ing domain, howe e , di e en sca e ing beha iou s and hus di e en
ends a e expec ed acco ding o he inciden ene gy and he in e ac ion adius o he sys em
conside ed [112]. As highligh ed by Goodman in [97], o gas-su ace sys ems ha a e no
cha ac e is ed by a s ong pe iodici y in he in e ac ion po en ial, low alues o inciden kine ic
ene gy (Ei<kBTw) and la ge in e ac ion dis ances 1de ine he so-called he mal sca e ing
egime. In hese condi ions, he impinging gas-pa icles can no see he su ace co uga ion
and he su ace appea s la and smoo h. Du ing he in e ac ion, he angen ial componen o
momen um is gene ally conse ed and he sca e ing dynamics a e domina ed by he su ace
he mal mo ion in he di ec ion no mal o he su ace. The mal sca e ing s udies [106, 109, 113–
116] show he ollowing common ea u es o he angula dis ibu ions (Fig. 9):
1. ∂θ ,max/∂Tw≤0: he sca e ing angle co esponding o he peak in he dis ibu ion sligh ly
mo es owa ds he no mal o he su ace wi h inc easing Tw[117]. Mo eo e , a high
alues o Tw, o which he su ace appea s o be ee o adso ben s, he wid h o he
dis ibu ion expe iences a sligh inc ease wi h dec easing Ti/Tw[106, 113]. Highe Tw
also induces lowe sca e ing in ensi y a he peak o he dis ibu ion [113, 116]. Some
au ho s, howe e , ob ained he opposi e end o he sca e ing o A [118], Xe [116, 118]
and K [113, 118] on a ious me al su aces a Ti=Tamb. Gene ally speaking, high wall
empe a u es a e e icien in educing he s icking p obabili y and he ime equi ed o he
in e ac ion, hus p e en ing comple e accommoda ion;
2. ∂θ ,max/∂θi≥0: he sca e ing angle co esponding o he peak in he dis ibu ion mo es
owa ds he su ace angen as he incidence angle inc eases [81, 106, 114, 115]. Mo eo e ,
∂θ ,max/∂mg≤0, i.e. he sca e ing angle o which he dis ibu ion p esen s a peak mo es
owa ds he no mal o he su ace as he mass o he inciden gas a om inc eases [81, 106,
119];
As he inciden kine ic ene gy o he beam inc eases (wi hin he limi s o he he mal sca e -
ing domain), he e ec o su ace he mal ib a ion on he angula sca e ing dis ibu ion becomes
less dominan . As a consequence, he lobal dis ibu ion becomes na owe and mo e symme ical
in shape, he sca e ed in ensi y a he peak inc eases [113] and θ ,max gene ally mo es owa ds
he angen o he su ace [108, 112, 119, 120] (Figu e 10, op igh ). When his condi ion is
obse ed, he e-emission dis ibu ion is said o be supe specula . The e ec seems o be mo e
1The in e ac ion dis ance is de ined he e ollowing he de ini ion o he non-dimensional adius pa ame e Rp o ided
by Goodman in [97]. The dis ance o in e ac ion is hus de ined as he a io be ween wo quan i ies: he closes dis ance
ha sepa a es he cen e o he impinging a om om he cen e o he su ace a om du ing he collision and he c i ical
alue o his dis ance o he gas-su ace sys em conside ed. When he c i ical dis ance alue is achie ed he impinging
pa icle en e s he su ace.
21
Figu e 9: Va ia ion o θ ,max - he angle a which he peak o he dis ibu ion occu s - in he he mal sca e ing egime.
Beha iou s obse ed wi h inc easing Tw( op le ), θi( op igh ) and mg(bo om) a e illus a ed.
no iceable o nea ly g azing angles o incidence [108]. Wi h ega ds o ansla ional ene gy
dis ibu ions, when he conse a ion o he angen ial momen um is obse ed, he ela i e a io
be ween he mean inal and inciden ene gies a ies acco ding o he pa allel momen um conse -
a ion cu e:
E
Ei
=sin2θi
sin2θ
(38)
The cha ac e is ics men ioned abo e a e well desc ibed by he cube models [117] e iewed in
he p e ious sec ion o his pape , because o he inhe en assump ions on which hese models
a e buil . Be e ag eemen , as expec ed, is ound wi h he SC [89, 105, 106] a he han wi h
he HC model [108, 112, 113, 116, 117, 121], no only because o he mo e ealis ic gas-su ace
in e ac ion po en ial assumed, bu also because adjus men s can be made h ough he pa ame e
W o he sys em conside ed. When he a ac i e well Wdomina es he dynamics, he epulsi e
po en ial assump ion loses accu acy and he HC model ails in desc ibing apping and pa ial
accommoda ion o he su ace. The limi a ions imposed on he gas pa icle-su ace a om mass
a io in he HC model a e likely o make he model unsui able o add essing a omic oxygen
sca e ing om mos su aces. Mo eo e , compa ison o he po en iali ies o hese wo mod-
els agains expe imen al da a is possible jus in he inciden sca e ing plane. Fo ou o plane
sca e ing conside a ions, echniques add essing su ace co uga ion in mo e han one dimension
need o be employed. While Maxwell’s model ails in ep oducing he pe al-shaped angula dis-
ibu ion obse ed in hese expe imen s, i appea s ha i s heo e ical appa a us and simpli ied
assump ions a e su icien o desc ibe some e-emission pola plo s showing a small nea ly spec-
ula and a la ge di use componen [104, 107]. Resul s p o ided by Meh a e al. [108] show ha
when he CLL model is adop ed some di icul ies a e encoun e ed in he a emp o selec ing he
22
Figu e 10: T ansi ion om he mal o s uc u e sca e ing wi h inc easing alues o Ei.
p ope combina ion o αnand σ o ep oduce he expe imen al condi ions. Excellen p edic-
ions o he posi ion o he peak (θ ,max) and o he dispe sion o he sca e ing dis ibu ion, o
selec ed alues o he accommoda ion coe icien s, seem o exclude an accu a e ep esen a ion o
he expe imen al E /Ei, and ice e sa.
As he kine ic inciden ene gy o he beam inc eases [117, 122, 123] wi h ega ds o he su -
ace a oms he mal ene gy (Ei>kBTw) and he adius o in e ac ion educes, ansi ion o he
s uc u e sca e ing egime is expe ienced: in his case, he su ace oughness sensed by he im-
pinging pa icles is no iceable because o he inc eased powe o pene a ion. The in e ac ion is
no longe domina ed by he su ace he mal mo ion bu by he su ace co uga ion. Because o
he mul iple collissions expe ienced by he pa icles wi h he ough su ace, he angula dis ibu-
ion in his egime becomes wide in shape, he alue o he peak sca e ed in ensi y dec eases
and a shi om supe specula o di ec ions close o he specula ange o θ ,max is obse ed
[115, 124]. The quali a i e e-emission beha iou expec ed o inc easing alue o Eiin he an-
si ion om he he mal o he s uc u e egime a e ep oduced in Figu e 10. In con as o he
he mal egime, he ene gy o he sca e ed beam (E ) inc eases wi h inc easing θ [117, 125].
Cube models elying on he la su ace app oxima ion, a e unable o ep oduce his scena io
and ag eemen is a he ound wi h he Ha d Sphe es model [126], he Washboa d model [120],
and, in gene al, mo e complex models. Acco dance o he HC model wi h some gas-beam da a
e e ed o hype he mal Eiseems o ui ous [127] and, acco ding o he au ho s, a ibu able o
he mo phology o he gas-su ace sys em conside ed. An in e es ing sca e ing phenomenon
obse ed in he s uc u e egime is ainbow sca e ing: when his p ocess occu s he ypically
wide spa ial dis ibu ion is cha ac e ised by wo peak lobes co esponding o di e en in ensi ies
in he lux measu emen s. This phenomenon was howe e , p edomina ly obse ed on he sca -
e ing o a e gases om LiF su aces [128, 129], wi h some e idence on sys ems composed by
me al su aces [120, 130]. Mo e de ails on he opic can be ound in [97, 98]. The appea ance
o ainbow sca e ing e ec s seems o be associa ed wi h su ace co uga ion. In his ega d, he
Washboa d model o e s a ela i ely simple o mula ion able o cap u ing mo e complica ed e-
23
emission mechanisms o which he HC and SC models a e no sui ed. Mo eo e , he Washboa d
model appea s o be mo e e ec i e in desc ibing gas pa icle a ac ion and su ace pene a ion as
well as sca e ing cha ac e is ic om so e su aces [121], especially i compa ed wi h he HC
model. These ea u es a e pa icula ly use ul when s udying gas-su ace in e ac ion speci ically
o o bi al ae odynamic applica ions. The in e ac ion o mos ma e ials wi h he o bi al en i on-
men is likely o cause a ia ions in he s a ic gas-su ace po en ial co uga ion [57] and, mo e
in gene al, deg ada ion o ma e ial pe o mances wi h ime. The la ice s uc u e is subjec ed o
become oughe as AO exe s i s e osion ac ion on he am su aces. As a consequence, ma e ials
wi h p omising ae odynamic pe o mances (i.e. nea -specula e-emissions) migh expe ience
a ia ions in he expec ed sca e ing beha iou wi hin a mission li e ime. In his scena io, he
p inciples on which he Washboa d model is buil could p o e use ul o add ess ma e ials deg a-
da ion and pe o mance a ia ion.
5. Conclusions
Renewed in e es in small sa elli es missions in he lowe egion o he Ea h’s a mosphe e
demands new ae odynamic echnologies capable o aking ad an age o he en i onmen in LEO.
Ae odynamic pe o mances a e dependen on he mechanisms ha ule he gas-solid in e ac ion,
bu g ea unce ain ies a e associa ed wi h he physical p ocesses occu ing a he wall in a e ied
and ex emely a e ied egimes. Gas-beam expe imen al esul s sugges ha eali y migh be
mo e complex han ha desc ibed by classical heo e ical kine ic models. The de elopmen o a
new gene a ion o ae odynamic ma e ials may he e o e equi e mo e accu a e p edic ions. The
numbe o unce ain ies in he sys em beha iou e lec s a as p oduc ion o models in li e a u e:
di icul ies howe e a ise in he a emp o combining e iciency wi h simplici y. In he p e ious
sec ions popula models which ha e ob ained conside able success o hei immediacy and abil-
i y o p edic sca e ing beha iou s ha e been e iewed. These models we e b oadly associa ed
wi h wo p incipal amilies acco ding o some common ea u es. Sca e ing-ke nel heo y based
GSI models a e buil on a s a is ical app oach, while physical GSI models p o ide a simple ools
o desc ibe he complex physical in e ac ion mechanisms obse ed a he wall. Whe e e possi-
ble, hei app op ia eness has been discussed agains ele an gas-beam expe imen al esul s o
he p oblem conside ed and physical anges o applica ion ha e been iden i ied.
The joined e o o se e al au ho s has esul ed in ema kable imp o emen s in he unde -
s anding o he phenomena in ol ed in he obse ed e-emissions. The p oblem is howe e
complex and mul idisciplina y in na u e. Despi e he challenges ha emain, hey ep esen a
s a ing poin o u u e de elopmen s. A he momen o w i ing, i appea s ha an easy- o-
implemen model applicable o di e en sca e ing egimes and o di e en gas-solid sys ems is
s ill o be de ined. Simila ly, a simple analy ical model capable o a mo e accu a e quan i a i e
desc ip ion o he beha iou s o bo h clean and con amina ed su aces migh be desi able. In his
ega d, he le el o echnological ad ancemen achie ed by gas-beam acili ies seems adequa e
o suppo a mo e c i ical analysis o he models de eloped so a . A mo e sc upulous exami-
na ion o he app oxima ions on which hese ely may help in iden i ying he poin s o s eng h
o each model and possibly expand hei ange o applicabili y. The e migh also be a chance
o iden i y wi h mo e accu acy he e-emission pa e ns ha an imp o ed GSI model should be
able o desc ibe. A possible s a egy may include en iching he p oposed sca e ing-ke nels o
GSI wi h some mo e ealis ical assump ions ega ding adso p ion and deso p ion phenomena.
Fu he compa ison o heo e ical models add essing su ace co uga ion wi h a b oade ange
o expe imen al da a migh be help ul as well. Due o he wide amoun o esul s conce ning
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