P ecision measu emen s o he magne ic pa ame e s o LISA Pa h inde es masses
M A mano,1H Audley,2J Bai d,3P Bine uy,3, ∗M Bo n,2D Bo oluzzi,4E Cas elli,5A Ca alle i,6
A Cesa ini,7A M C uise,8K Danzmann,2M de Deus Sil a,9I Diepholz,2G Dixon,8R Dolesi,5L Fe aioli,10
V Fe oni,5E D Fi zsimons,11 M F eschi,9L Gesa,12, 13, ∗D Gia dini,10 F Gibe ,5, 14 R Gius e i,2
C G imani,7J G zymisch,1I Ha ison,15 M-S Ha ig,2G Heinzel,2M Hewi son,2D Holling on,16 D Hoyland,8
M Huelle ,5H Inchausp´e,3, 17 O Jenn ich,1P Je ze ,18 N Ka nesis,3B Kaune,2N Ko sako a,19
C J Killow,20 L Liu,5J A Lobo,12, 13, ∗J P L´opez-Za agoza,12, 13, †R Maa schalke wee d,15 D Mance,10
V Ma ´ın,12, 13 J Ma ino,3L Ma in-Polo,9F Ma in-Po que as,9N Meshksa ,10 P W McNama a,1
J Mendes,15 L Mendes,9M No a ias,12, 13, ‡S Paczkowski,2M Pe eu -Lloyd,20 A Pe i eau,3P Pi a o,5
E Plagnol,3J Ramos-Cas o,21, 13 J Reiche,2D I Robe son,20 F Ri as,12, 13 G Russano,5L Sala,5
D Se ano,12, 13, §J Slu sky,22 C F Sopue a,12, 13 T Sumne ,16 D Texie ,9J I Tho pe,22 D Ve ugno,5
S Vi ale,5G Wanne ,2H Wa d,20 P J Wass,16, 17 W J Webe ,5L Wissel,2A Wi chen,2and P Zwei el10
1Eu opean Space Technology Cen e, Eu opean Space Agency, Keple laan 1, 2200 AG Noo dwijk, The Ne he lands
2Albe -Eins ein-Ins i u , Max-Planck-Ins i u ¨u G a i a ionsphysik und
Leibniz Uni e si ¨a Hanno e , Callins aße 38, 30167 Hanno e , Ge many
3APC, Uni Pa is Dide o , CNRS/IN2P3, CEA/l u, Obs de Pa is, So bonne Pa is Ci ´e, F ance
4Depa men o Indus ial Enginee ing, Uni e si y o T en o, ia Somma i e 9,
38123 T en o, and T en o Ins i u e o Fundamen al Physics and Applica ion / INFN
5Dipa imen o di Fisica, Uni e si `a di T en o and T en o Ins i u e o
Fundamen al Physics and Applica ion / INFN, 38123 Po o, T en o, I aly
6Is i u o di Fo onica e Nano ecnologie, CNR-Fondazione B uno Kessle , I-38123 Po o, T en o, I aly
7DISPEA, Uni e si `a di U bino “Ca lo Bo”, Via S. Chia a, 27 61029 U bino/INFN, I aly
8The School o Physics and As onomy, Uni e si y o Bi mingham, Bi mingham, UK
9Eu opean Space As onomy Cen e, Eu opean Space Agency, Villanue a de la Ca˜nada, 28692 Mad id, Spain
10Ins i u ¨u Geophysik, ETH Z¨u ich, Sonneggs asse 5, CH-8092, Z¨u ich, Swi ze land
11The UK As onomy Technology Cen e, Royal Obse a o y, Edinbu gh, Black o d Hill, Edinbu gh, EH9 3HJ, UK
12Ins i u de Ci`encies de l’Espai (ICE, CSIC), Campus UAB, Ca e de Can Mag ans s/n, 08193 Ce danyola del Vall`es, Spain
13Ins i u d’Es udis Espacials de Ca alunya (IEEC), C/ G an Capi `a 2-4, 08034 Ba celona, Spain
14isa dSAT SL, Ma ie Cu ie 8-14, 08042 Ba celona, Ca alonia, Spain
15Eu opean Space Ope a ions Cen e, Eu opean Space Agency, 64293 Da ms ad , Ge many
16High Ene gy Physics G oup, Physics Depa men , Impe ial College
London, Blacke Labo a o y, P ince Conso Road, London, SW7 2BW, UK
17Depa men o Mechanical and Ae ospace Enginee ing, MAE-A, P.O.
Box 116250, Uni e si y o Flo ida, Gaines ille, Flo ida 32611, USA
18Physik Ins i u , Uni e si ¨a Z¨u ich, Win e hu e s asse 190, CH-8057 Z¨u ich, Swi ze land
19Obse a oi e de la Cˆo e d’Azu , Boule a d de l’Obse a oi e CS 34229 - F 06304 NICE, F ance
20SUPA, Ins i u e o G a i a ional Resea ch, School o Physics and As onomy, Uni e si y o Glasgow, Glasgow, G12 8QQ, UK
21Depa men d’Enginye ia Elec `onica, Uni e si a Poli `ecnica de Ca alunya, 08034 Ba celona, Spain
22G a i a ional As ophysics Lab, NASA Godda d Space Fligh Cen e , 8800 G eenbel Road, G eenbel , MD 20771 USA
A p ecise cha ac e iza ion o he magne ic p ope ies o LISA Pa h inde ee alling es -masses
is o special in e es o u u e g a i a ional wa e obse a o y in space. Magne ic o ces ha e an
impo an impac on he ins umen sensi i i y in he low equency egime below he millihe z. In
his pape we epo on he magne ic injec ion expe imen s pe o med h oughou LISA Pa h inde
ope a ions. We show how hese expe imen s allowed a high p ecision es ima e o he ins umen
magne ic pa ame e s. The emanen magne ic momen was ound o ha e a modulus o (0.245 ±
0.081) nAm2, he x-componen o he backg ound magne ic ield wi hin he es masses posi ion
was measu ed o be (414 ±74) nT and i s g adien had a alue o (−7.4±2.1) µT/m. Finally, we
also measu ed he es mass magne ic suscep ibili y a 5 mHz o be (−3.3723 ±0.0069)×10−5. All
esul s a e in ag eemen wi h on-g ound es ima es.
∗Deceased
†jplop[email p o ec ed]
‡[email p o ec ed]
§[email p o ec ed]
I. INTRODUCTION
LISA Pa h inde (LPF) [1, 2] was an ESA mission de-
signed as a echnology demons a o o he u u e g a -
i a ional wa e obse a o y in space, LISA [3]. The main
goal o he mission was o demons a e key echnologies
equi ed o de ec g a i a ional wa es in space. In o -
de o do so, he ins umen on-boa d had o demon-
a Xi :2407.04431 2 [as o-ph.IM] 5 No 2024
2
s a e a ela i e accele a ion noise be ween i s wo es
masses (TMs) in nominal geodesic mo ion a a le el o
3×10−14 ms−2Hz−1/2a 1 mHz, a le el o p ecision im-
possible o achie e wi h on g ound g a i a ional wa e
de ec o s.
LPF launched on Decembe 3 d, 2015 and s a ed i s
scien i ic ope a ions on he Ma ch 1s , 2016 a e each-
ing he Lag ange poin L1 o he Ea h-Sun sys em. The
mission was di ided in o wo di e en expe imen s on-
boa d, he Eu opean Space Agency LISA Technology
Package (LTP) and he NASA Dis u bance Reduc ion
Sys em (DRS). A e se en een mon hs o scien i ic op-
e a ions, he mission success ully demons a ed i s main
scien i ic goal, su passing i s equi emen s and achie ing
a le el o accele a ion noise below he LISA equi emen s
in i s en i e measu emen equency band [4, 5].
As impo an as achie ing his demanding le el o
geodesic ee all was he de elopmen o an unde s and-
ing all he di e en con ibu ions ha build he noise
model o he ins umen . Se e al expe imen s we e
planned du ing he LPF ope a ions in o de o isola e
and e alua e he mos impo an con ibu ions o he
accele a ion noise budge . Wi h ha objec i e, LISA
Pa h inde ca ied he Da a and Diagnos ics Subsys em
(DDS), which included a empe a u e measu emen sub-
sys em [6, 7], a magne ic diagnos ic subsys em [8, 9] and
a adia ion moni o [10–13].
In his wo k we will ocus on he esul s o he magne ic
diagnos ics and, speci ically, on he expe imen s un o
cha ac e ize he magne ic pa ame e s o he es masses
on-boa d LPF. P ecise knowledge o such alues is c ucial
o he u u e space-bo ne g a i a ional wa e obse a o-
ies, since any magne ic pe u ba ion can ha e a po en ial
impac on he ins umen pe o mance h ough magne ic
pa asi ic o ces.
This wo k is o ganized as ollows. In sec ion II we
desc ibe he magne ic diagnos ic sys em on-boa d de-
signed o s udy and disen angle he na u e o he mag-
ne ic o ces, in oduced in sec ion III, ha can pe u b
he es mass mo ion. In sec ion IV we desc ibe he
in- ligh magne ic expe imen s pe o med o ex ac he
TMs magne ic pa ame e s and we p esen ou conclu-
sions in sec ion V.
II. EXPERIMENTAL SETUP
A. The magne ic diagnos ics subsys em
The magne ic diagnos ics subsys em on-boa d LISA
Pa h inde was esponsible o moni o ing he magne ic
en i onmen and c ea ing con olled magne ic ields o
pe u b he es mass mo ion in o de o p ope ly cha -
ac e ize he con ibu ion o magne ic o ces o he o al
ins umen noise budge . To achie e hese goals, he sub-
sys em was composed by ou iaxial magne ome e s and
wo induc ion coils.
FIG. 1. Coo dina e e e ence sys em o he coil and he es
mass. The con en ion used o he h ee angles o o a ions
along each es mass axis is also shown.
The coils—see Fig. 1—we e able o p oduce a con-
olled magne ic ield a he TM loca ions as well as in
he magne ome e s closes o hem. Bo h ci cula in-
duc ion coils, wi h an a e age adius o 56.5 mm, we e
loca ed 85.5 mm away om he es masses and hey
we e a ached o he ex e nal wall o each acuum enclo-
su e. The wi e winding used o build up he coils we e
made o a Ti anium alloy (Ti6Al4V) and loop a ound he
s uc u e o a o al o 2400 u ns. The cen e s o bo h
coils we e aligned wi h he axis, x, joining bo h TMs cen-
e s so ha he induced magne ic ield had axial symme-
y. The ou magne ome e s we e aligned by pai s in he
x−yplane o he TMs in o de o be able o measu e
g adien s wi hin he spacec a in bo h he xand ydi ec-
ions. Magne ic ield g adien s along he zdi ec ion could
no be measu ed. The magne ome e s con inuously mea-
su ed he e olu ion o he on-boa d magne ic ield wi h a
p ecision o 10 nT Hz−1/2. Each luxga e magne ome e
measu emen axis consis ed o a sensing coil su ound-
ing a second inne d i e coil a ound a high pe meabili y
magne ic co e ma e ial. This mean ha all ou magne-
ome e s con ained ac i e magne ic senso s ha had o
be loca ed a enough om he es masses o hem no
o con ibu e as a sou ce o magne ic pa asi ic o ces.
B. Magne ic en i onmen on-boa d
The backg ound magne ic ield measu ed on-boa d
was comple ely domina ed by he con ibu ion om he
elec onics o he spacec a uni s. Among hem, he
h us e sys ems we e a majo con ibu o , bo h he
cold gas high p essu e la ch al es ( he ones used by
ESA) and he colloidal h us e s ( he ones ope a ed by
NASA). Cold gas h us e s o , mo e p ecisely, some pe -
manen magne s in he cold gas h us e subsys em, con-
ibu ed wi h oughly he 80% o he measu ed magne ic
ield. Al hough a s ong con ibu ion, his emained con-
s an h oughou he mission—pa ially hanks o he
high he mal s abili y eached on-boa d [7]. This was
3
no he case o he colloidal h us e s, whe e a pe -
sis en slow d i o a ound 150 nT in he span o 100
days was obse ed [9]. The main con ibu ion o he
magne ic-induced o ce noise is below he millihe z [14].
In his equency egime, magne ic ield luc ua ions a e
domina ed by he in e plane a y magne ic ield con i-
bu ion, which can show an impo an non-s a iona y
componen associa ed wi h changes in he in e plane-
a y plasma. Fo ins ance, a ia ions in he ange o
300−500 km s−1in he sola wind eloci y we e ound o
be co ela ed o a ia ions in he magne ic ields ampli-
ude spec al densi y in he ange 20 −50 µHz o a ound
170 −750 nT Hz−1/2[9].
In wha e e s o ou analysis in he ollowing, we
assume ha in all he in- ligh expe imen s whe e we
induced magne ic ields wi h he coils, he backg ound
magne ic ield (ei he gene a ed by he spacec a o
due o he in e plane a y con ibu ion) can be sa ely ne-
glec ed as i was a leas one o de o magni ude smalle
han he ones induced by he coils. The same is ue o
he g adien s o he magne ic ields.
C. Fo ces on-boa d LPF
We e alua e he induced o ce in he es mass h ough
he ∆g a iable— he p incipal scien i ic ou pu o he
mission—nominally de ined as he di e en ial accele a-
ion be ween he wo TMs in hei nominal posi ion [4].
Since he main objec i e o he ∆gmeasu emen is he
e alua ion o he ee all o he es masses, hose o ces
a ising due o he spacec a dynamics con ol loop o
o he o ces caused by spacec a non-ine ial e e ence
ame a e sub ac ed in he de ini ion o his pa ame-
e [15]. We will assume ha du ing magne ic injec ions
he dominan o ces ha TMs will eel will be pu ely
o magne ic o igin. ∆g, by cons uc ion, is he di e -
ence in accele a ion be ween TMs along he axis joining
hem, he same as he spacec a xaxis by de ini ion and
aligned wi h he xaxis om Fig. 1,
∆g=F2,x
MTM1 −F1,x
MTM2
,(1)
whe e MTM is he mass o each TM and Fis he o al
o ce each one eels. When calcula ing he o ce measu ed
by each TM du ing magne ic injec ions, we assume ha
he TM u hes om he injec ing coil will eel a neg-
ligible o ce when compa ed o he one nea es o he
coil. So, when injec ing magne ic signals wi h he coils,
we will es ima e he magne ic o ce ha he nea es TM
eels as Fx≡MTM∆g.
The same is ue o he o que such ha Nη,ϕ ≡
I∆gη,ϕ, whe e Iis he momen o ine ia o a cube and
∆gη,ϕ a e he angula accele a ions measu ed by he in-
e e ome ic sys em. Only o a ions along he yand z
axes, ηand ϕ espec i ely in Fig 1, a e ob ainable since
he ins umen is no sensi i e o o a ions a ound he x
axis as i is aligned wi h he lase beam.
III. MAGNETIC-INDUCED FORCES AND
TORQUES IN A FREE FALLING TEST MASS
Magne ic luc ua ions can couple in o he dynamics o
he ee alling es masses on-boa d he sa elli e. In he
ollowing we de elop he basic equa ions needed o de-
sc ibe he expe imen s ca ied ou wi h he induc ion
coils in LISA Pa h inde .
A. A magne ic dipole in a su ounding magne ic
ield
In a i s app oxima ion, he ee- alling es masses
inside LISA Pa h inde can be conside ed as a magne ic
dipole wi h o al magne ic momen densi y minside o a
su ounding magne ic ield B. Pa ame e s in bold e e
o ec o s. The dipole would he e o e eel an associa ed
o ce and o que gi en by
F=⟨(m·∇)B⟩V,(2a)
N=⟨m×B+ ×(m·∇)B⟩V,(2b)
whe e deno es he dis ance o he TM wi h espec o
he coil. We use he con en ion ⟨. . . ⟩ ≡ 1
VRV(. . .)d3x o
deno e he a e age o he enclosed quan i y o e he TM
olume, V. The o al magne ic momen densi y mis he
sum o wo componen s: he emanen magne ic momen
densi y m which depends on he ma e ial and manu ac-
u ing p ocess and he induced magne ic momen densi y,
mi. The induced magne ic dipole densi y o any ma e ial
is p opo ional o he applied magne ic ield, i.e.
mi=χ/µoB,(3)
whe e χis he magne ic suscep ibili y and µ0is he
acuum pe meabili y. The es mass composi ion is
73% gold, diamagne ic, and 27% pla inum, pa amagne ic.
Gi en ha he dominan ma e ial is gold we should ex-
pec he TM o beha e like any diamagne ic ma e ial,
opposing he ex e nal magne ic ield hus, ha ing a neg-
a i e magne ic suscep ibili y. Despi e his, we lea e he
sign unde e mined in he ollowing de i a ion. Also, we
ha e implici ly assumed he e an iso opic es mass which
allows he use o a scala suscep ibili y in he p e ious
equa ion. Nex , we assume he es mass magne ized by
a slowly oscilla ing ield
B( ) = BAC sin(ω ),(4)
whe e BAC is he ampli ude o he oscilla ing magne ic
ield and ωi s angula equency, we will ob ain a mag-
ne iza ion ha a ies wi h ime acco dingly, mi( ). In
diamagne ic, pa amagne ic and many e omagne ic ma-
e ials, he magne iza ion also a ies sinusoidally and in
phase wi h he applied magne ic ield wi h a cons an a-
io gi en by he magne ic suscep ibili y. Howe e , some
e omagne ic ma e ials show a delayed esponse ha is
4
no in phase wi h he applied ield. This phenomena is
ypically desc ibed by conside ing he in-phase o eal,
χ , and he ou -o -phase o imagina y, χi, componen s
o he magne ic suscep ibili y such ha χ=χ +iχi,
whe e i=√−1. Fo he case o LPF expe imen s he
mos ele an physical mechanism in ol ed in he imag-
ina y suscep ibili y a e eddy cu en s since his con i-
bu ion becomes inc easingly impo an wi h inc easing
conduc i i y o he ma e ial. Howe e , o mos o he
expe imen s in he low equency egime his componen
is expec ed o be o de s o magni ude smalle han he
eal pa hus, i s con ibu ion can be neglec ed. In he
ollowing, he e o e, we e e o he magne ic suscep i-
bili y χas equi alen o i s eal componen χ , unless
explici ly s a ed o he wise.
Conside ing bo h con ibu ions o he magne iza ion
( he emanen and he induced magne ic momen s) we
expand Eqs. (2) in o
F=(m ·∇)B+χ
µ0
[(B·∇)B]V,(5a)
N=m ×B+ ×(m ·∇)B+χ
µ0
(B·∇)BV.
(5b)
In o de o desc ibe ou expe imen s in he ollowing
sec ions, we need o de elop he equa ions u he . Fi s ,
we will conside he magne ic ield as composed by an ap-
plied, oscilla ing magne ic ield BAC , and a s able mag-
ne ic ield B0, di ided in o an applied ime independen
DC magne ic ield BDC and some en i onmen al back-
g ound Bback.
B=B0+BAC sin(ω )
=Bback. +BDC +BAC sin(ω ).(6)
By subs i u ing in Eq. (5a) and ac o ing ou he compo-
nen s in e ms o hei equency esponse o he inpu
signal, we ind ha he o ce can be di ided in o h ee
componen s: a cons an DC e m, a e m ha oscilla es
a he same equency o he induced magne ic ield 1 ω
and a e m oscilla ing a wice he equency 2 ω
F=FDC +F1ω+F2ω,(7)
wi h
FDC ="⟨(M ·∇)B0⟩
+χV
µ0⟨(B0·∇)B0⟩+1
2(BAC ·∇)BAC#,
(8a)
F1ω="(M ·∇)BAC
+χV
µ0(B0·∇)BAC+BAC ·∇B0#
×sin(ω ),
(8b)
F2ω=−χV
2µ0BAC ·∇BACcos(2 ω ),(8c)
whe e M =m V is he emanen magne ic momen
and we ha e assumed homogenei y and s a iona i y o
he es mass p ope ies. Conside ing ha he ela i e
accele a ion measu emen s in LISA Pa h inde a e in he
xdi ec ion, he only componen o he o ce om Eq. 7
ha will be needed is i s xcomponen . Analogously, i
we manipula e he o que equa ions a simila esul wi h
he h ee e ms be o e men ioned should appea .
B. Es ima e o es mass magne ic pa ame e s
The e alua ion o bo h he o ce and o que exp es-
sions, Eqs. (5a) and (5b) espec i ely, implies he calcu-
la ion o he a e age o an ex e nal magne ic ield and i s
g adien wi hin he TMs olume as exp essed by ⟨. . . ⟩.
Making use o he induc ion coils om Fig. 1 we can
con ol he injec ed ield (BAC,DC and ∇BAC,DC) as i
can be calcula ed by means o Amp`e e’s induc ion laws
unde he assump ions o coils wi h negligible hickness
and a wi e winding o N u ns. Thanks o he symme y
o ou sys em, only he xcomponen s o he a e aged
induced ields a e non-ze o, BAC, DC
x, and hei g adi-
en s along he yand zaxes a e 3 o de s o magni ude
smalle han ∇xBAC, DC
x. Fu he mo e, he magne ic
ield in he xdi ec ion and i s g adien along xa e p o-
po ional o one ano he a any gi en poin in space,
ha is: BAC, DC
x=κ∇xBAC, DC
x. This ac o con-
s an κonly depends on he coil dimensions and he dis-
ance om he coil cen e . I s alue can be ound an-
aly ically o he simple on-axis magne ic ield o a coil
bu i is ha de o ob ain o he gene al o -axis mag-
ne ic ield o mula in ol ing ellip ic in eg als. Thus, i s
alue was calcula ed nume ically o be κ=−0.04487 m
o ou pa icula con igu a ion, wi h negligible unce -
ain y o igina ed only due o nume ical e o . We e e
5
he in e es ed eade o Appendix A o mo e de ail on
he calcula ions in ol ed a he TMs loca ion. Finally,
he magne ic o ce is ob ained by applying a he e odyne
demodula ion a he di e en equencies o in e es o
he on-boa d measu emen s o he s ay TM o ce (mo e
de ail on his in he upcoming sec ion) esul ing in he
es ima o s ˆ
FDC,x ,ˆ
F1ω,x and ˆ
F2ω,x. We now p oceed
o desc ibe how we will es ima e he es mass magne ic
pa ame e s om he p e ious gene ic exp essions.
a. Magne ic suscep ibili y The coupling be ween an
induced magne ic ield and i s g adien wi h he mag-
ne ic suscep ibili y o he es mass is esponsible o he
appea ance o a o ce componen a wice he injec ed
modula ion equency in Eq. (8c). Ou analysis can ake
ad an age o his by ex ac ing he signal a 2ω om he
measu ed es mass o ce, i.e.
χ2ω=−2µ0
V
ˆ
F2ω,x
⟨BAC
x⟩·⟨∇xBAC
x⟩.(9)
This equa ion p o ides a di ec es ima e o he es mass
suscep ibili y decoupled om any o he o he magne ic
pa ame e s. The no a ion in Eq. (9) shows explici ly ha
he es ima e o he suscep ibili y is ob ained a wice he
injec ed equency by demodula ing he encoded in o -
ma ion in ∆gand compa ing i wi h he p edic ed TM
a e age magne ic ield and g adien .
We no ice ha , in p inciple, we could use he signal
a 2ω o ob ain bo h eal and imagina y con ibu ions o
he magne ic suscep ibili y. To es ima e he imagina y
con ibu ion we would need o look o a 2ωcon ibu ion
wi h a π/2 phase shi wi h espec he o iginal injec ion.
We will explo e his in he discussion o ou esul s in
Sec ion IV.
b. Remanen magne ic momen The componen o
he o ce a he injec ion equency, F1ω,x, depends on all
he magne ic pa ame e s o he TM. Taking ad an age o
he ac ha all e ms in Eq. (8b) depend on BAC
xo
∇xBAC
x, we can ew i e he exp ession as ollows
ˆ
F1ω,x =M ,x +χV
µ0
(B0,x +κ∇xB0,x)∇xBAC
x.
(10)
The e m in b acke s can be ela ed o an e ec i e mag-
ne ic momen such ha
Me ,x =M ,x +χV
µ0"Bback.,x +BDC
x
+κ∇xBback.,x +BDC
x#,
(11)
whe e we ha e expanded B0,x as explained in Eq. (6).
Bback.,x can be conside ed negligible compa ed o he in-
jec ed magne ic ield BDC
xas i s alue is expec ed o
be an o de o magni ude smalle han he injec ed ields
h ough he coils. Thus, we ha e
Me ,x ≃M ,x +2χV
µ0BDC
x.(12)
I he only a iable in Eq. (12) is BDC
x, hen plo ing
Eq. (10), we will ob ain a s aigh line wi h an o se ha
co esponds o he emanen magne ic momen M ,x and
a slope ha is p opo ional o he magne ic suscep ibili y
χa 1ω. Fu he mo e, when we induce a magne ic ield in
he TM posi ion, using he coils, apa om di ec o ces
in he xdi ec ion, we a e also gene a ing o ques, as de-
sc ibed in Eq. (5b). Due o he symme y o he sys em
he e m in ol ing he c oss p oduc wi h will in eg a e
o ze o acc oss he TM olume due o he alignmen be-
ween he coil axis and he TMs cen e esul ing in only
wo componen s, see Appendix B o de ails, whe e only
he 1ω e m will be o in e es leading o he ollowing
equa ions
ˆ
Nϕ,1ω=−M ,y BAC
x;ˆ
Nη,1ω=M ,z BAC
x.(13)
We can conclude ha he 1ωoscilla ion o he o que
in ϕis di ec ly ela ed o he emanen magne ic momen
along y,M ,y, while he 1ωoscilla ion o he o que in
ηis di ec ly ela ed o he emanen magne ic momen
along z,M ,z. The e o e, by demodula ing he o que a
1ω o ηand ϕand conside ing he alues o he injec ed
magne ic ield BAC
x, we will be able o de e mine M ,y
and M ,z.
c. Backg ound es ima es Simila ly o he 1ω e m,
in Eq. (8a), he exp ession o he DC o ce componen
o he signal in ol es again all he unknown pa ame e s.
I we g oup all he e ms o Eq. (8a) as a unc ion o
BDC
x, we can ew i e i as
ˆ
FDC,x ≃χV
µ0κBDC
x2
+M ,x
κ+χV
µ0∇xBback.,x +Bback.,x
κBDC
x
+(M+∇xBback.,x +χV
µ0"3Bback.,x∇xBback.,x
+1
2BAC
x∇xBAC
x#),
(14)
whe e M+=M ,x +M ,y +M ,z. We ha e made he as-
sump ion ha he backg ound magne ic ield is he same
in all di ec ions, Bback.,x ≃Bback.,y ≃Bback.,z, because
we don’ ha e any a p io i in o ma ion abou i s alue.
We also assume ha he h ee componen s o ∇Bback.,x
a e equal: ∇xBback.,x ≃ ∇yBback.,x ≃ ∇zBback.,x, which
is a wo s case scena io since all componen s con ibu ing
6
o he backg ound g adien would add up when in eali y
hey could pa ially cancel each o he . I he only a i-
able is BDC
x, we can obse e ha ˆ
FDC,x has a quad a ic
o m.
IV. IN-FLIGHT EXPERIMENTAL CAMPAIGN
Soon a e LPF s a ed scien i ic ope a ions, on Ma ch
1s , 2016, magne ic expe imen s we e scheduled o ex-
ac he magne ic pa ame e s ela ed o he TMs. The
expe imen s consis ed in applying an elec ic cu en
h ough he coils o induce a magne ic ield in he po-
si ion o he TMs. The applied cu en in he coils was
a sinusoidal signal I( ) = IDC +IAC sin(ω ), whe e IDC
was a cons an o se , IAC he ampli ude o he sinusoidal
signal and ωi s angula equency. The cu en induces a
magne ic ield in he su oundings o he coil o he same
ype B( ) = BDC +BAC sin(ω ).
A he beginning o he commissioning pe iod, all sub-
sys ems wen h ough an ini ial checkou p ocedu e. In
his ini ial phase, coil #2 — he one closes o TM2—
showed a mal unc ion. Due o his ac , he injec ions
pe o med du ing he ope a ions pe iod we e on coil #1,
and he only se o injec ions pe o med in coil #2 we e
done wi h low cu en s o p e en any possible cu en
leak o o he sys ems. This esul ed in a educ ion o
he p ecision achie able wi h coil #2 expe imen s. Thus,
mos o he esul s ha will be shown he e will be o
TM1 i no speci ied o he wise.
We ca ied ou a o al o h ee se s o magne ics in-
jec ions. The i s se was pe o med on he Ap il 28 h
and 29 h, 2016. The injec ions o Ap il 28 h consis ed o
applying a sinusoidal signal h ough he coil #1 wi h di -
e en DC o se s and di e en AC ampli udes. The injec-
ions o Ap il 29 h we e exac ly he same bu h ough coil
#2. The second se o injec ions we e done on June 18 h,
2016. They consis ed o a se ies o sinusoidal injec ions a
a wide ange o bo h DC o se s and AC ampli udes han
he p e ious ones bu pe o med exclusi ely in coil #1.
The hi d, and las , se o injec ions we e pe o med on
Ma ch 14 h, 15 h and 16 h , 2017. They consis ed o e y
long-las ing signals a high DC o se s and wi h small
AC ampli udes applied exclusi ely h ough coil #1. The
comple e lis o expe imen s is shown in Appendix C.
A ypical un o magne ic expe imen s is shown in
Fig. 2, whe e we show he esul s om he hi d se o in-
jec ions du ing June 18 h. The applied cu en s h ough
he coil a 5 mHz can be seen in he Appendix wi hin
Table VI. The h ee panels display he main a iables
o in e es in ou analysis, hese a e he magne ic ield in
he x di ec ion, as measu ed by he closes magne ome e
o each coil, he accele a ion p oduced be ween he TMs
due o he p esence o hese injec ions and he o que
being induced be ween bo h TMs along he yaxis.
The es ima ion o magne ic o ces expe ienced by he
es masses du ing he magne ic injec ions is pe o med
by demodula ing he ∆gmeasu emen s. Te ms ˆ
FDC,x,
FIG. 2. Expe imen s wi h coil #1, June 18 h, 2016. Top:
Bxas measu ed in he magne ome e closes o he coil, PX.
Middle: ∆g.Bo om: Angula accele a ion along he o a ion
angle η.
ˆ
F1ω,x and ˆ
F2ω,x p e iously de ined in Sec ion III B a e
es ima ed by applying a he e odyne demodula ion a he
co esponding equencies and escaling he ampli udes
ob ained by means o he mass o he TMs, MTM =
(1.928 ±0.001) kg. Analogously, he same p ocedu e
can be ex apola ed o he o que by using he momen
o ine ia o a cube wi h he side leng h o he TMs,
lTM = (46.000 ±0.005) mm.
Since he magne ic ield can no be di ec ly measu ed
in he es mass posi ion we mus e e o he magne ome-
e s ead-ou o calib a ion. We measu ed he ampli ude
in he +xmagne ome e o each o he 20 injec ions o
June 18 h, 2016 o coil #1. To do so we demodula e
he ead-ou a he injec ion equency. By compa ing
hese ampli udes o he ones p edic ed by Amp`e e’s law
wi h o igin a he coil loca ion we ound a sys ema ic
disc epancy o 11.85 ±0.45%, wi h he p edic ed mag-
ne ic ield being la ge han he measu emen s om he
7
FIG. 3. ˆ
F1ω,x on TM1 as a unc ion o he applied AC mag-
ne ic ield g adien . The di e en colo s co espond o ixed
DC alues o he injec ed signal.
magne ome e s. This sys ema ic disc epancy can ha e
many o igins. Fo he magne ic ield p edic ion, we a e
assuming a pe ec alignmen o bo h he coil and mag-
ne ome e , as well as no il s. In eali y, du ing bo h
manu ac u ing and in eg a ion o he magne ic diagnos-
ics i ems (coils and magne ome e s) he e a e una oid-
able mechanical ole ances. Fu he mo e, he launch i -
sel adds mo e unce ain y on he ela i e dis ance and
il s be ween he diagnos ics i ems, being his a sys em-
a ic e o ha is di icul o quan i y a p io i. All he
e ec s combined can add up o he alue epo ed. Since
he magne ome e accu acy is 0.5%, we assume ha he
misma ch o igina es in he gene a ion o he magne ic
ield by he coils and we apply his co ec ion o all he
calcula ed magne ic ield alues ha in e ene in all he
equa ions de i ed in he p e ious sec ion, ede ining hem
o simplici y, i.e. BAC
x≡0.8815 BAC
x.
A. Remanen magne ic momen
The es ima e o he emanen magne ic momen is ob-
ained h ough he dependence wi h he 1ωcomponen o
he o ce exp essed in Eq. (10) oge he wi h he app ox-
ima ion o Eq. (11), ha we ecall he e o con enience:
ˆ
F1ω,x =M ,x +2χV
µ0
BDC
x∇xBAC
x.
In o de o e alua e his e m, du ing he June 18 h un
se e al injec ions wi h di e en AC ield g adien s we e
applied o he TMs. By doing so we can e alua e he
e m in b acke s abo e a di e en alues o he g adien
∇xBAC
x. This is shown in Fig. 3 whe e we display how
he 1ωcomponen o he o ce changes depending on he
in ensi y o he injec ed AC magne ic ield g adien , o
ixed DC o se s. Each esul in he plo ep esen s an
FIG. 4. E ec i e emanen magne ic momen plo ed as a
unc ion o he injec ed DC magne ic ield. The esul o
he i , o an equa ion o he ype y = mx + n, is m =
(−3.380±0.027) ×10−5and n = (0.140 ±0.138) ×10−9Am2.
injec ion a 5 mHz wi h AC ampli udes applied o he
coil o IAC = 0.5,0.8,1.0,1.5 mA.
As explained in Sec ion III B, by unning he expe -
imen a di e en DC le els we can u he disen angle
he dependencies o he pa ame e s inside he b acke s
and ob ain he es ima e o he emanen magne ic mo-
men . Table I ga he s he linea i s o he esul s ha
we also show in Fig. 4. The o se pa ame e co esponds
o he emanen magne ic momen o es mass #1 in he
xdi ec ion, M ,x = 0.140 ±0.138 nAm2. The slope pa-
ame e is di ec ly ela ed o he magne ic suscep ibili y
a 1ω. Howe e , he alues ha we e used o DC o se s
o ±0.1,0.2 mA came om he injec ions o Ap il 28 h
which we e pe o med a di e en equencies (3 mHz),
han he es a 5 mHz. Thus, o ob ain he alue o he
suscep ibili y a 5 mHz, we will use he i om Fig. 4,
bu wi h only he DC alues o ±0.75,1.5 mA gi ing a
esul o χ5mHz = (−3.3723 ±0.0069) ×10−5.
Analogously, o ob ain M ,y and M ,z, we demodu-
TABLE I. Coe icien s o he i ed lines o Fig. 3 o he ype
y=Ax +B. The e o s o he i s o IDC =±0.1,0.2 a e 0
because he e we e only wo poin s o i he line.
IDC [mA] A [nAm2] B [ N]
1.50 (−31.24 ±0.25) (−31 ±23)
0.75 (−15.48 ±0.15) (−13 ±13)
0.20 (−4.39 ±0) (−23 ±0)
0.10 (−2.04 ±0) (−14 ±0)
-0.10 (2.60 ±0) (−1.3±0)
-0.20 (5.00 ±0) (13 ±0)
-0.75 (15.58 ±0.21 (−12 ±19)
-1.50 (31.09 ±0.23) (−1.3±21)
8
la e he ampli udes o he o que measu emen s a ound
he equi ed angles and apply Eq. (13) o he espec-
i e injec ed magne ic ields. This way, we ob ained
M ,y = 0.178 ±0.025 nAm2and M ,z = 0.095 ±0.010
nAm2which we will need o he backg ound es ima ions.
No e he la ge unce ain y ob ained o M ,x compa ed
wi h M ,y and M ,z. The di e ence is due o he di e en
me hods used o ex ac hem. The o me is calcula ed
indi ec ly as he o se o he linea i o he slopes o he
o ce a 1ωwhile he o he wo componen s o he ema-
nen magne ic momen a e calcula ed di ec ly om he
measu ed o ques along ηand ϕ. Ro a ions along θwould
ha e allowed a mo e p ecise alue o M ,x bu he in e -
e ome ic sys em is no sensi i e along such axis as i is
aligned wi h he lase beam. The esul s lead o a o al
emanen magne ic momen o : |M |= (0.245 ±0.081)
nAm2.
B. Backg ound magne ic ield
The induc ion o o ces in he es mass by means o
he con olled injec ion o magne ic ields allow he de e -
mina ion no only o he es mass magne ic pa ame e s
bu also o en i onmen pa ame e s ha con ibu e o
he magne ic o ce, which is he case o he backg ound
magne ic ield in he es mass posi ion. We emphasize
he e ha his is he way o es ima e o his pa ame e
since he magne ome e s a e loca ed oo a away om
he es masses o gua an ee a p ecise es ima e o his
a iable.
In o de o do so we e alua e Eqs. (14) using he in-
jec ions o June 18 h, 2016. We exp ess he in o ma ion
p o ided by hese uns by displaying he DC componen
o he measu ed o ce as a unc ion o he DC componen
o he applied magne ic ields. As p edic ed, we ob ain
he pa abolas in Fig. 5 o di e en alues o he AC am-
pli ude om whe e we de i e he pa abola coe icien s o
Table II.
As de i ed om Eq. (14), he Acoe icien p o ides a
di ec es ima e o he magne ic suscep ibili y
A=χV
µ0κ(15)
which, in con as wi h o he al e na i e es ima es, is no
dependen o he injec ion modula ion equency. The
TABLE II. Coe icien s o he i ed pa abolas o Fig. 5 o he
ype y=A x2+B x +C.
IAC[mA] A [N/T2](10−2) B [N/T](10−8) C [N](10−12)
1.5 (5.99 ±0.11) (−0.67 ±0.39) (1.047 ±0.025)
1.0 (6.003 ±0.025) (0.265 ±0.089) (0.4948 ±0.0058)
0.8 (5.64 ±0.16) (−1.73 ±0.58) (0.323 ±0.038)
0.5 (5.50 ±0.22) (−2.46 ±0.79) (0.144 ±0.052)
FIG. 5. ˆ
FDC,x on TM1 as a unc ion o he injec ed DC mag-
ne ic ield oge he wi h hei espec i e i s o an equa ion o
he ype y = Ax2+Bx +C. The di e en colo s co espond
o ixed AC alues o he injec ed signal.
alue ob ained using Eq. (15) is χDC = (−3.35 ±0.15) ×
10−5.
Wi h wo emaining e ms, Band C, we can build a
sys em o equa ions, being he wo unknowns he pa am-
e e s ha de ine he backg ound magne ic ield in he
es mass posi ion, i.e. Bback.,x and ∇xBback.,x
C=M+∇xBback.,x +χV
µ0"3Bback.,x∇xBback.,x
+1
2BAC
x∇xBAC
x#,
B=M ,x
κ+χV
µ0∇xBback.,x +Bback.,x
κ.(16)
Sol ing his quad a ic sys em o equa ions, we ob ain
an exp ession o he backg ound magne ic ield and i s
g adien in he xdi ec ion a he loca ion o he TMs
ha we can e alua e o each o he ou i alues.
F om he wo ma hema ically a ailable solu ions we se-
lec he one close o he es ima es o he magne ic ield
and ield g adien ob ained du ing on-g ound cha ac e -
iza ion o he spacec a , which we e 267 nT and -7575
nT/m, espec i ely [16]. The alues ha we ob ain o
he in- ligh es ima es a e Bback.,x = 414 ±74nT and
∇xBback.,x =−7400 ±2100nT/m.
C. Magne ic Suscep ibili y
We ha e al eady es ima ed he es mass magne ic
suscep ibili y as a by-p oduc o he es ima e o he e-
manen magne ic momen and he backg ound magne ic
ield and ield g adien .
9
FIG. 6. Magne ic suscep ibili y o bo h es masses a di e -
en equencies oge he wi h he model p edic ed by Eq. (17).
The DC alue o he suscep ibili y, ha is he e-
quency independen pa , was ob ained in sec ion IV B
and he alue o he suscep ibili y a 5 mHz was mea-
su ed in sec ion IV A. The es o he measu emen s o he
magne ic suscep ibili y o he TMs ha e been ob ained
using he 2ωcomponen o he o ce and Eq. (9). These
alues co espond o wice he equency a which he in-
jec ion was pe o med. The esul s o he suscep ibili ies
can be seen in Table III wi h he alue o he equen-
cies a which hey co espond. Using all he injec ions,
om he h ee di e en magne ic expe imen s uns, he
equencies ha could be calcula ed o he magne ic sus-
cep ibili y we e 2, 5, 6, 10 and 30 mHz. In coil #2, o
TM2 esul s, only he injec ions om he 29 h o Ap il
we e pe o med, which we e a equencies 1, 3, 5 mHz
meaning ha only h ee suscep ibili y alues could be
ob ained a wice hei equency.
Acco ding o [17], he AC magne ic suscep ibili y o
LPF TMs can be app oxima ed a low equencies as
χ(ω)≃χDC +−iωτe
1 + iωτe
,(17)
whe e χDC is he equency independen e m o he sus-
cep ibili y and wi h τebeing he magne ic suscep ibil-
i y cu , i.e., he equency a which he eal and imag-
ina y pa o he magne ic suscep ibili y ha e he same
TABLE III. Suscep ibili y alues o bo h TMs ob ained a 2ω
using Eq. (9) o all he injec ion equencies.
F equency [mHz] χTM1 (10−5)χTM2 (10−5)
DC (−3.35 ±0.15) -
2 (−3.43 ±0.58) (−4.0±2.3)
5 (−3.3723 ±0.0069) -
6 (−2.65 ±0.62) (−2.64 ±0.92)
10 (−3.35 ±0.12) (−3.833 ±0.057)
30 (−4.73 ±0.34) -
alue. Fo LPF, his alue was measu ed on-g ound o
be τe= (2π630)−1Hz−1[18].
I we now plo he measu ed alues o he magne ic
suscep ibili y o TMs 1 and 2 along wi h he cu e in
Eq. (17), we ob ain he plo shown in Fig. 6. We can see
ha all magne ic suscep ibili y esul s om Table III a e
compa ible wi hin hei unce ain y anges as p edic ed
om Eq. (17), excep o he one a 30 mHz. The e
we e only h ee signal injec ed a 15 mHz, which had a
low SNR and, also, showed an unexpec ed linea d i
ha had o be sub ac ed. We hink his explains he
sys ema ics a ec ing he demodula ion o i s second ha -
monic a 30 mHz. Wi h espec o he esul s o TM2,
we compu ed hem o comple eness bu we ecall ha
coil 2 su e ed a mal unc ion a he beginning o ope a-
ions which could explain he sys ema ic e o obse ed
in he de i ed pa ame e s.
The esul s shown we e ob ained by he e odyne de-
modula ion o he ∆gsignal wi h a sinusoidal signal in-
phase. Howe e , i we made he la e be ou o phase
by π/2 one would expec he ampli udes measu ed by
his me hod o be ze o only i he e we e no imagina y
componen . Thus, he imagina y suscep ibili y can be
ob ained demodula ing a 2ω, in quad a u e. In he e-
quency egime ha we a e wo king in, his alue is ex-
pec ed o be o de s o magni ude smalle han he eal
suscep ibili y. The esul s ha we ob ain o he imagi-
na y suscep ibili y a 10 mHz we e consis en wi h ze o,
χi= (0.0±1.8) ×10−6. Wi h he limi s de e mined
by he o ce sensi i i y o he demodula ion a ound he
en hs o em oNew ons. Thus, we can only con i m ha
he alues o he imagina y suscep ibili y o he TMs a e
below |χi|<1.8×10−6a 10 mHz. A he es o e-
quencies he esul ga e a less p ecise uppe bound o
he imagina y suscep ibili y.
Finally, we can e alua e he p edic ion o ou o ce
model om Eq. (7) in compa ison wi h he measu ed
accele a ion du ing injec ions by making use o he ex-
ac ed magne ic pa ame e s wi hin his a icle. To do
so, we ha e selec ed a single injec ion om all he ones
o June 18 h, 2016 wi h a DC o se o 0.75 mA and an
AC ampli ude o 1.5 mA. We ha e used hese alues o
he co esponding magne ic ield DC and AC calcula-
ions oge he wi h he magne ic pa ame e s ound in
he p e ious sec ions and subs i u ed in o Eqs. (8). The
esul s can be seen in Fig. 7 whe e, oge he wi h he
o al model o ce, we can see plo ed he di e en o ce
con ibu ions FDC,F1ωand F2ω. The p edic ed o ce
and he da a ma ch wi h a esidual di e ence be ween
hem o (0.0±1.9) ×10−13 N. Due o he p esence o
a linea d i wi hin he da a o igina ed om he desyn-
ch oniza ion be ween he clocks o he LPF measu emen
sys ems, one o he diagnos ics and a di e en one o
he op ical me ology sys em, we a e able o obse e a
le o e s signal wi hin he esidual da a.
