Binary evolution theory predicts that the second common envelope ejection can produce low-mass (0.32–0.36 M⊙) subdwarf B (sdB) stars inside ultrashort-orbital-period binary systems, as their helium cores are ignited under nondegenerate conditions. With the orbital decay driven by gravitational-wave (GW) radiation, the minimum orbital periods of detached sdB binaries could be as short as ∼20 min. However, only four sdB binaries with orbital periods below an hour have been reported so far, and none of them has an orbital period approaching the above theoretical limit. Here we report the discovery of a 20.5-min-orbital-period ellipsoidal binary, TMTS J052610. 43+593445.1, in which the visible star is being tidally deformed by an invisible carbon–oxygen white dwarf companion. The visible component is inferred to be an sdB star with a mass ∼0.33 M⊙ approaching the helium-ignition limit, although a He-core white dwarf cannot be completely ruled out. In particular, the radius of this low-mass sdB star is only 0.066 R⊙, about seven Earth radii. Such a system provides a key clue in mapping the binary evolution scheme from the second common envelope ejection to the formation of AM CVn stars having a helium-star donor. It may also serve as a crucial verification binary of space-borne GW observatories such as LISA and TianQin in the future.

1. Nature Astronomy
natureastronomy
https://doi.org/10.1038/s41550-023-02188-2
Article
Aseven-Earth-radiushelium-burningstar
insidea20.5-mindetachedbinary
Jie Lin 1,2,3,20
, Chengyuan Wu 4,5,6,20
, Heran Xiong7,20
, Xiaofeng Wang 1,8
,
Péter Németh 9,10
, Zhanwen Han 4,5,6
, Jiangdan Li4,5
, Nancy Elias-Rosa 11,12
,
Irene Salmaso 11,13
, Alexei V. Filippenko14
, Thomas G. Brink 14
, Yi Yang14
,
Xuefei Chen 4,5,6
, Shengyu Yan1
, Jujia Zhang4,5,6
, Sufen Guo 15
, Yongzhi Cai4,5,6
,
Jun Mo1
, Gaobo Xi1
, Jialian Liu1
, Jincheng Guo8
, Qiqi Xia1
, Danfeng Xiang 1
,
Gaici Li1
, Zhenwei Li4,5
, WeiKang Zheng14
, Jicheng Zhang16,17
, Qichun Liu1
,
Fangzhou Guo1
, Liyang Chen 1
& Wenxiong Li18,19
Binaryevolutiontheorypredictsthatthesecondcommonenvelope
ejectioncanproducelow-mass(0.32–0.36 M⊙)subdwarfB(sdB)stars
insideultrashort-orbital-periodbinarysystems,astheirheliumcoresare
ignitedundernondegenerateconditions.Withtheorbitaldecaydrivenby
gravitational-wave(GW)radiation,theminimumorbitalperiodsofdetached
sdBbinariescouldbeasshortas∼20 min.However,onlyfoursdBbinaries
withorbitalperiodsbelowanhourhavebeenreportedsofar,andnoneof
themhasanorbitalperiodapproachingtheabovetheoreticallimit.Herewe
reportthediscoveryofa20.5-min-orbital-periodellipsoidalbinary,T­MT­S
J­05­26­10­.43+593445.1,inwhichthevisiblestarisbeingtidallydeformedby
aninvisiblecarbon–oxygenwhitedwarfcompanion.Thevisiblecomponent
isinferredtobeansdBstarwithamass∼0.33 M⊙ approachingthe
helium-ignitionlimit,althoughaHe-corewhitedwarfcannotbecompletely
ruledout.Inparticular,theradiusofthislow-masssdBstarisonly0.066 R⊙,
aboutsevenEarthradii.Suchasystemprovidesakeyclueinmappingthe
binaryevolutionschemefromthesecondcommonenvelopeejectiontothe
formationofAMCVnstarshavingahelium-stardonor.Itmayalsoserveasa
crucialverificationbinaryofspace-borneGWobservatoriessuchasLISAand
TianQininthefuture.
Since the beginning of minute-cadence observations with Tsinghua
University–MaHuatengTelescopesforSurvey(TMTS)1,2
,wehavedis-
covered a dozen unusual short-period objects3,4
in the Galaxy. TMTS
J052610.43+593445.1(J2000coordinatesrightascensionα = 81.5434,
declinationδ = +59.5792;hereafterJ0526)isanewlydiscoveredvariable
star with a dominant photometric period of only 10.3 min (Extended
DataFig.1).Theperiodicitywascross-checkedbyphotometricobser-
vations from the Zwicky Transient Facility (ZTF)5,6
and the Yunnan
Faint Object Spectrograph and Camera (YFOSC) mounted on the
Lijiang 2.4 m Telescope (LJT)7,8
(Fig. 1). Time-resolved spectroscopic
observations from the Keck I Low-Resolution Imaging Spectrometer
(LRIS)9,10
and the Gran Telescope Canarias (GTC)/Optical System for
Imagingandlow-ResolutionIntegratedSpectroscopy(OSIRIS)11
yielded
a dozen single-line spectra with various radial velocities (RVs; Fig. 2).
TheRVcurveismodulatedbya20.5-minperiodandreachesitspeaks
and valleys at the phases of maximum light (Fig. 1), which proves that
J0526isanultracompactellipsoidalbinary.Theunequalmaximainthe
light curves are caused by the relativistic Doppler beaming effect12,13
ofthevisiblecomponent,consistentwithitslargeRVamplitude.This
object was also recently identified as a candidate verification binary
Received: 21 August 2023
Accepted: 20 December 2023
Published online: xx xx xxxx
Check for updates
A full list of affiliations appears at the end of the paper. e-mail: wang_xf@mail.tsinghua.edu.cn

2. Nature Astronomy
Article https://doi.org/10.1038/s41550-023-02188-2
GTC/OSIRIS spectra using non-local thermodynamic equilibrium
(non-LTE)spectralmodelsobtainedfromTLUSTYandSYNSPECsoft-
ware20,21
(Methods). The best-fitting model reproduces well the main
Balmer lines and He i λ4471 seen in the observed spectra, which gives
estimates of the effective temperature Teff = 25,480 ± 360 K, surface
gravity log g = 6.355 ± 0.068 , helium abundance log y = log NHe
/NH = −2.305 ± 0.062 and projected rotational velocity
ν sin i = 220+140
−90
km s−1
forthevisiblecomponent.
Among the large samples obtained from current spectroscopic
surveys,itisveryrareforWDstohavehydrogen-richspectracontami-
nated by He i lines22–24
, but such spectra are common for subdwarf B
(sdB) stars25–27
. Observationally, the helium abundance of hot subd-
warfsisoverallpositivelycorrelatedwiththeireffectivetemperatures.
However, this correlation tends to show two distinct branches for
He-rich and He-weak sequences, especially at the high-temperature
end28,29
. In comparison with the existing sdB sample, J0526B is on the
low-temperature end of the He-rich sequence in the Teff versus log y
diagram(ExtendedDataFig.2).
Stellarradiusandmass
The broadband SED is a very useful tool for constraining the stellar
radius and luminosity if the distance is available and reliable. With
prior knowledge of the effective temperature and surface gravity of
the visible star (Teff,B and log (g)B ) derived from the spectroscopic
analysis above, we fitted the SED (Fig. 3) using archival multicolour
photometry and the distance inferred from the Gaia DR3 parallax30
(Methods).Asthisobjectisnotincludedintheultraviolet(UV)-source
catalogueoftheGalaxyEvolutionExplorer31
,wealsousedUVobserva-
tions made with the Swift Ultraviolet/Optical Telescope (UVOT). The
best-fittingmodelsuggestsaradius RB = 0.0681+0.0059
−0.0049
R⊙andabolo-
metricluminosity Lbol =1.70+0.31
−0.24
L⊙ forthevisiblestar,andisusedto
update the effective temperature and surface gravity (Table 1). The
model also yields an estimate of the line-of-sight extinction as
E(B − V ) = 0.387+0.008
−0.009
mag,wellconsistentwiththeGalacticvalueof
E(B − V) = 0.385 magasqueriedfromathree-dimensionaldustextinc-
tion map32
. With the surface gravity and radius, we computed the
mass of the visible star as MB = 0.361+0.094
−0.073
M⊙ using Newton’s law of
gravity. The radius and mass obtained from the SED suggest that
J0526B is more likely a hot subdwarf with an extremely thin
hydrogen-richenveloperatherthanahelium-coreWD(seethemass–
radius relation below). The latter scenario is tenable only when this
visible component is an inflated WD during hydrogen shell flashes,
althoughsuchflashesaretheoreticallyshort-livedorevenabsentfor
stars having such a mass33–35
.
Orbitaldynamics
TheDopplershiftofspectrallinesprovideskeycluesforustoinvesti-
gatetheorbitaldynamicsofJ0526.Byassumingthattheorbitiscircu-
larized, the RV curve can be modelled well by a sinusoidal curve
(Fig. 1), with a semi-amplitude of 559.6+6.4
−6.5
km s−1
inferred for the
visible component. Hence, the mass function of the invisible compo-
nent was computed:
f(MA) ≡
M3
A
sin
3
i
(MA + MB)
2
=
K3
B
Porb
2πG
= 0.259 ± 0.009 M⊙, (1)
where MA and MB represent the masses of the invisible and visible
components (respectively), i is the inclination of the orbital plane,
KB is the semi-amplitude of the RV curve and G is the gravitational
constant. As the RV semi-amplitude and orbital period are both well
constrained from the observations, the mass function bridges a tight
relation between the masses of the binary stars and the inclination
angle,andthusaidsbinaryparameterestimationfromthelight-curve
fit below.
(ZTFJ0526+5934)ofgravitationalwaves(GWs)bytheZTFDR8database
and Gaia EDR3 catalogue14
.
Although J0526 has been included in white dwarf (WD) cata-
logues15,16
,theprobabilityofitbeingaWD(PWD)givenbytheprobabil-
ity map17
is only PWD = 0.0046 (ref. 15), suggesting that J0526 should
have large differences from those WDs. Thus, we present a detailed
analysisofJ0526inthispaper.Asthenatureofnon-eclipsingbinaries
is usually not well constrained from light curves alone, the physical
parameters of J0526 were determined using a combination of spec-
troscopy, broadband spectral energy distribution (SED), RV curve
and multicolour light curves. According to the prevailing workflow
for the analysis of ellipsoidal binaries, the properties of visible stars
are obtained before the orbital solutions18,19
. Prior knowledge of the
visiblecomponenthelpsdeterminetheinclinationoftheorbitalplane
andthemassoftheinvisiblecomponentfromthelightcurvesandRV
curves. We refer to the invisible component of this binary as J0526A
andthevisiblecomponentasJ0526B.
Atmosphericparameters
As shown in the dynamical spectra (Fig. 2), the Balmer lines and He i
λ4471showsynchronousshiftsagainsttheorbitalphase,whichsuggests
thatbothHandHefeaturesarisefromthevisiblestarinthebinarysys-
tem.BecausetherearenotanysignificantH/Helinestracingthemotion
of the invisible star, the invisible component must be very faint and is
assumedtobenegligibleinthespectralfitbelow.Asnoemissionlines
are visible in the spectra, mass accretion should not occur in the two
componentsofJ0526,andwe,thus,assumethatthebinarysystemisstill
detached. To verify this assumption, we further compared the radius
ofthevisiblestarwithitsRoche-lobesizeinthefollowingdiscussion.
AstheH/Heabsorptionlinesinthespectracarrykeyinformation
about the atmospheric properties of the visible star, we fitted all the
400 Keck I
GTC
ZTF g
LJT g
ZTF r
LJT r
200
0
–200
–400
–600
1.10
1.05
1.00
0.95
0.90
1.10
1.05
1.00
0.95
0.90
–0.75 –0.50 –0.25 0
Orbital phase
Normalized
flux
Normalized
flux
Radial
velocity
(km
s
−1
)
0.25 0.50 0.75
Fig.1|Phase-foldedRVcurveanddouble-bandlightcurvesforJ0526.Top,RV
curvederivedfromKeck/LRISandGTC/OSIRISobservations.Thedotted-dashed
lineisthebest-fittingsinusoidalmodel.Middleandbottom,g-andr-bandphase-
foldedlightcurvesprovidedbyLJTandZTF.Thepurplesolidlinesrepresentthe
best-fittinglight-curvemodelsobtainedfromtheellcpackage36
.Theunequal
maximaareduetotherelativisticDopplerbeamingeffect12,13
.Orbitalphaseϕ = 0
representstheepochofsuperiorconjunctionwhenthevisiblestarisclosest
totheobserver.TheRVdatapointsanderrorbarsindicatethebest-fitting
valuesand68%confidenceintervalsintheχ2
fitting.Thephotometricdataare
presentedasmeanvalues± 1σ.

3. Nature Astronomy
Article https://doi.org/10.1038/s41550-023-02188-2
Ellipsoidalvariationsandcompactobject
Ellipsoidalmodulationinthelightcurvesisinducedbytidaldeforma-
tionandrotationofthevisiblecomponent.Owingtothesynchroniza-
tion between the rotation and orbital motion, different geometric
cross sections of the visible star emerge throughout the orbit. The
large amplitude of the ellipsoidal modulations suggests that the vis-
ible star almost fills its Roche lobe (for example, fR = RB/RL,B ≳ 0.80).
We modelled the g- and r-band light curves of J0526 using the ellc
package36
(Methods). The values of RB, MB and f(MA) obtained above
wereincludedaspriorparameterdistributions.TheDopplerbeaming
effect was also included in the model to offset the unequal maxima.
Gravity-andlimb-darkeningcoefficientsandDopplerbeamingfactors
wereobtainedbyinterpolatingthegrids37
withthesurfaceparameters
derived from the spectroscopy and SED. The best-fitting model
gives i = 68.2+3.7
−5.2
deg and MA = 0.735+0.075
−0.069
M⊙ , and updates other
physicalparameterswithBayes’theorem(Table1).Themasssuggests
that the invisible star is a carbon–oxygen (CO) WD. Through the
mass–radiusrelationofCOWDs(ref.38),weestimatethattheradius
of the invisible star is about 0.011 R⊙, which favours the
noneclipsing scenario for J0526. With the updated radius of J0526B
(0.0661 ± 0.0054 R⊙), we derived the projected rotational velocity
(νB sin i)cal = 2πRB/Porb = 216+18
−19
km s−1
, consistent with the result
obtained above from the spectroscopy.
Giventheultracompactorbitandrelativelyhighmassesofthetwo
components, it is predicted that J0526 will be detected by the Laser
InterferometerSpaceAntenna(LISA)39
withinthefirst3 monthsofits
operation (Methods), and thus, it will serve as a verification binary of
GWs in the future. The GW characteristic strains of J0526 and dozens
of other verification/detectable binaries of GWs are presented in
ExtendedDataFig.3.
Beyondthestripselectioneffects
Figure4showsthatJ0526Bisinaninterlacedzonebetweenhotsubd-
warfs25
and(extremely)low-massWDs40
intheKieldiagram(Teff versus
log g). As the cooling sequences of WDs are widely distributed on the
bluesideofthemainsequence(MS),thesurfacegravitiesandeffective
temperaturesofhotsubdwarfsarecompatiblewiththe(pre-)WDs.We
can further cross-check the nature of J0526B by comparing the spec-
trophotometric parallax, derived from the atmospheric parameters
and hypothetical nature, against the astrometric parallax obtained
fromGaiaDR3(ref.30).ByassumingthatJ0526Bisahelium-coreWD,
weestimatedthespectrophotometricparallaxϖspec = 1.542 ± 0.136 mas
forJ0526(Methods),whichshowsadeviationfromtheGaiaDR3paral-
lax ϖGaia = 1.183 ± 0.091 mas by 2.2σ.
Followingthetheoreticalpredictionsfrombinaryevolutiontheory
and binary population synthesis41,42
, the second common envelope
(CE)channelisresponsiblefortheformationofsdBbinarieswithvery
shortorbitalperiods(typicallyPorb < 1 h)42
.BecausethesesdBstarsare
producedfromnondegenerateHecores,theirmassesareexpectedto
beonly∼0.33 M⊙ (refs.42,43).However,theextremehorizontalbranch
in the Kiel diagram, corresponding to hot subdwarfs with canonical
masses of ∼0.48 M⊙, is widely applied to confirm the natures of hot
subdwarfs.Thisselectioneffect(theso-calledstripselectioneffect42
)
wouldleadtoasystematicabsenceoflow-masssdBstarswithveryshort
orbits.Additionally,foradetachedsdBbinaryhavinganorbitalperiod
of only 20 min, the hydrogen-rich envelope must be extremely thin
1.0
0.8
Normalized
flux
Normalized
flux
Orbital
phase
Velocity (km s−1
)
Wavelength (Å)
0.6
6,550
GTC/OSIRIS, Hα GTC/OSIRIS, Hβ GTC/OSIRIS, Hγ GTC/OSIRIS, He I 4,471 Keck/LRIS, Hγ Keck/LRIS, He I 4,471
–1,000 0 1,000 –1,000 0 1,000 –1,000 0 1,000 –1,000 0 1,000 –1,000 0 1,000 –1,000 0 1,000
6,600 4,840 4,860 4,880 4,320 4,340 4,360 4,460 4,480 4,320 4,340 4,360 4,460 4,480
1.2
1.0
0.8
0.6
0.4
0.2
1.2
1.0
0.9
0.8
0.7
0.6
0.8
0.4
1.6
1.4
1.2
1.0
0.8
0.6
0.4
Fig.2|DynamicalspectraofJ0526fromGTC/OSIRISandKeck/LRISobservations.Top,lineprofilesofHα,Hβ,HγandHei λ4471atallepochsofobservations.
Bottom,spectrallinesphasedwiththeorbitalperiodof20.5 min.Colourscalesindicatethecontinuum-normalizedfluxandredsolidlinesrepresentthebest-fitting
RVcurveofthevisiblestarofJ0526.

4. Nature Astronomy
Article https://doi.org/10.1038/s41550-023-02188-2
(for example, 10−6
M⊙; refs. 44,45) to avoid a Roche-lobe overflow
(RLOF).Withthispriorknowledge,werantheModulesforExperiments
inStellarAstrophysics(MESA)46
codetoreproduceevolutionarytracks
forsdBstarsinextremelyshort-orbital-periodbinarysystems(Meth-
ods).Althoughthesetrackscoveronlyaverysmallarearelativetothe
regions of hot subdwarfs or WDs, as shown in Fig. 4, the atmospheric
parametersofJ0526BoverlapexactlyonthesdBtracksof0.32–0.33 M⊙.
ByassumingthatJ0526BisansdBstarof0.33 M⊙,thespectrophotomet-
ricparallaxcanbere-estimatedasϖspec = 1.30 ± 0.10 mas,approximat-
ingtheastrometricparallax.
Mass–radiusrelation
As diverse equations of state from different classes of stars lead to
distinctmass–radiusrelations,themass–radiusdiagram(MRD)47–49
is
avalidtoolfordistinguishingdifferenttypesofstars.Aversionofthe
MRDextendedtowardsthelow-radiusendisshowninFig.5,wherethe
three dominant types of stars are CO-/He-core WDs, MS stars and hot
subdwarfs.Thesethreetypesofstarsareinseparateregionsandthey
hardly overlap in the MRD. Being supported by electron-degeneracy
pressure,WDstendtohavesmallerradiiatlargermasses,theopposite
of nondegenerate stars. The hot subdwarfs cover a wide area in the
MRD owing to diverse hydrogen-envelope masses generated from
differentinitialbinariesandevolutionarychannels.Theobservational
constraintsfromspectroscopy,theSEDandlightcurvessupportthat
J0526Bislocatedexactlyatthelowertipofthehot-subdwarfdomain.In
otherwords,J0526Bcouldbealow-masscore-helium-burning(CHeB)
starwithanextremelythinhydrogenenvelope,asalsosuggestedbythe
analysis of the Kiel diagram. As a hydrogen-exhausted star inside the
20.5-min orbit, the size of J0526B is smaller than that of all previously
knownnondegeneratestars,evenbrowndwarfsandgasplanets48,50,51
.
However, J0526B has an average density ∼1,200 times greater than
that of the Sun!
SecondCEejectionandAMCVnstars
FollowingthetheoreticalpredictionsfromthesecondCEejectionchan-
nelofsdBstars41,42
,somesdBstarsarebornfromapaircomprisingaWD
andan(evolved)MSstar.Thisbinaryisthestellarremnantthatsurvived
the first CE ejection52
or stable RLOF (ref. 53). In this channel, the WD
companionhasaverysmallradiusandcanpenetratedeeplyintotheCE
beforeCEejection,whichallowstheformationofansdBbinarywitha
veryshortorbitalperiod.Inparticular,iftheMScomponenthasanini-
tialmasslargerthanthecriticalmassforastartoexperiencethehelium
flashattheendofitsfirstgiantbranch(alsocalledthered-giantbranch),
forexampleMMS,0 ≳ 2.0 M⊙ (refs.41,43),itsprimaryfusionreactionsare
1.0
UVM2
UVW2 UVW1 W1 W2 W3 W4
g r i z y
Observed
Synthetic
TMAP spectrum
0.5
10
–15
10
–16
10
–17
10
–18
10
–19
10
–20
10
–21
0.4
0.2
(obs.–syn.)/obs.
Wavelength (Å)
F
λ
(erg
cm
–2
s
–1
Å
–1
)
Transmission
0
–0.2
–0.4
10,000 100,000
Fig.3|BroadbandSEDofJ0526.Top,transmissioncurvesforthefiltersofSwift
UVbands73
,Pan-STARRSgrizybands75,76
andAllWISEW1–W4bands77,78
.Allcurves
arenormalizedatthemaximum.Toavoidovercrowding,thetransmissioncurves
ofotherfiltersarenotdisplayedhere.Middle,broadbandSEDoverUV,optical
andinfraredbands.GreenpointsarethephotometricfluxesprovidedfromSwift
UVW2/M2/W1bands,GaiaDR3BP,GandRPbands30,74
,Pan-STARRSgrizybands,
ZTFgrbandsandAllWISEW1–W4bands.Thebluesolidcurverepresentsthe
best-fittingsyntheticspectrumfromtheTübingenNon-LTEModel-Atmosphere
Package(TMAP)80
.Diamondsindicatethesyntheticfluxesderivedfromthe
modelspectrumandtransmissioncurves.Thephotometricdataarepresented
asmeanvalues± 1σ,andarrowsrepresent3σupperlimits.Bottom,relative
residuals.obs.,observed;syn.,synthetic.
Table 1 | Physical parameters of J0526
α (J2000) 05h26m10.416s
δ (J2000) + 590
34′
45.305′′
d (kpc) 0.847+ 0.071
− 0.060
Porb (min) 20.5062426±0.0000053
E(B−V) (mag) 0.385
Spectroscopic (GTC)
Teff,sd (K) 25,480±360
log (g)sd (cms−1
) 6.355±0.068
log (NHe/NH)sd
−2.305±0.062
νsd sin i (kms−1
) 220+ 140
− 90
Orbital dynamics (GTC+Keck I)
Ksd (kms−1
) 559.6+ 6.4
− 6.5
γ (kms−1
) −35.6±4.4
f(MWD) (M⊙) 0.259±0.009
Spectral energy distribution
Rsd (R⊙) 0.0681+ 0.0059
− 0.0049
Msd (M⊙) 0.361+ 0.094
− 0.073
Lbol (L⊙) 1.70+ 0.31
− 0.24
E(B−V)SED (mag) 0.387+ 0.008
− 0.009
Teff,sd,re (K) 25,410±370
log (g)sd,re (cms−1
) 6.352±0.068
Light-curve analysis
i (deg) 68.2+ 3.7
− 5.2
MWD (M⊙) 0.735+ 0.075
− 0.069
Msd,re (M⊙) 0.360+ 0.080
− 0.071
Rsd,re (R⊙) 0.0661±0.0054
T0,re (BJDTDB) 2459933.175697+ 0.000016
− 0.000017
a (R⊙) 0.255±0.011
Derived parameters
(νsd sin i)cal (kms−1
) 216+ 18
− 19
Porb /Porb
.
(yr−1
) − 1.72+ 0.40
− 0.47
×10− 7
𝒜𝒜 2.49+ 0.72
− 0.60
×10− 22
4-year LISA SNR 33.6+ 9.7
− 8.0
3-month LISA SNR 3.3+ 1.0
− 0.8
4-year TianQin SNR 24.5+ 7.1
− 5.9
The subscript ‘sd’ denotes the parameters for J0526B, and ‘WD’ indicates the parameters for
J0526A. The subscript ‘re’ denotes those parameters refined by Bayes’ theorem. The bold
titles indicate the corresponding methods of obtaining parameters.

5. Nature Astronomy
Article https://doi.org/10.1038/s41550-023-02188-2
through the CNO cycle instead of the proton–proton chain during its
MS stage. Consequently, the MS component can retain a higher cen-
tral temperature and, thus, produce a more massive, nondegenerate
helium core (∼0.2 M⊙) when leaving the MS, compared to those with
lowerinitialmasses.Giventhehighercentraltemperature,thehelium
corecouldpotentiallybeignitedundernondegenerateconditions41,54
even if the envelope is lost during passage through the Hertzsprung
gap. SdB stars formed through this subchannel would have very low
masses (Msd = 0.32–0.36 M⊙)42,43
. Envelopes of Hertzsprung-gap stars
aregenerallymoretightlyboundthanthoseofstarsnearthetipofthe
firstgiantbranch.Consequently,theinnerbinarysystemmustrelease
more orbital energy to counterbalance the binding energy, resulting
inanextremelyshortfinalorbitalperiod(typicallyoftheorderoftens
of minutes) after the second CE ejection.
J0526 is such an ultracompact binary. Its extremely short orbital
period, WD companion and low-mass sdB component exactly follow
the theoretical predictions from the second CE ejection channel of
sdB stars41,42
. By assuming that the mass of J0526B is equal to 0.33 M⊙,
itsRoche-loberadiusisestimatedasabout0.079 R⊙,whichisproperly
wider than the size of J0526B and consistent with the above inference
thatthisbinarysystemiscurrentlydetached.Withorbitalcontraction
drivenbygravitational-waveradiation(GWR),thenafterabout1.5 Myr,
J0526BwilloverflowitsRochelobeandtransfermasstowardstheWD
at an orbital period of around 14 min (Extended Data Fig. 4), leading
to the formation of an AM CVn star through the helium-star chan-
nel55,56
. Owing to the nondegenerate nature of J0526B, its radius will
shrink in response to mass loss induced by RLOF, supporting further
orbitalcontractiondrivenbyGWR.Asthemasstransferquenchesthe
helium burning, J0526B will begin a transition to a degenerate state,
for example becoming a He-core WD. When the electron-degeneracy
pressure becomes dominant, J0526B will reach the minimum orbital
periodof∼9 minanditwillstarttoexpandasitlosesmass,leadingtoan
increasing orbital period as predicted by binary evolution theory.
Ultimately,thedonorstarinsuchanAMCVnsystemwilleitherevolve
into a planet orbiting the WD companion56,57
or become tidally dis-
rupted by the WD accretor when its mass becomes smaller than
∼0.002 M⊙ (ref.58).
In summary, J0526 could be the shortest-orbital-period
single-degenerate detached binary, so that it could provide crucial
observationalevidencesupportingthecompleteevolutionaryscheme
rangingfrominitialbinaryMSstarstoMS+WDbinarytosdB+WDbinary
to AM CVn star55,56
. With the operations of the Large Synoptic Survey
Telescope59
,WideFieldSurveyTelescope60
andspace-borneGWobser-
vatory39,61
, more previously unknown extremely short-orbital-period
sdBbinarieswillbediscovered,whichmaythusaidourunderstanding
oftheformationofsdBstarsandAMCVnstars.
Methods
Photometric observations and orbital period
Initsfirst2-yearsurvey,TMTShasdiscoveredmorethan1,100variable
stars with periods shorter than 2 h (ref. 4). J0526 is one of the
shortest-periodvariablestarsinthecatalogue.Its10.3 minperiodicity
was first revealed by the Lomb–Scargle periodogram62
derived from
the12 hminute-cadenceobservationson18December2020(UTCdates
are used throughout this paper; Extended Data Fig. 1). The TMTS
Light-curveAnalysisPipelineautomaticallyestimatedthedereddened
colour (Bp − Rp)0
= −0.41 ± 0.02 mag and absolute magnitude
MG = 6.86 ± 0.21forJ0526usingitsembeddedGaiaDR2database63
and
DUSTMAPSPythonpackage64
.Acolour–magnitudediagramforJ0526
withsomesdBstarsandlow-massWDsispresentedinSupplementary
Fig.1.ThedataarefromtheGaiaDR3database30
.Theultrashortperiod
and extraordinary colour of J0526 drove us to trigger further photo-
metricandspectroscopicobservationsofthisobject.
Theg-andr-bandphotometricdatawereobtainedfromZTFPublic
Data Release 14 (DR14)5,6
and LJT/YFOSC observations7,8
conducted
on 19 December 2022. The LJT observations lasted for 60 min in r and
4.5
5.0
5.5
6.0
log
g
(cm
s
–2
)
log Teff (K)
6.5
7.0
7.5
8.0
4.7 4.6
(ELM) WDs from B20
He-core WD cooling sequence
Low-mass hot subdwarf sequence
(core helium-burning stage)
0.435 M
0.32 M
0.33 M
0.34 M
0.35 M
0.36 M
ZAHeMS
ZAEHB
TAEHB
0.363 M
0.321 M
0.272 M
0.239 M
0.203 M
0.187 M
0.176 M
0.165 M
0.155 M
sdBs from G20
J0526B
4.5 4.4 4.3 4.2 4.1 4.0
Fig.4|Kieldiagramforhotsubdwarfs,low-massWDsandJ0526B.
Atmosphericparametersofhotsubdwarfsand(extremely)low-mass(ELM)WDs
aretakenfromG20(ref.25)andB20(ref.40),respectively.Thebluedotted-
dashedlinesaretheoreticalevolutionarytracksofhelium-coreWDs34
,whereas
theshellflashloopsareclippedforclarity.Thepurplesolidlinesrepresentthe
evolutionarytracksoflow-massCHeBstarswithaverythinhydrogenenvelopeof
10−6
M⊙ (Methods).Weoverplotthezero-ageheliummainsequence(ZAHeMS;
Methods),zero-ageextremehorizontalbranch(ZAEHB)andterminal-age
extremehorizontalbranch(TAEHB)105
asdarkdashedlines.Theerrorbarsof
J0526Bindicate1σuncertaintiesin log g andTeff.

6. Nature Astronomy
Article https://doi.org/10.1038/s41550-023-02188-2
44 min in g, with a common exposure time of 30 s and a readout time
of ∼3 s. All LJT photometric data were reduced according to stand-
ard procedures, including bias subtraction, flat-field correction and
cosmic-rayremoval.ForZTFdata,measurementswithcatflag = 32,768
wereexcluded.AllmodifiedJuliandaysinbothZTFandLJTdatawere
converted into barycentric Julian dates in the Barycentric Dynamical
Time (BJDTDB). All observed fluxes were normalized by average fluxes
ineachband.
We computed Lomb–Scargle periodograms from light curves of
ZTF and LJT/YFOSC observations, and thus confirmed the periodic-
ity of J0526. Thanks to the long-term observational coverage from
ZTF, we obtained a precise photometric period Pph = 10.2531213 ±
0.0000026 min from the g-band light curve, and thus, the orbital
periodisPorb = 20.5062426 ± 0.0000053 min.Astheuncertaintyinthe
orbital period is tiny, Porb was fixed to 20.5062426 min for the analys
isbelow.
Spectroscopicobservations
We observed two series of spectra for J0526 within two independent
observations runs, one with the 10 m Keck I Telescope equipped with
LRIS (blue grism 600/4000, R ≈ 1,000 and red grating 1200/7500,
R ≈ 2,000)9,10
,andtheotherwiththe10.4 mGTCplusOSIRISinstrument
(grismR1000B,2 × 2binning,R ≈ 1,000)11
.TheKeck/LRISspectrawere
observedatsixsequentialorbitalphaseson23September2022,withan
exposuretimeof180 sforthefirstspectrumand240 sfortheothers.
AtotalofsevenGTC/OSIRISspectrawereobservedon26January2023,
witheachhavinganexposuretimeof180 sandareadouttimeof∼25 s.
BecauseofterribleweatherconditionsduringtheKeckIobservations,
thesignal-to-noiseratio(SNR)oftheKeck/LRISspectraissignificantly
lowerthanthatoftheGTC/OSIRISspectra.
TheGTC/OSIRISspectrawerereducedfollowingstandardtasksin
IRAF with the graphical user interface FOSCGUI, which was designed
toextractsupernovaspectraandphotometryobtainedwithFOSC-like
instruments.ItwasdevelopedbyE.Cappellaro,andapackagedescrip-
tion can be found at http://sngroup.oapd.inaf.it/foscgui.html. The
raw images were first corrected for bias, overscan, trimming and flat
fielding, and subsequently, one-dimensional spectra were optimally
extracted from the two-dimensional images. Wavelength calibration
wasperformedusingthespectraofcomparisonlampsthatwerepro-
duced 2 d earlier than the observation night, whereas the flux was
calibratedusingobservationsofspectrophotometricstandardstars.
Thesecalibrationimagesweretakenwiththesameinstrumentalcon-
figuration and on the same night as the spectra of J0526. Finally, the
J0526spectrawerefine-tunedwiththecoevalbroadbandphotometry
data,andthebroadabsorptionbands(forexample,H2OandO2)dueto
Earth’satmospherewereremovedusingthespectrumofthestandard
star.TheKeck/LRISspectrawerereducedthroughadedicatedpipeline
LPipe(ref.65)followingsimilarprocedures.
Bayesianinference
ToconstrainthephysicalparametersofJ0526fromspectra,broadband
SED, RV curve and light curves together, we linked the model param-
etersderivedfromeachobservationalclueusingBayes’theorem66
:
p(θ|𝒟𝒟) 𝒟 𝒟𝒟𝒟𝒟𝒟θ)p(θ), (2)
0.20
0.15
0.10
Radius
(R
)
Mass (M )
0.05
0.1 0.2 0.3 0.4 0.5
He WD
Radius of J0523
Radius of Jupiter
Thinner
H
envelope
Core
helium-burning
stars
Thicker
H
envelope
M
a
i
n
s
e
q
u
e
n
c
e
CO WD
8,500 K
10,000 K
15,000 K
J0526B
WD inside UCB
Constraint from surface gravity
90% of Roche-lobe radius
He main sequence
M H, env
= 10
–6 M
80% of Roche-lobe radius
sdB inside UCB
Hot subdwarf
White dwarf
Main-sequence star
25,000 K
25,000 K
Constraint from SED
40,000 K
0.6
Fig.5|MRDforMSstars,WDs,hotsubdwarfsandJ0526B.Massesandradiiof
MSstars,WDsandhotsubdwarfswerecollectedfromC17(ref.48),B20(ref.40)
andS22(ref.106),respectively.SixsdBstarsinsideultracompactbinaries(UCBs)
havingorbitalperiodsbelow100 min(refs.107–113)areshown.Forcomparison,
18WDcomponentsinside11ultracompactbinaries107,109,114–116
discoveredby
photometricsurveysarealsoincluded.Weoverplottheoreticalmass–radius
relationsforCOWDs,HeWDs,MSstarsandHeMSstars.Thedegenerateand
nondegeneratemodelsaredistinguishedbysolidanddashedlines,respectively.
ThepurpleareaabovetheHeMScorrespondstothehotsubdwarfs,whichare
CHeBstarswithhydrogen-richenvelopes,inwhichthickerenvelopesleadto
largerstellarradii.AsequenceforCHeBstarswithanextremelythinhydrogen-
richenvelope(MH,env = 10−6
M⊙)ishighlightedusingapinkdashedline.Wepresent
theconstraintsfromsurfacegravityandSED,derivedfromcurrentobservations,
accompaniedwiththemass–radiusrelationsofan80%/90%Roche-lobe-filling
starinsidea20.5-min-orbital-periodbinaryusingPaczyński’sapproximation85
(Methods).TheradiiofJupiterand2MASSJ0523−1403(ref.50;thesmalleststar
amongknownMSstars)arelabelledontheleftofthefigure.Thedatapoints
anderrorbarsrepresentthebestestimatesand68%confidenceintervals,
respectively.

7. Nature Astronomy
Article https://doi.org/10.1038/s41550-023-02188-2
where p(θ|𝒟𝒟) is the posterior distribution of model parameters θ
with given data 𝒟𝒟, 𝒟(𝒟𝒟|θ) is the likelihood function (also sampling
distribution)for 𝒟𝒟withgivenθandp(θ)isthepriordensitydistribution
ofmodelparametersθ.
Followingprevailingmethodsforresolvingthenatureofellipsoi-
dal binaries18,19
, we first determined the physical parameters of the
visiblecomponent,whichprovidesbetterparameterconstraintswhen
obtaining correct orbital solutions. The GTC spectra were fitted with
non-LTE model atmospheres to obtain the atmospheric parameters
(including log g andTeff)ofthevisiblecomponentindependently.Then
theseatmosphericparametersweretakenaspriordensitydistributions
(p(θ))andusedtoderivetheradiusandmassofJ0526Bfromthebroad-
bandSED.WealsofittedtheRVcurvetoobtaintheRVsemi-amplitude
(KB) and epoch of superior conjunction (T0) when the visible star was
closest to the observer. The model parameters (log (g)B, RB, KB and T0)
were used to construct prior density distributions for light-curve
modellingandwererefinedbythefinalposteriordistribution.
Atmosphericmodel
BecausetheSNRoftheLRISspectraistoolow(owingtopoorweather)
toyieldcorrectatmosphericparametersforJ0526B,weusedonlythe
OSIRISspectratoderivetheatmosphericparameters.Toreducesmear-
ingeffectsandpotentialcontributionsfromtheinvisiblecomponent,
wefittedallGTCspectrasimultaneously(SupplementaryFig.2)using
metal-freenon-LTETLUSTY(v.207)modelatmospheresandsynthetic
spectra with SYNSPEC (v.53)20,21
. The best-fitting model was obtained
fromaniterativespectralanalysisprocedure(XTGRID)29
,whichapplies
asteepest-descentχ2
minimizationtosimultaneouslyoptimizeallfree
parameters,includingeffectivetemperature,surfacegravity,chemical
abundances and the projected rotational velocity. All comparisons
wererungloballyoverthespectralrange3,780–6,850 Å,andthemodel
waspiecewisenormalizedtotheobservations.Inparallel,theRVswere
also determined by shifting each observation to the model. The pro-
cedure converges once the relative changes of all model parameters
andχ2
dropbelow0.5%inthreeconsecutiveiterations.Finally,param-
eter uncertainties were calculated by mapping the parameter space
around the best solution. Notice that when modelling the spectra of
sdBstars,thesystematicdifferencescausedbyreplacingthemetal-free
non-LTEmodelatmosphereswiththemetal-lineblanketedLTEmodel
atmospheres(with[Fe/H] = −2to+1)couldbeatthelevelsΔTeff ≈ ±500 K
and Δ log g ≈ ±0.05 (refs. 67–69), which are comparable to the uncer-
tainties quoted in our results. All atmospheric parameters of J0526B
aresummarizedinTable1.
Spectrophotometricparallax
Acommonmethodfortestingthehypotheticalnatureofsourcesisto
comparetheirspectrophotometricparallaxeswithavailableastromet-
ric parallaxes. Following the approach of calculating the spectropho-
tometric parallax40,70
, we first assumed J0526B to be an extremely
low-mass WD or a sdB star. We can obtain the mass of an extremely
low-mass WD M′
B
= 0.237 ± 0.018 M⊙ by interpolating the grids from
evolutionarytracksofHe-coreWDs34
usingTeff and log g obtainedfrom
spectra. The grids include the evolutionary tracks of shell flash loops
and, thus, lead to multiple solutions at same region of the Teff versus
log g diagram.Themassuncertaintyhereincludestheerrorsofsurface
parametersandmultiplesolutionsofmodeltracks.ThemassofansdB
star was assumed to be 0.33 M⊙ based on the position of J0526B in the
Teff versus log g diagram(Fig.4).UsingNewton’slawofgravityandthe
Stefan–Boltzmann law, we estimated its stellar radius and luminosity
fromtheatmosphericparametersandtheassumedmasses.Theabso-
lutemagnitudeintheGaiaGfiltercanbeobtainedfrom
Gabs = −2.5 log (
LB
L⊙
) + Mbol,⊙ − BCG, (3)
whereMbol,⊙ = 4.74,andBCG isthebolometriccorrectionfortheGband.
Hence,thespectrophotometricparallaxwascalculatedusing
Gabs = G + 5 log (
ϖspec
100 mas
) − AG, (4)
where G = 17.563 ± 0.003 mag, and AG is the interstellar dust extinc-
tionintheGband.BothBCG andAG wereobtainedbyinterpolatingthe
bolometric correction tables from MESA Isochrones & Stellar Tracks
(MIST)71,72
withtheatmosphericparametersandsolarmetallicity.
Radial-velocitycurve
As introduced above, the RVs were determined by shifting each
observed spectrum to the TLUSTY/SYNSPEC synthetic spectrum.
All RV measurements were corrected to the barycentric rest frame. It
is reasonable to assume that such a compact orbit is highly circular-
ized with eccentricity e = 0. Therefore, the RV curve can be fitted by
a sinusoidal function. The likelihood function for RV measurements
wascalculatedas
ln 𝒟ν = −
1
2
∑
j
(
Vr,obs, j − Vr,model,j
σν,obs, j
)
2
, (5)
where Vr,model,j = KB sin[
2π
Porb
× (tν,obs,j − T0)] − γ is the model RV at time
tν,obs,j andγistheRVofthebarycentreofthebinarysystem.Also,tν,obs,j,
Vr,obs,j andσν,obs,j aretheBJD,observedRVanduncertaintyobtainedfrom
thejthspectroscopicobservation,respectively.
Spectralenergydistribution
ThebroadbandSEDwasconstructedfromSwift73
target-of-opportunity
observations (UVW2/M2/W1 bands) and archival photometry from
Gaia DR3 (BP, G and RP bands)30,74
, Panoramic Survey Telescope and
RapidResponseSystem(Pan-STARRS;grizybands)75,76
,ZTF(grbands)5,6
and AllWISE (W1–W4 bands)77,78
. The photometric measurements in
three Swift UV bands were obtained by running the HEASOFT (v.6.31)
command UVOTPRODUCT. As J0526 was accidentally located in the
bad area of the detector during the observation on 15 March 2023
(ID 00015916001), we used measurements from the observations on
22 March 2023 (ID 00015916002). The optical and infrared photo-
metric fluxes were directly obtained from the Virtual Observatory
SED Analyzer (VOSA)79
online tool. We did not use the upper-limit
measurements in the W2–W4 bands produced by Wide-field Infrared
SurveyExplorer(WISE)intheSEDfitting.
Weconstructedthegridofsyntheticphotometrybysequentially
integrating extinction factors and filter transmission curves over the
TMAP synthetic spectra80
. The extinction factors were derived from
the Fitzpatrick extinction curve81,82
with reddening law RV = 3.1. The
gridofreddeningE(B − V)spans0.00–1.00 magwithastepof0.05 mag.
ThetransmissioncurveswereobtainedfromtheFilterProfileService
of the Spanish Virtual Observatory83
. The TMAP spectral grid covers
20,000 K ≤ Teff ≤ 66,000 Kwithastepof2,000 K,and 4.50 ≤ log g ≤ 6.50
withastepof0.25.Weappliedathree-dimensionallinearinterpolation
to approximate the synthetic flux at arbitrary [Teff, log g, E(B − V)]
coordinateswithinthegrid.
To consider additional flux uncertainties caused by ellipsoidal
variations of J0526, a free parameter σF,sys was included in the likeli-
hoodfunction:
ln 𝒟F = −
1
2
∑
j
⎧
⎨
⎩
[Fobs,j − (RBϖ)
2
× Fsyn, j]
2
σ2
F,obs,j
+ σ2
F,sys
+ ln [2π × (σ2
F,obs, j
+ σ2
F,sys
)]
⎫
⎬
⎭
,
(6)
where Fobs,j and σF,obs,j represent the observed flux and uncertainty
in the jth photometry, respectively. Fsyn,j, the synthetic flux in the jth
band,isafunctionofTeff, log g andE(B − V).

8. Nature Astronomy
Article https://doi.org/10.1038/s41550-023-02188-2
Lightcurves
To obtain the inclination angle (i) of the orbital plane and, thus,
calculate the mass of the primary MA through the mass function
(equation (1)), we modelled all g- and r-band light curves from both
ZTF and LJT observations simultaneously using the ellc (v.1.8.7)
package36
. For an ellipsoidal binary, the flux variation can be approxi-
matedas
ΔFell
F
≈ 0.15 ×
(15 + μB)(1 + τB)
3 − μB
MA
MB
(
RB
a
)
3
sin
2
i, (7)
whereΔFell/Frepresentsthefractionalsemi-amplitudeoftheellipsoidal
modulation, and a is the orbital separation. μB and τB are the
limb-darkening coefficient and gravity-darkening coefficient of the
visible star, respectively. Grids for the limb-darkening and
gravity-darkening coefficients were generated from the reference
tables37
.Weappliedtwo-dimensionalcubicinterpolationstoapproxi-
matethegandrcoefficientsatarbitrary (Teff, log g)coordinatesinside
thegrids.WithTeff and log g determinedasabove,weobtainedτB = 0.45
and μB = 0.31 for the g light curves, and τB = 0.41 and μB = 0.26 for the r
band. The Doppler beaming effect leads to a higher flux (ϕ ≈ −0.25)
when the visible component is approaching the observer than when
movingaway(ϕ ≈ 0.25).Wesetthebeamingfactorasb = 1.30fortheg
bandandb = 1.47fortherband37
.
To respect the RV semi-amplitude KB and epoch of superior con-
junctionT0 obtainedfromtheRVcurves,weintroducedapriordensity
distributionofthemassfunctionf(MB)(equation(1))tobridgethetight
relationamongMA,MB andinclinationi.Inaddition,theT0 derivedfrom
the RV curve was also included as a prior distribution for the epoch
ϕ = 0. Because the mass of the visible star (MB) in an ellipsoidal binary
is poorly constrained by both orbital dynamics and ellipsoidal varia-
tions, we adopted the prior density distribution of MB derived from
spectroscopyandthebroadbandSED.
Asthephotometricdatawereobtainedwithvariousinstruments
and filters, we introduced four free parameters σf,sys,k (k = 1, 2, 3, 4)
to offset the systematic errors in the ZTF g, ZTF r, LJT g and LJT r
light curves, respectively. Hence, the likelihood function was
expressedas
ln 𝒟f = −
1
2
∑
k
Nk
∑
j=1
{
(fobs, j,k − fellc, j,k)
2
σ2
f,obs, j,k
+ σ2
f,sys,k
+ ln [2π × (σ2
f,obs, j,k
+ σ2
f,sys,k
)]} , (8)
whereNk isthenumberofphotometrypointsinthekthlightcurve,and
fellc,j,k representstheellcmodelfluxforthekthbandattimetj,k.Alsotj,k,
fobs,j,k andσf,obs,j,k representtheBJD,observedfluxanduncertaintyofthe
jthphotometrypointinthekthlightcurve,respectively.
Mass–radiusrelationwithinRochelobe
Given the orbital period and fillout factor (the ratio between stellar
radius to the Roche-lobe radius, fR = RB/RL,B), a mass–radius relation
forthestar49,84
canbeobtainedfromKepler’sthirdlaw:
a3
P2
orb
=
G
4π2
(MA + MB) =
GMB
4π2
1 + q
q
, (9)
andthePaczyńskiapproximation(preferredforq = MB/MA ≲ 0.8)85
for
theRoche-loberadius(inunitsoforbitalseparation):
f(q) =
RL,B
a
≈ 0.462(
q
1 + q
)
1/3
, (10)
where a is the orbital separation. With RB = fRRL,B = fRf(q) × a to bridge
the relation between equation (9) and equation (10), the terms q in
the two equations cancel each other out, leading to a q-independent
expression(ifq ≲ 0.8)forthemass–radiusrelation:
RB = 0.067 R⊙ × (
MB
0.33 M⊙
)
1/3
(
Porb
20.5 min
)
2/3
(
fR
0.85
) . (11)
Galacticorbit
WecomputedthecomponentsoftheGalacticvelocityforJ0526.Weset
the Sun at a distance of R0 = 8.27 ± 0.29 kpc from the Galactic centre86
and its peculiar motion in relation to the Local Standard of Rest87
at
(U⊙, V⊙, W⊙) = (11.1, 12.24, 7.25) km s−1
. The rotational speed of the
Milky Way at the Solar circle86
was set to Vc = 238 ± 9 km s−1
. The com-
puted Galactic velocity components resulted in (U = 43 ± 4 km s−1
,
V = 233 ± 9 km s−1
, W = 5 ± 3 km s−1
), suggesting a membership in the
thin-diskcategory88
.
VerificationbinaryofGWs
GWs are thought to dominate the orbital angular momentum loss
(AML) of ultracompact binaries and, thus, lead to orbital contrac-
tion. Thanks to the compact orbit, the GWs generated from J0526 are
expected to be detected by space-borne GW observatories, such as
LISA(ref.39)andTianQin(ref.61).Withthecomponentmasses,orbital
period and distance of J0526 (Table 1), we can estimate the detection
SNRofitsGWsignalafter3 months89
and4 yearsofobservations.
Forabinarysystem,theGWstrainamplitudecanbecalculated90
as
𝒜𝒜 =
2(GMc)
5/3
(πfGW)
2/3
c4d
, (12)
where c is the speed of light in a vacuum, d is the source distance,
fGW = 2/Porb representstheGWfrequencyand Mc = (MAMB)
3/5
/(MA + MB)
1/5
is the chirp mass. The characteristic strain hc is, thus, given by
hc = √fGWTobs𝒜𝒜,whereTobs istheintegrationobservationtime,typically
4 yearsforLISAandTianQin.Wepresentthecharacteristicstrainsfor
dozens of verification/detectable binaries with detection sensitivity
curvesfromLISA(ref.91)andTianQin(ref.92)inExtendedDataFig.3.
ThechirpmassofJ0526iscalculatedusingthebinarymassesgivenby
the light-curve analysis, as shown in Table 1. Benefiting from its close
distance, J0526 is one of the ultracompact binaries that can generate
the strongest GW characteristic strains. For the first 3 months of GW
observationsbyLISA,theSNRofJ0526couldreach∼3(Table1).
Owingtotheabsenceofdenselysampledobservations,theorbital
contractionrateofJ0526isnotyetavailable.ByassumingthattheAML
is driven only by GWR, we can obtain a theoretical decay rate of the
orbitalperiodusing
̇
Porb
Porb
= −
96
5
× (
GMc
c3
)
5/3
(πfGW)
8/3
. (13)
Hence, the theoretical period derivative of J0526 is ̇
Porb/Porb
= −1.72+0.40
−0.47
× 10−7
yr−1
, implying a characteristic decay timescale of
τc =
3
8
Porb/| ̇
Porb| ≈ 2.2 × 106
years.
Evolutionarymodels
J0526isanexcellentobjectfordevelopingGWastronomyandinvesti-
gatingkeyprocessesinbinaryevolution,suchassecondCEejection.To
figureoutthenatureofJ0526,weemployedthestellarevolutioncode
MESA(refs.46,93–96)toinvestigateitsoriginandfinalfate.
Single-star evolution models. According to the observational clues
introducedabove,J0526probablyconsistsofaCOWDandalow-mass
sdBstar.ToconstructmodelsforthesdBstar,wefirstcreatedaseries
of He-pre-MS stars whose core masses ranged from 0.32 to 0.36 M⊙ in
steps of 0.01 M⊙ with an He-mass fraction of 0.98 and metallicity of

9. Nature Astronomy
Article https://doi.org/10.1038/s41550-023-02188-2
0.02. The nuclear reaction network adopted in our simulations was
approx21.net,whichisa21-isotopeα-chainnuclearnetwork.TheOPAL
type2radiativeopacitiestablesforacarbon-andoxygen-richmixture
with a base metallicity Z = 0.02 was employed in our simulations97,98
.
The adopted metallicity is consistent with that of Galactic thin-disk
membership.Somephysicalprocesseswerenotincludedinourmod-
els,suchasovershooting,rotation,diffusion,etc.
TogeneratetheheliumMS(HeMS)models(CHeBmodelswithout
aHenvelope),wecreatedaseriesofpre-MSmodelswithinitialHeabun-
dance0.98andmetallicity0.02(werefertothesemodelsasHe-pre-MS
models)andevolvedthemuntilcentral-Heignition.
TogeneratethesdBmodels(CHeBmodelswithaverythinHenve-
lope), we evolved the He-pre-MS models until their central tempera-
turesreached log Tc = 7.8,andthenturnedoffallthenuclearreactions
and implanted different masses of a pure H shell onto the surface of
theHecoreusingamass-accretionrateof10−7
M⊙ yr−1
.Themassofthe
H shell was in the range 10−6
to 10−2
M⊙. After the formation of the H
shell, we restored the nuclear reactions to evolve the sdB models for-
wardwithtime.Whenthenuclearreactionswererestored,allthesdB
models underwent Kelvin–Helmholtz contraction and, thus, ignited
thecentral-Heburning.Attheonsetofcentral-Heburning,theircentral
temperatures reached log Tc = 8.0, the central densities ranged from
log ρc = 4.6 to 4.8 and the central-He mass fractions were about 0.96.
sdB stars with core masses of 0.32–0.33 M⊙ and H-envelope mass of
10−6
M⊙ areconsistentwithJ0526B(Fig.4).
For the CHeB models with 0.32–0.33 M⊙, the stars will evolve
towardtheWDcoolingsequenceifthecentralHeisexhausted,whereas
othermodels(0.34–0.36 M⊙)experienceunstableshell-Heburningand
finally become a WD when the shell-He abundance cannot maintain
furtherHeburning.
Binary evolutionary models.ThestrongGWRfromJ0526willleadto
orbitalcontractionandRLOFofJ0526Binthefuture.Topredictitsfinal
fate,weevolvedbinarymodelswithaninitialorbitalperiodof20.5 min.
Inthesemodels,thedonorstarsarethezero-ageCHeBstarsobtained
above from single-star evolution. The accretor in these binaries is a
0.735 M⊙ COWD,whichistreatedasaparticleinthemodels.Tocalcu-
latetherateofchangeoftheorbitalangularmomentum,wetookinto
accountcontributionsfrombothGWRandmassloss.TheAMLdriven
byGWRisintroducedusingthestandardformula99
:
d ln JGW
dt
= −
32G3
5c5
MWDMsd(MWD + Msd)
a4
, (14)
where MWD and Msd are the masses of the WD accretor and sdB donor,
respectively.
MasstransferbeginsafterthedonorfillsitsRochelobe.However,
theWDcannotaccumulateallthematerialtransferredfromthedonor.
Moreover,somematerialislostfromtheWDthroughthestellarwind,
leading to the mass loss and AML of the system. The AML caused by
masslosscanbecalculatedasfollows:
̇
JML,WD = −(1 − ηHe) ̇
Macc(
aMsd
MWD + Msd
)
2
2π
Porb
, (15)
where ̇
Macc is the mass-accretion rate of the WD, which equals the
mass-transfer rate from the donor to the WD. Referring to previous
work100–103
,themass-accumulationefficiencyofaWDforhelium(ηHe)
canbeapproximatedas
ηHe =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
̇
Mcr,He
̇
MHe
, ̇
Macc > ̇
Mcr,He,
1, ̇
Mst,He ≤ ̇
Macc ≤ ̇
Mcr,He,
η′
He
, ̇
Mlow,He < ̇
Macc < ̇
Mst,He,
0, ̇
Macc ≤ ̇
Mlow,He,
(16)
where ̇
Mlow,He and ̇
Mcr,He are the lowest and highest critical accretion
rates,respectively,and ̇
Mst,He istheminimumaccretionrateforstable
He-shellburning.If ̇
Macc ≤ ̇
Mlow,He,alltheaccretedmaterialwillbeblown
away by the strong stellar wind, which means that the
mass-accumulation rate of the WD is almost negligible. If ̇
Mlow,He
< ̇
Macc < ̇
Mst,He,themass-accumulationefficiencyunderweakHe-shell
flashes (η′
He
) was adopted from the simulation results104
. For
̇
Mst,He ≤ ̇
Macc ≤ ̇
Mcr,He , the He-shell burning is stable, and thus, all
accretedmaterialisaccumulatedontothesurfaceoftheWD(ηHe = 1).
If ̇
Macc > ̇
Mcr,He, the WD accumulates material with its extreme rate of
̇
Mcr,He,andexcessmaterialisblownawaybytheopticallythickwind.
The sdB stars in our models begin to transfer material to the WD
aftertheirRLOF.Weevolvedthesemodelsuntiltheluminositiesofthe
donorswerebelow10−6
L⊙.AllthesemodelsexperienceAMCVnphases
and then the mass-transfer rates begin to decrease with increasing
orbital periods, as can be seen from Extended Data Fig. 4. The final
outcomeisprobablyaWD+planetsystem56,57
orasingleWDwhenthe
donoristidallydisruptedbytheaccretor58
.
Dataavailability
TheZTFg-andr-bandphotometrycanbeobtainedfromtheNASA/IPAC
Infrared Science Archive (https://irsa.ipac.caltech.edu). The optical
and infrared photometric fluxes in the SED can be obtained from the
VOSAonlinetool(http://svo2.cab.inta-csic.es/theory/vosa).Thebolo-
metric correction tables can be downloaded from MIST (http://waps.
cfa.harvard.edu/MIST/model_grids.html). All light curves, observed
and synthetic spectra, RV curve, photometric and synthetic fluxes
in the SED, and the stellar/binary evolutionary models used for this
work are available from our Zenodo page (https://www.zenodo.org/
record/8074854orhttps://doi.org/10.5281/zenodo.8074854).Source
dataareprovidedwiththispaper.
Codeavailability
The codes TLUSTY (v.207) and SYNSPEC (v.53) that were used for
generating (non-LTE) model atmospheres and producing synthetic
spectra are available at https://www.as.arizona.edu/∼hubeny, and
the services for online spectral analyses (XTGRID) are provided by
Astroserver(www.Astroserver.org).ThePythonpackageellc(v.1.8.7),
which was used for modelling light curves, can be obtained from
https://pypi.org/project/ellc.ThesensitivitycurveofLISAcanbecom-
puted using the codes from https://github.com/eXtremeGravityIn-
stitute/LISA_Sensitivity. The software MESA (v.12778) used for stellar
evolutionary calculations is available at http://mesastar.org, and the
full inlists for evolutionary models used for this work are available
from our Zenodo page (https://www.zenodo.org/record/8074854 or
https://doi.org/10.5281/zenodo.8074854).
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Acknowledgements
We acknowledge the support of the staff of the 10.4m GTC, Keck I
10m telescope, LJT and Swift/UVOT. The work of X.W. is supported
by the National Natural Science Foundation of China (NSFC; Grant
Numbers 12033003, 12288102 and 11633002), the Ma Huateng
Foundation, the New Cornerstone Science Foundation through the
XPLORER PRIZE, China Manned-Spaced Project (CMS-CSST-2021-A12)
and the Scholar Program of the Beijing Academy of Science and
Technology (DZ:BS202002). J. Lin is supported by the Cyrus Chun
Ying Tang Foundations. C.W. is supported by the NSFC (Grant Number
12003013) and the Yunnan Fundamental Research Projects (Grant
Number 202301AU070039). C.W., Z.H., X.C., Jujia Zhang and Y.C.
are supported by International Centre of Supernovae, Yunnan Key
Laboratory (Grant Number 202302AN360001). P.N. acknowledges
support from the Grant Agency of the Czech Republic (GAČR 22-
34467S). The Astronomical Institute in Ondřejov is supported by
project RVO:67985815. N.E.-R. acknowledges partial support from
the Research Projects of National Relevance (PRIN) for 2017 as funded
by the Italian Ministry of Education, University and Research (MIUR;
Grant Number 20179ZF5KS, The new frontier of the Multi-Messenger
Astrophysics: follow-up of electromagnetic transient counterparts
of gravitational wave sources), from the Italian National Institute for
Astrophysics (INAF) through PRIN-INAF 2022 (Shedding light on the
nature of gap transients: from the observations to the models), from
the Spanish Ministry of Science, Innovation and Universities (Grant
Number PID2019-108709GB-I00) and from the European Regional
Development Fund. I.S. is supported by funding from MIUR through
PRIN 2017 (Grant Number 20179ZF5KS) and PRIN-INAF 2022 (Shedding
light on the nature of gap transients: from the observations to the
models) and acknowledges the support of the doctoral grant funded
by Istituto Nazionale di Astrofisica through the University of Padova
and MIUR. A.V.F.’s group at the University of California, Berkeley, has
received financial assistance from the Christopher R. Redlich Fund,
Alan Eustace (W.Z. is a Eustace Specialist in Astronomy), Frank and
Kathleen Wood (T.G.B. is a Wood Specialist in Astronomy), Gary and
Cynthia Bengier, Clark and Sharon Winslow, and Sanford Robertson
(Y.Y. is a Bengier-Winslow-Robertson Postdoctoral Fellow), and
many other donors. This research is based on observations made
with the GTC, installed at the Spanish Observatorio del Roque de
los Muchachos of the Instituto de Astrofísica de Canarias on the
island of La Palma. This research is based on data obtained with
the instrument OSIRIS, which was built by a consortium led by
the Instituto de Astrofísica de Canarias in collaboration with the
Instituto de Astronomía of the Universidad Nacional Autónoma de
Mexico. OSIRIS was funded by GRANTECAN and the National Plan of
Astronomy and Astrophysics of the Spanish Government. Some of the
data presented herein were obtained at the W. M. Keck Observatory,
which is operated as a scientific partnership among the California
Institute of Technology, the University of California and the National
Aeronautics and Space Administration (NASA). The observatory was
made possible by the generous financial support of the W. M. Keck
Foundation. We acknowledge the target-of-opportunity observations
supported by the Swift Mission Operations Center. This research has
used the services of www.Astroserver.org under references T4JRRH
and Y75AKG and was based in part on observations obtained with
the Samuel Oschin 48inch Telescope at the Palomar Observatory as
part of the ZTF project. ZTF is supported by the US National Science
Foundation (NSF) under grant AST-1440341 and a collaboration
including Caltech, Infrared Processing and Analysis Center (IPAC), the
Weizmann Institute for Science, the Oskar Klein Center at Stockholm
University, the University of Maryland, the University of Washington,
Deutsches Elektronen-Synchrotron and Humboldt University, Los
Alamos National Laboratory, the TANGO Consortium of Taiwan,
the University of Wisconsin at Milwaukee and Lawrence Berkeley
National Laboratory. Operations were conducted by COO, IPAC and
the University of Wisconsin. This work has made use of data from the
European Space Agency’s Gaia mission (https://www.cosmos.esa.int/
gaia) processed by the Gaia Data Processing and Analysis Consortium
(DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium).
Funding for DPAC has been provided by national institutions, in
particular the institutions participating in the Gaia Multilateral
Agreement. The Pan-STARRS1 Surveys (PS1) and the PS1 public
science archive have been made possible through contributions by
the Institute for Astronomy, the University of Hawaii, the Pan-STARRS
Project Office, the Max-Planck Society and its participating institutes
(the Max Planck Institute for Astronomy, Heidelberg, and the Max
Planck Institute for Extraterrestrial Physics, Garching), Johns Hopkins
University, Durham University, the University of Edinburgh, Queen’s
University Belfast, the Harvard-Smithsonian Center for Astrophysics,
Las Cumbres Observatory Global Telescope Network Incorporated,
the National Central University of Taiwan, the Space Telescope
Science Institute, NASA (under Grant Number NNX08AR22G issued
through the Planetary Science Division of the NASA Science Mission
Directorate), NSF (Grant Number AST-1238877), the University of
Maryland, Eotvos Lorand University, Los Alamos National Laboratory,
and the Gordon and Betty Moore Foundation. This publication
makes use of data products from the WISE, which is a joint project
of the University of California, Los Angeles, and the Jet Propulsion
Laboratory/California Institute of Technology as funded by NASA.
This publication makes use of VOSA, developed under the Spanish
Virtual Observatory (https://svo.cab.inta-csic.es) project funded by
MCIN/AEI/10.13039/501100011033/ through Grant Number PID2020-
112949GB-I00. VOSA was partially updated using funding from the

13. Nature Astronomy
Article https://doi.org/10.1038/s41550-023-02188-2
European Union’s Horizon 2020 Research and Innovation Programme
(Grant Agreement Number 776403, EXOPLANETS-A).
Authorcontributions
J. Lin, C.W., H.X. and X.W. drafted the paper. Z.H. and A.V.F. edited
the paper in detail. P.N., N.E.-R., X.C., Y.C. and S.G. also helped with
the paper. X.W. is the principal investigator of TMTS and led the
discussions. J. Lin discovered this source by analysing a large volume
of data from TMTS observations and performed a detailed analysis
of the spectroscopy, SED, orbital dynamic and light curves. C.W.
computed the stellar and binary evolution models for low-mass sdB
stars. H.X. provided some key ideas for these models. P.N. determined
the atmospheric parameters from the GTC/OSIRIS spectra and
computed RVs from both the GTC/OSIRIS and Keck I/LRIS spectra.
J. Li and Q.X. helped with the analysis of SED and light curves. The
GTC/OSIRIS spectra were provided and reduced by N.E.-R. and I.S.
A.V.F., T.G.B., Y.Y. and W.Z. obtained and reduced the Keck I data. Jujia
Zhang obtained and reduced the high-cadence observations of the
LJT. S.G. computed the Galactic orbit. J. Liu reduced and analysed the
observations made by Swift/UVOT. S.Y., Y.C., J.G., D.X. and G.L. assisted
in the spectral observations and analysis. J. Lin, C.W., H.X., X.W., P.N.,
Z.H., J. Li, X.C., J.G., Q.X. and Z.L. contributed to beneficial discussions.
X.W., Jicheng Zhang, J.M., G.X. and J. Lin contributed to the building of
the TMTS and developing of its pipeline and database. G.X., J.M., J.G.,
Q.X., Q.L., F.G., L.C. and W.L. contributed to the operations and data
products of TMTS.
Competinginterests
The authors declare no competing interests.
Additionalinformation
Extended data Extended data are available for this paper at
https://doi.org/10.1038/s41550-023-02188-2.
Supplementary information The online version
contains supplementary material available at
https://doi.org/10.1038/s41550-023-02188-2.
Correspondence and requests for materialsshould be addressed to
Xiaofeng Wang.
Peer review information Nature Astronomy thanks Kevin Burdge and
the other, anonymous, reviewer(s) for their contribution to the peer
review of this work.
Reprints and permissions informationis available at
www.nature.com/reprints.
Publisher’s note Springer Nature remains neutral with regard to
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Springer Nature or its licensor (e.g. a society or other partner) holds
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the author(s) or other rightsholder(s); author self-archiving of the
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terms of such publishing agreement and applicable law.
© The Author(s), under exclusive licence to Springer Nature Limited
2024
1
Physics Department and Tsinghua Center for Astrophysics, Tsinghua University, Beijing, People’s Republic of China. 2
CAS Key Laboratory for Research
in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China, Hefei, People’s Republic of China. 3
School of
Astronomy and Space Sciences, University of Science and Technology of China, Hefei, People’s Republic of China. 4
Yunnan Observatories, Chinese
Academy of Sciences, Kunming, People’s Republic of China. 5
Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy
of Sciences, Kunming, People’s Republic of China. 6
International Centre of Supernovae, Yunnan Key Laboratory, Kunming, People’s Republic of China.
7
Research School of Astronomy & Astrophysics, Australian National University, Canberra, Australian Capital Territory, Australia. 8
Beijing Planetarium,
Beijing Academy of Sciences and Technology, Beijing, People’s Republic of China. 9
Astronomical Institute of the Czech Academy of Sciences, Ondřejov,
Czech Republic. 10
Astroserver.org, Malomsok, Hungary. 11
INAF-Osservatorio Astronomico di Padova, Padua, Italy. 12
Institute of Space Sciences (ICE, CSIC),
Campus UAB, Carrer de Can Magrans s/n, Barcelona, Spain. 13
Dipartimento di Fisica e Astronomia ‘G. Galilei’, Università degli Studi di Padova, Padua,
Italy. 14
Department of Astronomy, University of California, Berkeley, Berkeley, CA, USA. 15
School of Physical Science and Technology, Xinjiang University,
Urumqi, People’s Republic of China. 16
Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing, People’s Republic of China.
17
Department of Astronomy, Beijing Normal University, Beijing, People’s Republic of China. 18
National Astronomical Observatories, Chinese Academy
of Sciences, Beijing, China. 19
The School of Physics and Astronomy, Tel Aviv University, Tel Aviv, Israel. 20
These authors contributed equally: Jie Lin,
Chengyuan Wu, Heran Xiong. e-mail: wang_xf@mail.tsinghua.edu.cn

14. Nature Astronomy
Article https://doi.org/10.1038/s41550-023-02188-2
Extended Data Fig. 1 | TMTS light curve and Lomb-Scargle periodogram for
J0526. Upper panel: the TMTS L-band light curve over a 12 hr night on 18
December 2020. The magnitudes are presented as mean values ± 1σ. Middle
panel: a 3000 s subset of the TMTS L-band light curve. The solid red line
represents the best-fit sinusoidal model with a period of 10.3 min. Lower panel:
the Lomb-Scargle periodogram (LSP) computed from the TMTS light curve.
The vertical dashed line indicates the frequency corresponding to maximum
power (fmax). The purple dot-dashed line represents the confidence level of
0.1%, and the red arrow shows the frequency corresponding to the orbital
period (that is, fmax/2).

15. Nature Astronomy
Article https://doi.org/10.1038/s41550-023-02188-2
ExtendedDataFig.2|RelationbetweenHeabundanceandeffectivetemperatureforhotsubdwarfs.Allatmosphericparametersofhotsubdwarfsaretakenfrom
reference25
.Thepurpledashedlineandgreendotted-dashlinerepresentthecorrelationfortheHe-richsequence28
andtheHe-weaksequence29
,respectively.

16. Nature Astronomy
Article https://doi.org/10.1038/s41550-023-02188-2
ExtendedDataFig.3|ThecharacteristicstrainsofJ0526accompanied
withdozensofverification/detectablebinariesofGWs.Thecharacteristic
strainswerecalculatedfromthecomponentmassesanddistancesprovided
fromreference113
.Thebluedashedlineandreddotted-dashlinerepresentthe
detectionsensitivitycurvesfromLISA91
andTianQin92
,respectively.TheLISA
sensitivitycurvehereincludestheinstrumentalnoisetheforegroundconfusion
noise,whiletheTianQinsensitivitycurveincludesonlytheinstrumentalnoise.
Theerrorbarsrepresent68%credibleintervals.

17. Nature Astronomy
Article https://doi.org/10.1038/s41550-023-02188-2
ExtendedDataFig.4|Binaryevolutionmodelsforextremely-short-orbital-periodsdBbinaries.Twomodelsaredifferentiatedowingtothedifferentcoremasses
ofsdBstars.Masstransfersareexpectedtobeginataround14and17minforMsd =0.33M⊙ andMsd =0.36M⊙,respectively.Theredarrowsdenotethedirectionof
evolution.