As onomy &As ophysics manusc ip no. ou pu ©ESO 2025
No embe 20, 2025
A Flux-Based Es ima e o he In e s ella Objec Con ibu ion o he
Galac ic Ba yon Budge
Sub i le
Ma ian Siwiak1
Cogni ion Sha ed Solu ions LLC, 16192, Lewes, USA
e-mail: [email p o ec ed]
Recei ed No embe 20, 2025
ABSTRACT
Con ex . In e s ella objec s p o ide he only di ec obse a ional e idence o mac oscopic bodies unbound o he Sola Sys em. Thei
po en ial con ibu ion o he Galac ic ba yon budge has no p e iously been quan i ied using an obse a ion-ancho ed app oach.
Aims. We aim o de e mine whe he he popula ion implied by he h ee con i med in e s ella objec s can supply a signi ican ac ion
o he Milky Way’s missing ba yons.
Me hods. We de i e a lux-based es ima e o he local ISO numbe densi y using measu ed de ec ion adii, encoun e imes, and
hype bolic eloci ies, and con e his o a local mass densi y ia a non-pa ame ic boo s ap o e he ISO masses. This densi y is hen
coupled o se e al Galac ic spa ial empla es and in eg a ed ac oss he Milky Way o ob ain he co esponding Galaxy-wide ISO mass.
De ec ion comple eness is ea ed as a ee scaling pa ame e , and Poisson and mass unce ain ies a e p opaga ed independen ly.
Resul s. Fo he h ee shallow-g adien empla es conside ed—ba yonic, o al-mass, and uni o m—modes incomple eness is su i-
cien o ISOs o supply he mass needed o aise he Milky Way ba yon ac ion o he alue expec ed o haloes o simila mass.
S onge incomple eness allows he in e ed ba yon ac ion o app oach he cosmic alue. A s ella - acing empla e yields unphysical
Galaxy-wide masses and is he e o e uled ou .
Conclusions. In e s ella objec s may cons i u e a non-negligible and p e iously unaccoun ed componen o he Galac ic ba yon
budge . The amewo k de eloped he e is eadily upda ed as addi ional ISOs a e de ec ed, p o iding a di ec and obse a ion-ancho ed
means o assessing hei con ibu ion o he Milky Way’s ba yonic mass.
Key wo ds. ISM: gene al – Galaxies: ISM – Galaxy: s uc u e – Galaxy: undamen al pa ame e s – Me hods: s a is ical – Me hods:
nume ical
1. In oduc ion
The ba yonic mass o he Milky Way emains unce ain.
Measu emen s om he cosmic mic owa e backg ound and
big-bang nucleosyn hesis cons ain he cosmic ba yon ac-
ion o b≃0.158 (Aghanim e al. 2020; Fields e al. 2020),
whe eas dynamical es ima es o he Galaxy indica e a sub-
s an ially smalle alue, b≃0.05–0.06 (McMillan 2017;
Bland-Haw ho n & Ge ha d 2016). This di e ence implies a
missing ba yonic mass o o de 2–3×1010 M⊙, po en ially esid-
ing in ho ci cumgalac ic gas, cold o di use phases, o compac 10
ba yonic componen s (B egman 2007; Tumlinson e al. 2017;
P iyadha shini e al. 2025).
In e s ella objec s (ISOs) cons i u e a a ely examined ca -
ego y wi hin his b oade in en o y. P io o he disco e y
o 1I/’Oumuamua, cons ain s elied solely on su ey non-
de ec ions (Engelha d e al. 2017). The subsequen de ec ions o
1I/’Oumuamua, 2I/Bo iso , and 3I/ATLAS (Meech e al. 2017;
Guzik e al. 2020; Seligman e al. 2025) p o ide di ec e idence
o mac oscopic bodies on unbound ajec o ies h ough he So-
la Sys em and allow a lux-based es ima e o hei local num-20
be densi y (Flekkøy & Toussain 2023; Do sey e al. 2025). Be-
cause ISOs o igina e om ba yonic p ocesses associa ed wi h
plane o ma ion and ea ly dynamical ejec ion (Jackson e al.
2018; Raymond e al. 2018), hey ep esen a po en ially non-
negligible bu p e iously unquan i ied ba yonic ese oi .
In his pape we use only he empi ical p ope ies o he h ee
known ISOs—de ec ion adii, encoun e imes, heliocen ic e-
loci ies, and es ima ed masses— o de i e he co esponding lo-
cal numbe and mass densi ies. We p opaga e hese cons ain s
h ough a se o conse a i e Galac ic-scale spa ial empla es and 30
in eg a e he esul ing ISO densi y ields ac oss he Milky Way.
This app oach quan i ies how he Galaxy-wide ISO mass de-
pends on su ey comple eness while a oiding assump ions abou
he unde lying mass unc ion o ejec ion his o y.
The analysis shows ha , o spa ial dis ibu ions acing
ba yonic mass, o al mass, o a uni o m backg ound, mod-
es incomple eness is su icien o ISOs o supply a ba yonic
mass compa able o he p esen de ici . Mo e ex eme incom-
ple eness alues allow he ISO con ibu ion o app oach he
ba yon ac ion empi ically associa ed wi h haloes o Milky- 40
Way mass. A s ella - acing empla e, in con as , p oduces un-
physical Galaxy-wide masses and is he e o e excluded. ISOs
canno se e as a da k-ma e analogue, bo h because o hei
ba yonic o igin and because such an in e p e a ion would con-
lic wi h cosmological cons ain s; howe e , hey may o m a
p e iously unaccoun ed componen o he Galac ic ba yon bud-
ge .
A icle numbe , page 1
A&A p oo s: manusc ip no. ou pu
All analysis code and igu e-gene a ion sc ip s a e pub-
licly a ailable a h ps://gi hub.com/ma iansiwiak/
ISODensi y.50
2. Modelling
The h ee con i med in e s ella objec s—1I/’Oumuamua,
2I/Bo iso , and 3I/ATLAS—p o ide he empi ical ounda ion
o es ima ing he local ISO mass densi y and ex apola ing i o
he Galaxy. Thei disco e y ci cums ances (Meech e al. 2017;
Guzik e al. 2020; Seligman e al. 2025) and associa ed pho o-
me ic and dynamical cha ac e isa ion (Jewi e al. 2017, 2020;
Jewi 2024; Bolin e al. 2020) supply he obse ed encoun e
da es i, heliocen ic de ec ion adii Ri, es ima ed masses mi,
and cha ac e is ic hype bolic eloci ies ∞(Table 1). These ob-60
se a ional inpu s, oge he wi h su ey sensi i i y conside a-
ions (Engelha d e al. 2017; Chambe s e al. 2019; Ton y e al.
2018), de ine he local no malisa ion used in he emainde o
he analysis.
Table 1: Adop ed pa ame e s o he h ee con i med in e s ella
objec s.
Objec
Disco e y
da e
Rde
(AU)
Mass
(kg)
∞
(km s−1)
1I 19/10/17 0.25 1.0×1011a26.3b
2I 30/08/19 2.98 2.6×1011c32d
3I 01/07/25 3.20 2.6×1014e58
No es. (a)Jewi (2024) (b)NASA Science (2017) (c)Jewi e al. (2020)
(d)Bolin e al. (2020) (e)Seligman e al. (2025) ( )NASA Science (2025)
2.1. Local ISO encoun e a e and numbe densi y
The empo al baseline be ween he i s and las de ec ion is
Tobs = max − min,(1)
yielding he maximum-likelihood Poisson a i al a e
λbes =Nde
Tobs
.(2)
Unce ain y in he a e is quan i ied using he exac Ga wood
in e al (Ga wood 1936), which p o ides he 95% bounds
(λlow, λhigh), exp essed h ough he mul iplica i e ac o s70
slow =λlow
λbes
,shigh =λhigh
λbes
.(3)
Because he h ee ISOs we e de ec ed a di e en heliocen-
ic dis ances and no uni o m su ey-sensi i i y model exis s o
all e en s, we adop he a i hme ic mean
Ra g =1
Nde X
i
Ri,(4)
and ea he su eyed egion as an e ec i e sphe e o c oss-
sec ion
A=πR2
a g.(5)
This choice p o ides a conse a i e, su ey-independen
no malisa ion. The co esponding lux- h ough-a-sphe e o -
mula ion o he local ISO densi y ollows he s anda d ap-
p oach used in ecen ISO isibili y and numbe -densi y analyses
(Flekkøy & Toussain 2023; Do sey e al. 2025). 80
Equa ing he lux nISO ∞A o he obse ed a i al a e gi es
he local numbe -densi y es ima o
nISO =λbes
A ∞
.(6)
In his wo k, he de ec ion comple eness ac o Cde ∈(0,1] is
in oduced as a ee scaling pa ame e o p obe he sensi i i y
o he in e ed ISO mass o su ey incomple eness. Cde escales
lux calcula ed in ewqua ion 6 o
nISO, ue(Cde )=nISO
Cde
=λ
Cde A ∞
.(7)
2.2. Local ISO mass densi y and no malisa ion
The es ima ed masses o he h ee ISOs span se e al o de s o
magni ude, consis en wi h expec a ions om popula ion syn-
hesis and ejec ion models (Jackson e al. 2018; Raymond e al. 90
2018; T illing e al. 2017; Po egies Zwa e al. 2018). To a oid
imposing a pa ame ic mass dis ibu ion, we employ a non-
pa ame ic boo s ap (E on 1979; Feigelson & Babu 2012) o e
he mass se {m1,m2,m3}. This gene a es an empi ical dis ibu-
ion o mean masses and yields he low, bes , and high es ima es
ma g,low,ma g,bes ,ma g,high. The co esponding local mass densi y
is
ρISO, ue(Cde )=ma g nISO, ue(Cde )=ma g λ
Cde A ∞
.(8)
2.3. Galaxy-wide ISO mass ex apola ion
To ex apola e he local mass densi y o he Galaxy, we assume
ha he ISO spa ial dis ibu ion ollows one o ou empla es: 100
s ella mass, o al ba yonic mass, o al mass (ba yons +da k
ma e ), o a uni o m backg ound. These dis ibu ions a e de-
i ed om he Milky Way mass model o McMillan (2017) as
implemen ed in galpy (Bo y 2015). Fo any empla e ρT(R,z),
he ISO densi y no malised a he Sola posi ion (R⊙,z⊙)=
(8.2 kpc,0) (Abu e e al. 2019) is
ρISO(R,z;Cde )=ρISO(Cde )ρT(R,z)
ρT(R⊙,0).(9)
We e alua e he dis ibu ion on an axisymme ic cylind ical
g id spanning R∈[0.1,20] kpc and z∈[−3,3] kpc, using he
olume elemen dV =2πR dR dz. The o al ISO mass is app ox-
ima ed by 110
MISO(Cde )≈X
i,j
ρISO(Ri,zj;Cde )(2πRi∆Ri∆zj),(10)
and he co esponding ba yon ac ion is
b(Cde )=Mba y,known +MISO(Cde )
M o al
.(11)
All uni con e sions a e handled h ough he as opy ame-
wo k (Collabo a ion 2022).
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Ma ian Siwiak: A Flux-Based Es ima e o he In e s ella Objec Con ibu ion o he Galac ic Ba yon Budge
2.4. Unce ain y ea men
Two independen sou ces o unce ain y en e he model: he
Poisson a i al a e and he ISO masses. Ra e unce ain y is
cap u ed h ough he Ga wood scalings (slow,shigh) applied o
λbes . Mass unce ain y is cap u ed h ough he boo s ap-de i ed
iple (ma g,low,ma g,bes ,ma g,high). These a e p opaga ed inde-
penden ly h ough he numbe -densi y calcula ion, he local120
mass densi y, and he Galac ic in eg a ion.
The combined unce ain y en elope is he e o e ob ained by
applying all combina ions o he h ee mass scena ios, and he
wo a e scalings, explici ly:
ρcomb
ISO, ue =ρbes
ISO, ue × {slow,shigh} × (ma g,low
ma g,bes
,ma g,high
ma g,bes ).(12)
These cases a e p opaga ed independen ly h ough he
Galac ic in eg a ion, p oducing he ull unce ain y en elope o
he o al ISO mass MISO(Cde ) and he ba yon ac ion b(Cde ).
3. Resul s
Galaxy-in eg a ed ISO masses inc ease mono onically as he
de ec ion comple eness Cde dec eases, and his end di ec ly130
shapes he in e ed ba yon ac ion. As shown in Fig. 1, he h ee
shallow-g adien empla es (ba yonic, o al-mass, and uni o m)
p oduce nea ly iden ical ends ac oss he de ec ion comple e-
ness ange, whe eas he s ella empla e de ia es by se e al o -
de s o magni ude.
103
104
105
106
b
(s ella )
Ba yonic
To al MW Mass
Uni o m Dis ibu ion
S ella
10 210 1100
De ec ion comple eness
C
de
10 1
100
b
( iable empla es)
Bes -es ima e
b
(
C
de ) ac oss empla es
Fig. 1: Bes -es ima e b(Cde ) ac oss he galac ic empla es. Ho -
izon al lines ma k he Milky Way ba yon ac ion, he expec ed
ac ion o haloes o simila mass, and he cosmic ba yon ac-
ion.
3.1. C ossing poin s o cha ac e is ic ba yon ac ions
The comple eness alues a which he in e ed ba yon ac-
ion eaches h ee cha ac e is ic h esholds— he Milky Way
alue ( b≃0.056), he halo-mass-expec ed alue ( b≃0.08;
McGaugh 2007; McGaugh e al. 2009; P iyadha shini e al. 140
2025), and he cosmic alue ( b=0.158; Aghanim e al. 2020;
Fields e al. 2020)—a e summa ised in Table 2.
Table 2: Comple eness alues Cde a which each empla e
eaches selec ed ba yon- ac ion h esholds.
Templa e ( b) 0.056 0.08 0.158 1
S ella 1.0 1.0 1.0 1.0
Ba yonic 1.0 0.46 0.10–0.31 0.01–0.03
To al-mass 1.0 0.39–1.0 0.09–0.27 0.03
Uni o m 1.0 0.33–1.0 0.08–0.23 0.02–0.03
Ac oss he iable models, modes incomple eness is su i-
cien o each b≃0.08, while s onge incomple eness allows
b o app oach he cosmic alue.
3.2. Beha iou ac oss he shallow-g adien empla es
Figu e 1 shows ha he in e ed ISO con ibu ion inc eases
mono onically wi h dec easing Cde . This end ollows di ec ly
om he C−1
de scaling o he local mass densi y. E en o Cde =1,
ISOs con ibu e a he se e al-pe cen le el o he Milky Way 150
ba yonic mass.
The h ee iable spa ial empla es a e compa ed mo e closely
in Figu e 2. Each panel displays he bes -es ima e cu e oge he
wi h he unce ain y en elopes de i ed om he boo s ap mass
dis ibu ion and Poisson a e bounds. The simila i y o he h ee
panels shows ha he esul s a e d i en p ima ily by he local
ISO no malisa ion and only weakly by assump ions abou he
la ge-scale spa ial dis ibu ion.
3.3. Implica ions o he missing-ba yon budge
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A&A p oo s: manusc ip no. ou pu
10 210 1100
De ec ion comple eness
C
de
10 1
100
101
b
Ba yonic
Combined × mass CI
Mass CI ( ixed)
CI (mass ixed)
Bes es ima e
(a) Ba yonic empla e
10 210 1100
De ec ion comple eness
C
de
10 1
100
101
b
To al MW Mass
Combined × mass CI
Mass CI ( ixed)
CI (mass ixed)
Bes es ima e
(b) To al-mass empla e
10 210 1100
De ec ion comple eness
C
de
10 1
100
b
Uni o m Dis ibu ion
Combined × mass CI
Mass CI ( ixed)
CI (mass ixed)
Bes es ima e
(c) Uni o m empla e
Fig. 2: Unce ain y en elopes o he h ee shallow-g adien em-
pla es. The h ee panels show nea ly iden ical beha iou ac oss
he comple eness ange.
The gap be ween he Milky Way ba yon ac ion and he 160
alue expec ed o haloes o compa able mass co esponds o
oughly (2–3) ×1010,M⊙. As shown in Figu e 1, once he com-
ple eness alls below Cde ≲0.6, he in e ed ISO con ibu ion
exceeds his h eshold o all iable empla es. Fo Cde ≈0.1–
0.15, he co esponding ba yon ac ions app oach he cosmic
alue. These ends ollow di ec ly om he local ISO no mal-
isa ion implied by he de ec ions and a ise wi hou in oking
changes o he unde lying Galac ic mass model.
3.4. S ella - acing empla e
The s ella empla e beha es quali a i ely di e en ly due o he 170
s eep cen al concen a ion o he s ella densi y ield (McMillan
2017). When no malised o he Sola neighbou hood, his leads
o Galaxy-in eg a ed ISO masses ha exceed bo h he ba yonic
and dynamical mass o he Milky Way. E en he mos conse a-
i e unce ain y combina ions gi e b≫1, as shown in Fig. 3.
10 210 1100
De ec ion comple eness
C
de
10 1
100
101
102
103
104
105
106
b
S ella - acing ISO dis ibu ion ( uled ou )
Combined × mass CI
Mass CI ( ixed)
CI (mass ixed)
Bes es ima e
Fig. 3: S ella - acing empla e. The in e ed ba yon ac ion ex-
ceeds uni y o all comple eness alues, indica ing incompa ibil-
i y wi h he ISO no malisa ion.
4. Conclusions
We ha e de eloped a minimal, obse a ion-ancho ed amewo k
o es ima ing he possible Galac ic mass con ibu ion o in e -
s ella objec s, based solely on he measu ed p ope ies o he
h ee con i med de ec ions. A lux-based es ima e o he local 180
ISO numbe densi y, combined wi h a boo s ap cha ac e isa ion
o ISO masses and an exac Poisson desc ip ion o he a i al
a e, p o ides a da a-d i en no malisa ion o se e al conse a-
i e Galac ic densi y empla es de i ed om he McMillan mass
model.
The h ee shallow-g adien empla es—ba yonic, o al-mass,
and uni o m—p oduce mu ually consis en ou comes. Fo com-
ple eness alues Cde ≲0.6, he co esponding Galaxy-in eg a ed
ISO masses a e su icien o aise he Milky Way ba yon ac ion
om i s cu en ly in e ed alue o he b≃0.08 le el expec ed 190
o haloes o compa able mass. Fo Cde ≈0.1–0.15, he in e ed
ac ions ex end owa d he cosmic ba yon ac ion. These be-
ha iou s a e d i en by he local ISO densi y and a e la gely in-
sensi i e o he assumed la ge-scale spa ial dis ibu ion.
A icle numbe , page 4
Ma ian Siwiak: A Flux-Based Es ima e o he In e s ella Objec Con ibu ion o he Galac ic Ba yon Budge
The s ella - acing empla e yields ba yon ac ions well
abo e uni y owing o he s eep cen al concen a ion o he s ella
mass ield. This incompa ibili y wi h he obse ed ISO en i on-
men p o ides an in e nal consis ency check on he modelling
amewo k and mo i a es he exclusion o such cen ally con-
cen a ed dis ibu ions.200
Low comple eness alues allow he in e ed ISO mass o ap-
p oach he dynamical mass scale o he Galaxy. Unde he s an-
da d cosmological ba yon in en o y, his end canno be in e -
p e ed as e idence o a ba yonic al e na i e o da k ma e : ISO
o ma ion is inhe en ly ba yonic, and any associa ed inc ease
in bwould need o emain consis en wi h CMB- and BBN-
based cons ain s. Wi hin hose cons ain s, he esul s should
be iewed in e ms o Galac ic ba yon accoun ing; should u-
u e e isions o he cosmological ba yon ac ion a ise, he ISO
componen quan i ied he e would in eg a e na u ally in o any up-210
da ed budge .
The amewo k is eadily ex ensible. Addi ional ISO de ec-
ions will igh en he a i al- a e cons ain s, educe s a is ical
unce ain y, and p og essi ely cons ain he ISO mass dis ibu-
ion. Imp o emen s in su ey sensi i i y models, ISO physical
cha ac e isa ion, and Galac ic po en ial modelling can be inco -
po a ed di ec ly. As he sample g ows, his app oach p o ides
a s able means o quan i ying he ole o in e s ella objec s in
he Milky Way ba yon census, complemen ing ongoing e o s
o iden i y and cha ac e ise non-luminous ba yonic componen s.220
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