Magne ic-induced o ce noise in LISA Pa h inde ee- alling 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, ∗F Gibe ,5, 14 D Gia dini,10 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 J A Lobo,12, 13, ∗L Liu,5J P L´opez-Za agoza,12, 13, †R Maa schalke wee d,15 D Mance,10
N Meshksa ,10 V Ma ´ın,12, 13 L Ma in-Polo,9J Ma ino,3F Ma in-Po que as,9P 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
LISA Pa h inde was a mission designed o es key echnologies equi ed o g a i a ional wa e
de ec ion in space. Magne ically d i en o ces play a key ole in he ins umen sensi i i y in he
low- equency egime, which co esponds o he measu emen band o in e es o u u e space-
bo ne g a i a ional wa e obse a o ies. Magne ically-induced o ces couple o he es mass mo ion,
in oducing a con ibu ion o he ela i e accele a ion noise be ween he ee alling es masses.
In his Le e we p esen he i s comple e es ima e o his e m o he ins umen pe o mance
model. Ou esul s se he magne ic-induced accele a ion noise du ing he Feb ua y 2017 noise un
o 0.25+0.15
−0.08 m s−2/√Hz a 1 mHz and 1.01+0.73
−0.24 m s−2/√Hz a 0.1 mHz. We also discuss how he
non-s a iona i ies o he in e plane a y magne ic ield can a ec hese alues du ing ex eme space
wea he condi ions.
In oduc ion LISA Pa h inde [1, 2] was an ESA
mission wi h NASA con ibu ions designed o es key
echnologies o he u u e g a i a ional-wa e obse a-
o y in space, he Lase In e e ome y Space An enna
(LISA) [3]. LISA Pa h inde was launched on Decembe
3 d 2015 and, a e a one-mon h c uise phase, eached he
L1 Lissajous o bi whe e i ope a ed un il July 2017. The
main scien i ic esul o he mission was o demons a e
he le el o ela i e accele a ion be ween ee alling es
masses (TMs) equi ed o de ec g a i a ional wa e om
space [4, 5]. The main expe imen on-boa d he LISA
Pa h inde mission consis ed o wo gold-pla inum es
masses si ing wi hin a 6 deg ee-o - eedom capaci i e
posi ion senso and ac ua o , he G a i a ional Re e -
a Xi :2407.04427 2 [as o-ph.IM] 5 No 2024
2
ence Senso (GRS) [6, 7]. The posi ion and o ien a-
ion o he es masses was con inuously moni o ed by
a high p ecision in e e ome ic eadou sys em, he Op-
ical Me ology Sys em (OMS) [8, 9]. These measu e-
men s we e ed o he on-boa d con ol sys em in cha ge
o isola ing he es mass om he pe u ba ions coming
om ou e space [10]. This a chi ec u e, he so-called
d ag ee con ol loop, ensu ed he equi ed ee all pu-
i y o one o he es masses by means o a con inuous
ac ua ion on he spacec a a i ude h ough he mic o-
New on p opulsion sys em [11]. The d ag ee con ol
loop was no in ended o isola e he es mass ee all
om o ces a ising in e nally o he spacec a . Ins ead,
mission and payload we e designed o ensu e ha hese
e ec s did no a ec he ins umen pe o mance. This
had implica ions, o ins ance, on he selec ed o bi — he
L1 o bi allowed a e y he mally s able en i onmen as
well as educed g a i a ional ield g adien s— o se e al
design d i e s in he payload o isola e he es masses
om po en ial dis u bing o ces, ypically o he mal o
magne ic o igin.
The LISA Pa h inde Da a and Diagnos ics Subsys-
em (DDS) included a se o high p ecision senso s on-
boa d p ecisely o moni o en i onmen al dis u bances
wi h po en ial impac in he main scien i ic esul o he
mission, he ela i e accele a ion be ween he ee alling
es masses. The DDS was composed by a empe a u e
measu emen subsys em [12, 13], a magne ic diagnos ic
subsys em [14, 15], see Fig. 1 o he dis ibu ion o he
magne ic componen s, and a adia ion moni o [16–19].
A c ucial ole o he DDS was o spli up he expe imen
pe o mance in o di e en con ibu ions o help on he
design o u u e space-bo ne g a i a ional wa e de ec-
o s. Fo ha eason, he DDS also included a se o
hea e s and coils ha we e ac i a ed du ing ligh ope -
a ions o cha ac e ise he esponse o he ins umen o
in ended en i onmen dis u bances. In his Le e we
epo on he esul s ob ained in he es ima e o he
magne ic-induced con ibu ion o he di e en ial o ce
noise o he es masses.
Magne ically induced o ce noise con ibu ion Mag-
ne ic ield luc ua ions couple in o he dynamics o he
ee alling es masses on boa d he sa elli e, which ac
as a dipole embedded in a magne ic ield
F=D(M ·∇)B+χV
µ0[(B·∇)B]E,(1)
whe e in ou no a ion bold le e s s and o ec o s,
wi h M being he 3-dimensional in insic emanen mo-
men o he magne ic dipole, χi s magne ic suscep-
ibili y, B he su ounding magne ic ield and µ0is
he acuum pe meabili y. The igh angle b acke s de-
no e TM olume, V, a e age o he enclosed quan i y:
⟨. . . ⟩ ≡ 1
VRV(. . .)d3x. This exp ession can be u he
expanded [20], howe e in his Le e we will be in e -
FIG. 1: Magne ic elemen s o he LPF DDS: 2 coils and
4 i-axial luxga e magne ome e s. The axis ep esen s
he spacec a e e ence sys em ame.
es ed in he noise con ibu ions mainly ocusing on he
x-componen which is he mos sensi i e axis, connec -
ing bo h TMs as de ined by he spacec a ame om
Fig. 1. Unde easonable assump ions –homogenei y and
s a iona i y o he es mass p ope ies– we can exp ess
he main con ibu ions o he accele a ion noise budge
as
SF,x(ω) = χV
µo⟨∇Bx⟩2
SB(ω) +
+⟨M ⟩+χV
µ0⟨Bx⟩2
S∇Bx(ω) (2)
whe e S(ω) s ands o he Powe Spec al Densi y
(PSD). The i s e m in Eq. (2) is he dominan one
in he low equency egime, since i couples luc ua ions
o he magne ic ield —wi h a −1beha iou due o he
in e plane a y con ibu ion— h ough an e ec i e cou-
pling ∼χ|∇Bx|. This coupling ac o explains he s in-
gen equi emen s se on he magne ic ield g adien alue
gene a ed by uni s on he spacec a in he es mass lo-
ca ion. No ice ha while he magne ic suscep ibili y is
a es mass p ope y and, he e o e, di icul o modi y
a e he design phase o i o be as low as possible, e-
ducing a po en ial g adien in he es mass loca ion has
an equal di ec e ec in he educ ion o his leading e m
in he magne ic noise con ibu ion. The second e m in
3
Pa ame e Value
χ(×10−5)−3.3723 ±0.0069
M ,x[nAm2] 0.140 ±0.138
M ,y[nAm2] 0.178 ±0.025
M ,z[nAm2] 0.095 ±0.010
Bback.,x[nT] 414 ±74
∇xBback.,x[nTm−1]−7400 ±2100
TABLE I: TM1 magne ic pa ame e s as measu ed
in- ligh in [21].
Eq. (2) has a lesse impac in he low equency band
since he luc ua ions associa ed wi h he g adien o he
magne ic ield show a la spec um, i.e. S∇Bx(ω)∼c .;
we ind his con ibu ion o be o de s o magni ude be-
low.
De e mina ion o magne ic pa ame e s The magne ic
diagnos ic subsys em on-boa d LISA Pa h inde con-
sis ed o ou i-axial magne ome e s and wo induc ion
coils. The wo induc ion coils had a adius o 56.5 mm
and we e buil wi h 2400 windings o a coppe alloy
moun ed on a i anium suppo , 85.5 mm away om he
es mass. Bo h coils we e aligned wi h he axis join-
ing bo h es masses so ha he gene a ed magne ic ield
had axial symme y and d i en by a dedica ed high s a-
bili y cu en d i e o ensu e ha high p ecision mag-
ne ic o ces we e p oduced [14]. An ex ensi e campaign
o expe imen s we e conduc ed o s udy magne ically-
induced o ces on he es masses du ing ligh ope a-
ions by applying con olled magne ic-induced o ces in
he es mass [21]. The esul o hese sui e o expe i-
men s can be summa ised in wo ype o pa ame e s as
we show in Table I. A i s se co espond o in insic
p ope ies o he es mass, he magne ic suscep ibili y
(χ) and he emanen magne ic momen (M ) which ob-
iously de e mine he eac ion o he es mass o any
applied magne ic ield. A second se o pa ame e s co e-
spond o ex insic pa ame e s o he es mass, he back-
g ound magne ic ield Bback.,xand magne ic ield g adi-
en ∇xBback.,x, which also con ibu e o he es mass
o ce when an ex e nal magne ic ield is applied.
The magne ic suscep ibili y was es ima ed a di e en
equencies du ing in- ligh expe imen s. We no ice ha
he ele an impac in he noise con ibu ion will be en-
e ing ou es ima es in he LISA Pa h inde band, i.e.
in a low enough equency egime whe e we can sa ely
ea his pa ame e as a cons an alue since, o he
LISA Pa h inde es masses, he suscep ibili y cu -o
equency is expec ed a ound he 630 Hz [22, 23]. Hence,
o ou calcula ions we will conside ou bes es ima e,
ob ained a = 5 mHz. The alue o he h ee com-
ponen s o he emanen magne ic momen a e shown
in Table I. All esul s we e only measu ed o es mass
#1 due o he mal unc ioning o he coil nea he o he
es mass. We no e he e ha , despi e he high ela i e
e o ca ied by he measu emen hese a e he mos p e-
cise alues ob ained o he emanen momen o he es
masses. The dedica ed measu emen s on-g ound could
only ind an uppe limi o 4 nAm2[24], i.e. one o de o
magni ude abo e he ones p esen ed he e.
When exe ing a magne ic induced o ce wi h he coil
we a e able o eco e he backg ound magne ic ield
g adien , since i will add up o he magne ic ield in-
duced by he coil. The expe imen s on-boa d can only
es ima e he x-componen o his con ibu ion. The
alue ob ained o TM1 du ing in- ligh expe imen s,
∇xBback.,x= (−7400 ±2100) nT/m, poin s owa ds his
con ibu ion o igina ing in he he mis o s a ached o
he ex e nal wall o he GRS Elec ode Housing. These
NTC (Nega i e Tempe a u e Coe icien ) senso s a e
manu ac u ed using ma e ials ha can show e omag-
ne ic beha iou . Despi e unde going a degaussing p o-
cess p io o hei assembly in he sa elli e, hei close
p oximi y o he es mass — oughly 13 mm— makes
hem p one o c ea e a local g adien [25]. I is wo h
no ing he e ha he es ima es o Bback.,xand ∇xBback.,x
ob ained du ing he expe imen s wi h coils on-boa d a e
he mos p ecise a ained in he es mass loca ion, since
magne ome e s mus be loca ed su icien ly a om he
es mass so ha he magne ic ield hey gene a e do no
pe u b he es mass dynamics.
Gi en ha he X-componen o he local g adien is
he only one measu able, we a e o ced o ex apola e
hese esul s o he Y and Z componen and o he o he
es mass as well. To do so, a Mon e Ca lo (MC) sim-
ula ion was pe o med wi h es mass #1 and all eigh
NTCs su ounding i . Fo each NTC he emanence was
andomly selec ed wi hin he alues [5×10−7, 5×10−5]
Am2, in acco dance wi h he ange o magne ic momen s
measu ed o hese senso s du ing on-g ound cha ac e i-
sa ion, and i s o ien a ion chosen a andom in sphe ical
coo dina es. The magne ic ield g adien s ac oss he ol-
ume o he es masses we e ound o be Gaussian dis-
ibu ed a ound ze o wi h 5µT/m o s anda d de ia ion
( o all componen s). To accoun ha he LPF in e e o-
me ic measu emen s in ol ed he ela i e di e ences in
posi ions o bo h es masses, he MC unce ain ies we e
mul iplied by a ac o o √2. Thus, he es ima ed o al
magni ude o he backg ound magne ic ield g adien was
gi en mainly by ∇xBback.,x, as he o he wo componen s
(∇yBback.,xand ∇zBback.,x) ob ained om he simula-
ion we e negligible, bu i s s anda d de ia ion ook in o
accoun all MC esul s leading o |∇Bx|= 7.4+5.5
−2.1µT/m.
Resul s and Discussion The main scien i ic esul
om LISA Pa h inde is exp essed in e ms o he di e -
en ial o ce pe uni mass ac ing on he wo es masses,
he so-called ∆g [4]. In o de o ob ain his pa ame-
e , a se ies o signals we e injec ed du ing he in- ligh
ope a ions which allowed a comple e desc ip ion o he
dynamic model o he mo ion o he es mass inside he
spacec a [26]. The model de i ed in his way was hen
4
FIG. 2: Noise p ojec ion o he magne ic induced o ces in he ee- alling es masses ela i e accele a ion. In blue
he S1/2
∆g om Feb. 14 h o Feb. 27 h, 2017. In ed he S1/2
Baccele a ion p ojec ion om he measu emen s o he
magne ome e s on-boa d du ing he same noise un. Black and pu ple do s ep esen an a e aging o he spec a
and he la e includes he unce ain y p opaga ion a e applying Eq. (2).
used o ansla e he high p ecision in e e ome ic mea-
su emen o he dis ance be ween he es masses in o
ela i e accele a ion be ween hem. The e alua ion o
his ela i e accele a ion was pe o med in he so called
noise uns, i.e. pe iods whe e he ins umen was kep in
i s mos s able, low-noise con igu a ion o se e al days
in o de o ob ain a good es ima e o he ∆gspec um
down o he millihe z egime.
In Fig. 2 we show he spec um o luc ua ions o he
ela i e accele a ion be ween bo h es masses in he
noise un spanning om Feb. 14 h a 01:59:50 o Feb.
27 h a 09:53:29, 2017. The blue line ep esen s he Am-
pli ude Spec al Densi y (ASD) — he squa e oo o he
PSD— o he ∆g ime se ies, S1/2
∆g, which, in his seg-
men , is well below he 30 m s−2/√Hz a 1 mHz equi ed
o LISA Pa h inde and e en a aining he 3 m s−2/√Hz
a 1 mHz equi ed o LISA. The ed line co esponds o
he squa e oo o only he i s e m in Eq. (2). As p e-
iously explained, his is he leading con ibu ion in he
low equency egime due o i s coupling wi h he in e -
plane a y magne ic luc ua ions, S1/2
B, and hei ∝ −1
dependence wi h equency. Black do s wi h e o ba s
ep esen he a e aging o he spec a, S1/2
∆g, a he same
equency bins used in [5]. In pu ple, he same a e ag-
ing is applied o he magne ic p ojec ion. In he la e
case, howe e , he e o ba s include no only he s a-
is ical e o om he Welch pe iodog am es ima e bu
also he unce ain ies om |∇Bx|, which includes he
p opaga ion o he MC analysis p e iously explained o
Eq. (2). Acco ding o ou es ima e, he magne ic con i-
bu ion o ∆gdu ing he Feb ua y noise un accoun s o
a 1.46+3.73
−0.77% in noise powe a 0.1 mHz.
We use he same da a segmen o e alua e he second
e m in Eq. (2). These con ibu ions couple o he ∆g
h ough he luc ua ions o he g adien o he magne ic
ield. Being a g adien , luc ua ions show a la spec um,
educing hus i s ele ance a low equencies. We es i-
ma e i s alue o be ≃14 am s−2/√Hz which would s a
playing a signi ican ole abo e 10 mHz. Howe e , om
Fig. 2 we can see ha a a ound 7 mHz he spec um
o he i s e m la ens a ≃50 am s−2/√Hz ins ead o
ollowing he −1 endency. This pla eau is due o he
in insic ead-ou noise o he magne ome e s as i s a s
domina ing he spec um o e he in e plane a y mag-
ne ic luc ua ions.
5
FIG. 3: S a is ical dis ibu ion o he magne ic
con ibu ion o accele a ion noise (in ed) a 20µHz
( op), 0.1 mHz (middle), and 1 mHz (bo om). The
associa ed Gaussian dis ibu ion a 1σ, 2σand 3σ
anges is also shown. Fo compa ison, we display he
median alues o S1/2
∆g(in blue) o each o he 12 noise
uns pe o med. The explici alues measu ed o bo h
du ing he 2017 Feb ua y un (Feb) a e also shown.
In Fig. 2 we ha e con eyed ou comple e unde s anding
o he magne ic con ibu ion o accele a ion noise du ing
he Feb ua y 2017 noise un. Howe e , he low equency
luc ua ions o he in e plane a y magne ic ield show a
non-s a iona y beha iou which ollows closely he dy-
namics o he sola wea he pa ame e s, no ably o he
sola wind speed [15]. In o de o s udy his a iabili y
we ex end ou analysis o he comple e magne ome e s
ime se ies and, he e o e, ex apola e he impac o his
non-s a iona y beha iou in he es ima ion o he mag-
ne ic con ibu ion o he ela i e accele a ion be ween
es masses o he whole mission du a ion. To do so, we
compu e he ASD o he in e plane a y magne ic ield
a di e en equencies loga i hmically spaced wi hin he
e y low equency egime [20µHz, 2 mHz] in segmen s
o 8 days. This is o ha e mo e han 10 Welch pe i-
odog am a e ages a 20µHz, ensu ing ha he spec al
window is no biasing ou es ima es while s ill ha ing
enough segmen s in o al, 241, o he analysis o be s a-
is ically ele an . A e wa ds, we p ojec i in o o ce
noise by conside ing he i s e m in Eq. (2). In Fig. 3
we show his magne ic p ojec ion ASD du ing he en i e
LPF mission in ed by using he median alue o |∇Bx|
and only conside ing he i s e m in Eq. (2). We show
his analysis o he equency bins a 20µHz, 0.1 mHz
and 1 mHz, which co espond o he h ee panels in he
igu e. Fo compa ison, we also display in blue he 12
noise uns pe o med du ing he whole mission.
Gi en ha we ha e a su icien s a is ics o he mag-
ne ic con ibu ion o ∆g, we can compu e he associ-
a ed con idence in e als a 1σ, 2σand 3σby conside -
ing he 16 h and 84 h pe cen iles, he 2.5 h and 97.5 h
pe cen iles and he 0.15 h and 99.85 h pe cen iles, e-
spec i ely. The dis ibu ion o he magne ic noise con-
ibu ion implies ha he 1.01+0.73
−0.24 m s−2/√Hz a 0.1
mHz ha we ha e ob ained o he Feb ua y un condi-
ions could inc ease o 2.70+2.00
−0.77 m s−2/√Hz o ex eme
sola wind condi ions, whe e we ha e conside ed he un-
ce ain y in |∇Bx|in he e o es ima e.
Finally, we wan o emphasize ha he he s a is ical
dis ibu ion o he magne ic noise con ibu ion due o he
in e plane a y magne ic ield non-s a iona i y can no be
di ec ly associa ed wi h he a iabili y o he measu ed
accele a ion noise in he es mass since he magne ic
con ibu ion is only one o he con ibu ions –and no o
dominan one– o he o e all ∆gmodel. Fig. 3 compa es
his dis ibu ion o he ew measu ed S1/2
∆g o he sake
o comple eness, al hough one should no de i e a causal
connec ion be ween bo h.
Conclusions We ha e p esen ed he i s comple e
es ima e o he magne ic-induced o ce noise con ibu-
ion o he accele a ion noise be ween ee- alling es
masses in he con ex o a space-bo ne g a i a ional
wa e de ec o . The esul s a e based on a p ecise
in- ligh cha ac e isa ion o hose pa ame e s a ec ing
he magne ic beha iou o he es mass. Ou esul s
se a alue o he magne ic con ibu ion o he LISA
Pa h inde accele a ion noise du ing he noise un o
Feb ua y 2017 o 0.25+0.15
−0.08 m s−2/√Hz a 1 mHz con-
side ing he magne ic g adien unce ain ies, which is
close o he ini ial assessmen s [4] and well below he
equi emen s es ablished be o e launch o his con ibu-
6
ion o 12 m s−2/√Hz a 1 mHz. Ou es ima e a 0.1
mHz is 1.01+0.73
−0.24 m s−2/√Hz, a 1.46+3.73
−0.77% PSD con i-
bu ion o ∆g, which is in he o de o o he con ibu ions
such as he cha ging noise, wi h an es ima ed con ibu-
ion o 1 m s−2/√Hz a 0.1 mHz [27] o he ac ua ion
noise, which is expec ed o be he dominan con ibu-
ion wi h an expec ed alue o 4.5 m s−2/√Hz [7] a 0.1
mHz. We ind ha , as expec ed, he leading e m o he
magne ic-induced o ce noise o he es mass mo ion in
hese low equencies o igina es om he coupling o luc-
ua ions in he in e plane a y magne ic ield o he local
magne ic ield g adien h ough a cons an e m gi en by
∼χ|∇Bx|.
Magne ic ield luc ua ions ha e a non-s a iona y be-
ha iou ela ed o space wea he condi ions, p edomi-
nan ly he sola wind speed, which will induce excess
es mass accele a ion noise in he low equency band
o some ex eme condi ions o he sola wind. Those
non-s a iona i ies could ise he noise powe con ibu ion
o ∆go magne ic luc ua ions up o a ac o ≃4.6 a 0.1
mHz in he wo s case scena io, ha is du ing he mos
ex eme sola wind condi ion, measu ed du ing LPF, and
he la ges p edic ed alue o he g adien magne ic ield
|∇Bx|= 7.4+5.5
−2.1µT/m. This low equency magne ic
ield non-s a iona y beha iou is implici o he in e -
plane a y ield and, hence, a po en ial dis u bance o
u u e space-bo ne g a i a ional wa e obse a o ies, al-
hough no being he dominan one in he case o LISA.
The local magne ic ield g adien in he es mass lo-
ca ion is he design pa ame e ha hese u u e g a i a-
ional wa e obse a o ies in space can use o supp ess his
po en ial noise con ibu ion. Ou analysis shows ha ,
in LISA Pa h inde , he empe a u e senso s (NTCs) lo-
ca ed close o he es mass can be conside ed a po en ial
sou ce o he magne ic local g adien . Despi e ha i is
no expec ed o limi he ins umen pe o mance in mos
o i s ope a ions, his po en ial excess noise con ibu ion
could be u he supp essed by using Pla inum empe a-
u e senso s wi h no magne ic con ibu ion. The esul s
epo ed he e a e applicable o u u e space-bo ne de ec-
o s and o he missions sha ing he echnology ha was
pu o es by he LISA pa h inde mission.
Acknowledgemen s This wo k has been made pos-
sible by he LISA Pa h inde mission, which is pa
o he space-science p og am o he Eu opean Space
Agency. The F ench con ibu ion has been suppo ed by
CNES (Acco d Speci ic de p oje CNES 1316634/CNRS
103747), he CNRS, he Obse a oi e de Pa is and he
Uni e si y Pa is-Dide o . E. P. and H. I. would also like
o acknowledge he inancial suppo o he Uni Ea hS
Labex p og am a So bonne Pa is Ci e´e (ANR-10-
LABX-0023 and ANR-11- IDEX-0005-02). The Albe -
Eins ein-Ins i u acknowledges he suppo o he Ge -
man Space Agency, DLR. The wo k is suppo ed by
he Fede al Minis y o Economic A ai s and Ene gy
based on a esolu ion o he Ge man Bundes ag (FKZ
50OQ0501 and FKZ 50OQ1601). The I alian con i-
bu ion has been suppo ed by Agenzia Spaziale I al-
iana and Ins i u o Nazionale di Fisica Nuclea e. The
Spanish con ibu ion has been suppo ed by Con ac s
No. AYA2010-15709 (MICINN), No. ESP2013-47637-
P, ESP2015-67234-P (MINECO); PID2019-106515GB-
I00, PID2022-137674NB-I00 (MICIU). F. R. acknowl-
edges suppo om a Fo maci´on de Pe sonal In es igado
(MINECO) con ac . The Swiss con ibu ion acknowl-
edges he suppo o he Swiss Space O ice (SSO) ia he
PRODEX P og amme o ESA. L. F. acknowledges he
suppo o he Swiss Na ional Science Founda ion. The
UK g oups wish o acknowledge suppo om he Uni ed
Kingdom Space Agency (UKSA), he Uni e si y o Glas-
gow, he Uni e si y o Bi mingham, Impe ial College,
and he Sco ish Uni e si ies Physics Alliance (SUPA).
J.I.T. and J.S. acknowledge he suppo o he U.S. Na-
ional Ae onau ics and Space Adminis a ion (NASA).
∗Deceased
†jplop[email p o ec ed]
‡[email p o ec ed]
§[email p o ec ed]
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