# A Jovian analogue orbiting a white dwarf star

J.W. Blackman School of Natural Sciences, University of Tasmania, Private Bag
37 Hobart, Tasmania 7001 Australia  Sorbonne Universités, UPMC Université
Paris 6 et CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis bd Arago,
75014 Paris, France  J.-P. Beaulieu School of Natural Sciences, University of
Tasmania, Private Bag 37 Hobart, Tasmania 7001 Australia  Sorbonne
Universités, UPMC Université Paris 6 et CNRS, UMR 7095, Institut
d’Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France  D.P. Bennett
Laboratory for Exoplanets and Stellar Astrophysics, NASA/Goddard Space Flight
Center, Greenbelt, MD 20771, USA  Department of Astronomy, University of
Maryland, College Park, MD 20742, USA  C. Danielski Instituto de Astrofísica
de Andalucía (IAA-CSIC), Glorieta de la Astronomía s/n, 18008 Granada, Spain
UCL Centre for Space Exochemistry Data, Atlas Building, Fermi Avenue, Harwell
Campus, Didcot, OX11 0QR, UK  Sorbonne Universités, UPMC Université Paris 6 et
CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis bd Arago, 75014
Paris, France  C. Alard  Sorbonne Universités, UPMC Université Paris 6 et
CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis bd Arago, 75014
Paris, France  A.A. Cole School of Natural Sciences, University of Tasmania,
Private Bag 37 Hobart, Tasmania 7001 Australia  A. Vandorou School of Natural
Sciences, University of Tasmania, Private Bag 37 Hobart, Tasmania 7001
Australia  C. Ranc Laboratory for Exoplanets and Stellar Astrophysics,
NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA  S.K. Terry
Department of Astronomy, University of California Berkeley, Berkeley, CA
94701, USA.  A. Bhattacharya  Laboratory for Exoplanets and Stellar
Astrophysics, NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA
Department of Astronomy, University of Maryland, College Park, MD 20742, USA
I. Bond Institute for Natural and Mathematical Sciences, Massey University,
Private Bag 102904 North Shore Mail Centre, Auckland 0745, New Zealand  E.
Bachelet Las Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA
93117 USA  D. Veras Centre for Exoplanets and Habitability, University of
Warwick, Coventry CV4 7AL, UK  Department of Physics, University of Warwick,
Coventry CV4 7AL, UK  N. Koshimoto Laboratory for Exoplanets and Stellar
Astrophysics, NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA
Department of Astronomy, Graduate School of Science, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  V. Batista Sorbonne
Universités, UPMC Université Paris 6 et CNRS, UMR 7095, Institut
d’Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France  J.-B.
Marquette Associated with Sorbonne Universités, UPMC Université Paris 6 et
CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis bd Arago, 75014
Paris, France  Laboratoire d’astrophysique de Bordeaux, Univ. Bordeaux, CNRS,
B18N, allée Geoffroy Saint-Hilaire, 33615 Pessac, France

###### Abstract

Studies (Vanderburg et al., 2015; Manser et al., 2019) have shown that
remnants of destroyed planets and debris-disk planetesimals can survive the
volatile evolution of their host stars into white dwarfs (Villaver & Livio,
2007; Duncan & Lissauer, 1998), but detection of intact planetary bodies
around white dwarfs are few (Sigurdsson et al., 2003; Vanderburg et al., 2020;
Luhman et al., 2011; Gaensicke et al., 2019). Simulations predict (Madappatt
et al., 2016; Mustill et al., 2018; Nordhaus & Spiegel, 2013) that planets in
Jupiter-like orbits around stars of $\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}8M_{\odot}$ avoid being destroyed
by the strong tidal forces of their stellar host, but as yet there has been no
observational confirmation of such a survivor. Here we report on the non-
detection of a main-sequence lens star in the microlensing event
MOA-2010-BLG-477Lb (Bachelet et al., 2012) using near-infrared observations
from the Keck Observatory. We determine this system contains a $0.53\pm 0.11$
solar mass white dwarf host orbited by a $1.4\pm 0.3$ Jupiter mass planet with
a separation on the plane of the sky of $2.8\pm 0.5$ AU, which implies a semi-
major axis larger than this. This system is evidence that planets around white
dwarfs can survive the giant and asymptotic giant phases of their host’s
evolution, and supports the prediction that over half of white dwarfs are
predicted to have Jovian planetary companions (Schreiber et al., 2019).
Located at approximately $2.0\,$\mathrm{k}\mathrm{p}\mathrm{c}$$ toward the
center of our Galaxy, it likely represents an analog to the end stages of the
Sun and Jupiter in our own Solar System.

tablenum SIXtablenum

##

The microlensing event MOA-2010-BLG-477Lb was first detected by the
Microlensing Observations in Astrophysics collaboration on 2 August 2010
(Bachelet et al., 2012). Microlensing is a technique which is sensitive to
cold planets down to the mass of Earth (Bennett & Rhie, 1996) and can probe
objects around all kinds of stars, including white dwarfs, as unlike other
detection methods it does not rely on the light coming from the host.
MOA-2010-BLG-477 is a planetary microlensing event with a planet-host mass
ratio of $q=(2.61\pm 0.03)\times 10^{-3}$ and a large Einstein ring radius
$\theta_{E}=1.26\pm 0.06$ mas, which implies a host star that is relatively
massive or nearby.
The target was subsequently observed with the Near Infra-Red Camera-2 (NIRC2)
instrument on the Keck-II telescope on July 27, 2015, August 5, 2016 and May
23, 2018 in the $H$ and $K_{s}$ near-infrared bands ($1.5-2.3\,\mu$m) using
laser guide star adaptive optics. Data were obtained with the wide camera (40
arcsecond field-of-view) for calibration, and the narrow camera (10 arcsecond
field-of-view) in an attempt to resolve the source and lens. Assuming it sits
on the main sequence, we predict a host star with mass of
$0.15M_{\odot}<M_{*}<0.93M_{\odot}$, orbited by a planet with mass $0.5M_{\rm
Jup}<m_{p}<2.1M_{\rm Jup}$ at a distance of $0.7<D_{L}<2.7\,$kpc if all
stellar types are equally likely to host a planet. Modelled parameters of the
microlensing light curve, and the source magnitude and color (Boyajian et al.,
2014), yield a lens-source relative proper motion of $\mu_{rel,G}=11.53\pm
0.56\,$\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{/}\mathrm{y}\mathrm{r}$$. This
prediction, presented here in the geocentric reference frame that moves with
the Earth’s velocity at the time of the event, allows us to estimate the
future lens-source separation following the event’s peak on 12 August 2010.

Figure 1: H-band adaptive optics imaging of MOA 2010-BLG-477 from the KECK
observatory a. A crop of a narrow-camera H-band image obtained with the NIRC2
imager in 2015 centered on MOA 2010-BLG-477 with an 8 arcsec field of view. b.
A 0.36 arcsec zoom of the same image. The bright object in the center is the
source. To the north-east (the upper left) is an unrelated $H=18.52\pm 0.05$
star 123 mas from the source. c. The field in 2018. The contours indicate the
likely positions of a possible main sequence host (probability of 0.393,
0.865, 0.989 from light to dark blue) using constraints from microlensing
parallax and lens-source relative proper motion.

In our Keck images we find a $H=18.52\pm 0.05$ star located ${\sim}123\,$mas
to the north-east (upper-left in Fig. 1c) of the source. Given the large lens-
source relative proper motion, a predicted separation of
$57\,$\mathrm{m}\mathrm{a}\mathrm{s}$$ means the lens star should be
detectable under ideal observing conditions in 2015, 4.96 years after the
event peak. By 2018, this separation will have widened to
$90\,$\mathrm{m}\mathrm{a}\mathrm{s}$$. We find that the separation of the
${\sim}123\,$mas star to the north east has actually decreased by
$11.5\,$\mathrm{m}\mathrm{a}\mathrm{s}$$ between 2015 and 2018, instead of the
expected $33\,$\mathrm{m}\mathrm{a}\mathrm{s}$$ increase if it were the lens.
The direction of this proper motion indicates that this object is unrelated to
the lens and the planetary system.
The source magnitude was measured as $H_{s}=17.32\pm 0.03$ and $K_{s}=17.17\pm
0.04$ using data from the Cerro Tololo Inter-American Observatory (CTIO) and
the Vista Variables in the Via Lactea (VVV) survey. This brightness is
compatible with our NIRC2/KECK images, determined using the flux-ratio between
the unrelated ${\sim}123\,$mas companion and the star at the position of the
source. The sum of the two stars shown in Fig. 1c is $K=16.78\pm 0.02$ using
the Keck data and $K=16.79\pm 0.03$ using the CTIO data. This is consistent
with there being no excess flux within the point spread function (PSF) of the
source star. To confirm this, we model the PSF with the addition of weak
contributions from the wings of the unrelated companion, and find no
significant structure in the residuals. Hence, with no evidence of a
detectable lens in our Keck/NIRC2 images, the lens star must have a brightness
below the detection limit.
To determine this limit we define a detection to be above the noise at the
3$\sigma$ level. This corresponds to a threshold of H ${\sim}21.1$, which
means that any object brighter than this value should be detectable within our
Keck narrow frames. This limit includes an extinction correction of
$A_{H}=0.21$. The light curve of this event indicates a microlensing parallax
signal due to the orbital motion of the Earth, but it is not fully
constrained. This implies that for each light curve model in our Markov Chain
distribution, we can determine the lens mass
$M_{L}=(c^{2}/4G)(\theta_{E}/\pi_{E})$ (Gaudi, 2012), where $\pi_{E}$ is the
microlensing parallax. We combine these light curve constraints with limits on
the source and lens distances and velocities, together with an empirical mass-
luminosity relation (Bennett et al., 2014), and find that a main-sequence lens
star must have a brightness of $H<18.10$ at 99.0% confidence. This is largely
due to the light curve constraint on the angular Einstein ring radius,
$\theta_{E}$, derived from the angular radius of the source star and the
Einstein ring crossing time.

Figure 2: H-band brightness of possible main-sequence host lenses. In white is
the Einstein ring radius constraint, $\theta_{E}$, derived from finite source
effects in the event light curve. This constraint indicates that if the planet
host star was main-sequence, it should be visible with Keck adaptive optics as
the entire area lies above the detection threshold of our H-band images at
H${\sim}21.1$. The mass-luminosity relations for different main-sequence lens
masses are shown: 1.0$M_{\odot}$, 0.5$M_{\odot}$, 0.3$M_{\odot}$, with
0.16$M_{\odot}$ the mass lower limit derived from the mircrolens parallax,
$\pi_{E}$. Our null detection in our Keck images implies that the exoplanet
host must be a stellar remnant, most likely a white dwarf.

The microlensing parallax and lens-source relative proper motion measurements
constrain the location of a main sequence lens star to the interior of the
contours in Fig. 1c. The predicted brightness of a main sequence lens as a
function of lens distance can be see in Fig. 2. Since all possible main-
sequence lenses for the event are brighter than the Keck detection limit, and
no such star is observed, the lens cannot be a main-sequence star. This same
analysis also excludes brown dwarf lenses due to an upper limit on the
microlensing parallax parameter, $\pi_{E}<1.03$, which leads to an implied
limit on the lens system mass of $M_{L}>0.15M_{\odot}$. Similarly, the lower
microlensing parallax limit of $\pi_{E}>0.26$, implies an upper mass limit of
$M_{L}<0.78M_{\odot}$, which rules out neutron star and black hole host stars.
With main sequence stars, brown dwarfs neutron stars, and black holes ruled
out, we conclude that the lens must be a white dwarf.
To estimate the properties of a white dwarf host, we use a complete sample of
130 white dwarfs within 20 pc (Giammichele et al., 2012), excluding unresolved
double white dwarfs and double white dwarfs candidates identified by Toonen et
al. (2017), and the Galactic model from (Bennett et al., 2014). This
calculation was made under the assumption that all white dwarfs are equally
likely to host planets. Our results are summarized in Fig. 3 and Table. 1. We
find a likely white dwarf host mass $M=0.53\pm 0.11$ M⊙, which sits slightly
below the peak of the single white dwarf mass distribution at 0.57-0.58 M⊙ and
excludes the high mass tail (Tremblay et al., 2016). This implies a Jovian
planet of mass $m_{p}=1.43\pm 0.30$ MJ at a distance of $D_{L}=1.99\pm
0.35\,$\mathrm{k}\mathrm{p}\mathrm{c}$$.

Table 1: System Parameters of MOA 2010-BLG-477Lb Parameter | Units | Value ($\pm 1\sigma$) | 2$\sigma$ range
---|---|---|---
White Dwarf Lens Distance |  $D_{L}$ (kpc) | $1.99\pm 0.35$ |  $1.31$–$2.69$
White Dwarf Lens Mass | $M_{L}\;(M_{\odot})$ | $0.53\pm 0.11$ |  $0.32$–$0.74$
Planet Mass | $m_{p}\;(M_{J})$ | $1.43\pm 0.30$ |  $0.87$–$2.03$
Source Star Distance |  $D_{S}$ (kpc) | $7.8\pm 1.3$ |  $5.5$–$10.2$
2D star-planet separation |  $a_{\perp}$ (AU) | $2.8\pm 0.5$ |  $1.9$–$3.7$
3D star-planet separation |  $a$ (AU) | $3.4^{+1.8}_{-0.8}$ |  $2.1$–$12.1$
Mass Ratio | $q\;(10^{-3})$ | $2.61\pm 0.03$ |  $2.55$–$2.67$
White Dwarf Host $V_{L}$ | mag | $26.92\pm 1.80$ |  $23.38$–$29.44$
White Dwarf Host $J_{L}$ | mag | $24.93\pm 0.91$ |  $22.80$–$26.41$
Figure 3: Physical properties of MOA-2010-BLG-477 Presented here are the (a)
predicted distance to the host star/lens planetary system, (b) 3d star-planet
separation, (c) mass of the planet, (d) mass of the host, (e) predicted host
star brightness in V, and (f) predicted host star brightness in J. These
parameters were calculated using a Bayesian analysis with priors from the
relative proper motion of disk and bulge lenses, the lens star mass function,
and velocity distributions derived from a galactic model as in Bennett et al.
(2014)

With a separation from the white dwarf of $a=3.4^{+1.8}_{-0.8}\,$AU (assuming
a random orientation), it is likely that the planet MOA-2010-BLG-477Lb formed
at the same time as the host star and managed to survive the post-main-
sequence evolution. The mass loss experienced by a star on the giant and
asymptotic giant branches pushes the planet toward a wider orbit, but tidal
forces can have the opposite effect when the star expands to radii $\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}1\,$AU (Veras, 2016). In rare
cases, the tidal effect can nearly cancel the mass loss effect, leaving a
giant planet orbiting at a separation as small as $\sim 2\,$AU, but this
requires a fine tuning of parameters to prevent the planet from being engulfed
by the star. Most white dwarfs in the Galactic disk are thought to have formed
from stars with initial masses of 1–$2.5\,M_{\odot}$ (Veras, 2016), and the
measured separation means that to avoid tidal engulfment the mass of the
progenitor star is likely to be $<2\,M_{\odot}$. Jovian planets orbiting these
stars are generally thought to move to orbital separations $>5$ or $6\,$AU
around the remnant white dwarfs (Mustill & Villaver, 2012; Nordhaus & Spiegel,
2013). While this is larger than the $\sim 2.8$ AU projected separation of
MOA-2010-BLG-477Lb, in the region where microlensing has the highest
sensitivity, the reduced detection probability for white dwarf planets might
be compensated for by a higher intrinsic planet occurrence for gas giant
planets around massive stars (Ghezzi et al., 2018).

While Mercury and Venus will most likely be engulfed by the Sun when it
reaches the RGB phase, Jupiter is expected to survive (Veras, 2016).
MOA-2010-BLG-477Lb is an example of the possible fate of Jupiter and supports
the prediction that over half of white dwarfs are predicted to have Jovian
planetary companions (Schreiber et al., 2019). It joins 29 other planets
included in recent statistical sample of microlensing events (Suzuki et al.,
2016). This will be supplemented with detections made by the Galactic
Exoplanet Survey of the Nancy Grace Roman Space Telescope (Penny et al., 2019)
aimed at determining planet occurrence rates for white dwarf hosts.

## Methods

### Observations

The microlensing event MOA 2010-BLG-477 was originally detected using the 1.8m
telescope at Mt. John Observatory in New Zealand on August 2, 2010, and
subsequently observed by more than 20 telescopes (Bachelet et al., 2012). In
order to find the predicted magnitude of the source star we refer to the
H-band light curve obtained by the $\mu$FUN 1.3 m SMARTS telescope at the
Cerro Tololo Inter-American Observatory (CTIO). We calibrate the light curve
according to data from the VVV (Vista Variables in the Via Lactea) survey
(Vandorou et al., 2020). Using a single amplified CTIO H-band frame with an
epoch corresponding to available VVV data, we cross identify between the two.
We derive a source magnitude of $H_{CTIO,source}=17.32\pm 0.03$. Given the
intrinsic source color of $(H-K)_{0}=0.07$, and using extinction corrections
derived using the OGLE extinction calculator111OGLE Extinction Calculator,
http://ogle.astrouw.edu.pl/cgi-ogle/getext.py (Nataf et al., 2013),
$A_{H}=0.21$ and $A_{K}=0.13$, we find $K_{CTIO,source}=17.17\pm 0.04$.

Located at
($\alpha,\delta)=(18^{h}06^{m}07^{s}.47,-31^{\circ}27^{\prime}16.17\arcsec$,
J2000.0), we observed the event using the NIRC2 instrument on the Keck II
telescope located on Mauna Kea (Hawaii) with laser guide star adaptive optics
(LGSAO). Seven dithered H-band images were obtained with the narrow camera on
2015 July 27 (HJD = 2457230.7), 4.96 years following the event peak. These
images have a median full width at half maximum (FWHM) of 50 mas. During this
same epoch we obtained 14, 95 mas $K_{s}$-band images with the wide camera,
and 10, 50 mas $H$-band images with the narrow camera. The event was observed
twice more: in 2016 (HJD = 2457605.0) with 17 $H$-band 50 mas narrow images
and two $H$-band 90 mas wide, and in 2018 with 16 50mas $K_{s}$-band narrow
frames. These final 2018 observations were taken on 23 May 2018 (HJD =
2458262.4), 7.78 years after maximum magnification.

Extended Data Fig. 1: OGLE-III and Keck imaging of MOA-2010-BLG-477Lb. (a)
OGLE-III image of the OGLE-BLG176.8 field (b) H-band image of the same field
taken in 2015 with Keck/NIRC2 with the narrow camera.

The goal of these observations was to determine the composition of the source-
lens blend by obtaining a refined photometric and astrometric solution. We
apply flat-field and dark-field corrections using standard techniques
(Blackman et al., 2020; Beaulieu et al., 2016). The images were stacked using
SWARP (Bertin & Emmanuel, 2010). We identify the source+lens star as the
bright object in Fig. 1c. Comparing this with the OGLE-III reference image of
the BLG 176.8 field in Extended Data Fig. 1 we see that the OGLE star is a
blend of four dimmer stars, the second brightest of which is the source+lens
star. The OGLE star (number 119416) has an $I$-band magnitude of $I=17.446\pm
0.052$, which is consistent with the value of $I=17.443\pm 0.031$ from DoPhot
CTIO photometry (Bachelet et al., 2012).
We perform aperture photometry using SExtractor (Bertin & Arnouts, 1996) on
the wide K-band frame from 2015, using the narrow images to cross-calibrate
the narrow frames, and find a K-band magnitude of the blend at the position of
the source to be $K_{Blend}=16.78\pm 0.03$. The magnitude is the combination
of the flux from the source, the lens and the 123mas star located to the
north-east. Fortunately, data of this field was also captured as part of the
VVV survey while the event was still magnified. Data were obtained from the
Vista telescope in JHK at the epochs:

$\displaystyle\mathrm{{H},MJD-OBS=55423.1515}$ (1)
$\displaystyle\mathrm{{J},MJD-OBS=55423.1576}$ $\displaystyle\mathrm{{K},MJD-
OBS=55423.1546}$

We use these data to determine whether the blend at the position of the source
is consistent with the predicted source brightness. The image at the location
of the source in the VVV image is a combination of the source, the lens, the
$123\,$mas star to the north-east (upper left) and the star to the north-west
(upper right). The brightness of these unrelated stars in the PSF are
negligible when the source was amplified by 19.27 in H (where $\mathrm{MJD-
OBS=55423.15153784}$ and MJD = JD - 2400000.5). We compare the flux ratios
between the star at the position of the source and the $123\,$mas companion,
in the narrow Keck image and compare that to a calibrated wide Keck frame and
that determined by CTIO/VVV.
In our Keck wide image we crossmatch to stars in the VVV catalog (Minniti et
al., 2010). Both the wide and narrow cameras on NIRC2 result in an image with
dimensions of 1048 × 1048 pixels. The wide has a field of view (FOV) of 40
arcseconds while the narrow has a FOV of 10 arcseconds. The narrow images were
taken in sequence with a dither of 0.7 arcseconds. The flux ratio of between
the source star and the $123\,$mas companion is:

$\displaystyle\mathrm{{2018,K}=0.42}$ (2)
$\displaystyle\mathrm{{2015,H}=0.33}$ $\displaystyle\mathrm{{2016,H}=0.34}$

Using the flux ratio from the 2018 K data and the predicted source K
brightness of $K_{s}=17.17\pm 0.04$, we determine the sum of the source and
the $123\,$mas companion to be $K=16.79\pm 0.04$, which is compatible with the
the $K_{Blend}=16.78\pm 0.03$. determined from only the 2015 Keck wide co-
added frame. This indicates that the there is no additional flux at the
position of the source, and that all the photons from that object come from
the source.

### Light Curve Model

Since the publication of the MOA-2010-BLG-477Lb discovery paper (Bachelet et
al., 2012), the MOA group has developed detrending methods (Bennett et al.,
2012; Bond et al., 2017) that can remove systematic errors due to the color
dependence of atmospheric refraction. We expect a significant color dependent
differential refraction signal due to a bright main sequence star at $I=13.5$
that is only 3 arc seconds from the MOA-2010-BLG-477 source star, and we have
found a significant correction using the method of Bond (Bond et al., 2017).
The best fit model parameters from using this new data are shown in Extended
Data Table 1. Three of the parameters also apply to single lens models. These
are the Einstein radius crossing time, $t_{E}$, the time of closest approach,
$t_{0}$ between the source and the lens center-of-mass, and the distance of
this closest approach, $u_{0}$, in units of the Einstein radius. A binary lens
requires three additional parameters, the lens separation, $s$, in units of
the Einstein radius, the angle between the lens separation and source
trajectory, $\alpha$, and the planet-star mass ratio, $q$. Events like
MOA-2010-BLG-477 with caustic crossings require the source radius crossing
time, $t_{*}$, to account for finite source effects. Finally, it is also
important to include parameters to describe the orbital motion of the lens
system and the Earth-based observers. The orbital period of the planet is many
years, so the only detectable effect of orbital motion is the relative lens
velocity, $\dot{s}_{x}$ and $\dot{s}_{y}$, in the directions parallel and
perpendicular to the lens separation. The microlensing parallax is described
by the North and East components of the microlensing parallax vector,
${\mbox{\boldmath$\bf\pi$}_{E}}$. The inclusion of microlensing parallax
introduces an approximate degeneracy that corresponds to a flip in the
orientation of the lens system with respect to the orientation of the Earth’s
orbit. We label the solutions with the sign of the $u_{0}$ parameter. The
$u_{0}>0$ model is favored by $\Delta\chi^{2}=5.08$, and it is displayed in
Extended Data Fig. 5, but both models are compatible with our overall
conclusions.

Extended Data Fig. 2: Light Curve Data and Model for microlensing event
MOA-2010-BLG-477. The solid curve is the best fit model and the dashed grey
curve is the single lens model with the same single lens parameters. The
different colors represent different data sets from different telescopes. One
sigma error bars are shown. The data sets are explained in the discovery paper
(Bachelet et al., 2012). Extended Data Table. 1: Best Fit Model Parameters

| This Analysis | Discovery Paper
---|---|---
parameter | $u_{0}>0$ | $u_{0}<0$ | $u_{0}>0$ | $u_{0}<0$
$t_{E}$ (days) | $39.523(425)$ | $39.796(441)$ | $46.868$ | $47.560$
$t_{0}$ (${\rm HJD}^{\prime}$) | $5420.9381(3)$ | $5420.9376(3)$ | $5420.9382$ | $5420.9374$
$u_{0}$ | $0.003966(51)$ | $-0.003937(54)$ | $0.003430$ | $-0.003315$
$s$ | $1.12203(52)$ | $1.12278(51)$ | $1.12407$ | $1.12429$
$\alpha$ (rad) | $0.54126(176)$ | $-0.54137(174)$ | $0.54004$ | $-0.53836$
$q\times 10^{3}$ | $2.590(31)$ | $2.587(31)$ | $2.210$ | $2.168$
$t_{\ast}$ (days) | $0.02736(18)$ | $0.02739(18)$ | $0.02719$ | $0.02719$
$\dot{s}_{x}$ | $0.00039^{+0.00021}_{-0.00030}$ | $0.00030^{+0.00023}_{-0.00027}$ | $0.00223$ | $0.00016$
$\dot{s}_{y}$ | $-0.00192^{+0.00099}_{-0.00146}$ | $0.00274^{+0.00042}_{-0.00194}$ | $-0.00587$ | $0.00196$
$\pi_{\rm E,N}$ | $0.4450^{+0.0750}_{-0.1499}$ | $-0.5393^{+0.4053}_{-0.1524}$ | $0.5432$ | $0.0974$
$\pi_{\rm E,E}$ | $-0.3165^{+0.0723}_{-0.0250}$ | $-0.1467^{+0.0420}_{-0.0766}$ | $-0.1026$ | $-0.0134$
$\chi^{2}/{\rm dof}$ | $6420.91/6490$ | $6425.97/6490$ | - | -

Note. — All errors are given as $\pm 1\sigma$.

One major difference between the analysis in the discovery paper (Bachelet et
al., 2012) and our new analysis is that the Einstein ring crossing time,
$t_{E}$, has dropped by about 16%. This causes a similar increase in the mass
ratio, $q$, because the timescale of the planetary signal has not changed. The
other significant change in the model parameters can be found in the
microlensing parallax values. As shown in Extended Data Fig. 5a, only one
component of the microlensing parallax vector,
${\mbox{\boldmath$\bf\pi$}_{E}}$, is tightly constrained by the data. It is
quite common for the component parallel to the direction of the Earth’s
acceleration at the time of the event to be measured much more precisely than
the perpendicular component, and in this case the uncertainty in the
perpendicular component is increased by the degeneracy between microlensing
parallax and lens orbital motion. This parallel component is roughly in the
the East-West direction for events in the Galactic bulge. However, the full
${\mbox{\boldmath$\bf\pi$}_{E}}$ can be determined (Bhattacharya et al., 2018;
Bennett et al., 2020) with a measurement of the lens-source relative proper
motion, $\bm{\mu}_{\mathrm{rel,G}}$, although this must be be done in the
Geocentric frame that has been used for the light curve modeling.
While the changes in the model due to the improved MOA data are quite
noticeable, they have very little effect on our conclusions. We have performed
the full analysis using the light curve models from the discovery paper
(Bachelet et al., 2012), and these implied very similar results. The only
difference is that the old analysis shifted the $\pi_{E,E}$ toward more
positive values by 0.1–0.2. As a result, the old light curve analysis allowed
higher mass host stars.
We note that the discovery paper suggested that it might be sensible to search
for three body lens systems that could explain this light curve. However,
unlike the case of published triple lens events (Gaudi et al., 2008; Bennett
et al., 2010; Gould et al., 2014; Bennett et al., 2016), this light curve
shows no evidence of an unmodeled light curve feature that can be explained by
an additional lens. Thus, we can expect that the only effect of an additional
lens would be to increase the uncertainties in the properties of the lens
masses in the binary lens model (Zhu et al., 2014), usually by very small
amounts. Since our primary conclusions are based on the properties of the host
star, an additional lens that is consistent with the data would not affect our
conclusions.

### Bayesian Analysis Light Curve Models with a Galactic Prior

Our analysis makes use of the light curve models from the Markov Chain Monte
Carlo calculations from our reanalysis of the MOA data with our detrending
method (Bond et al., 2017), as well as other data sets from the discovery
paper (Bachelet et al., 2012). These models incorporate the effects of the
orbital motion of the Earth which are responsible for the microlensing
parallax, as well as the orbital motion of the planet. The microlensing
parallax is a two-dimensional vector, ${\mbox{\boldmath$\bf\pi$}_{E}}$,
parallel to the lens-source relative proper motion, which means the
distribution of microlensing parallax vectors from the light curve models
constrains the direction of the lens-source relative proper motion. The
amplitude of the lens-source relative proper motion vector is determined by
finite source effects in the microlensing light curve. However, the direction
and length of this lens-source relative proper motion vectors are not
determined in the Heliocentric reference frame that is appropriate for high
angular resolution Keck follow-up observations. The light curve models employ
a Geocentric reference frame that moves with the velocity of the Earth at the
time of closest lens-source alignment. For MOA-2010-BLG-477, this velocity is
$\bm{v}_{\earth N,E}=(-2.7933,19.5634)\>\mathrm{km\>s^{-1}}$, and the
transformation to the relative proper motion in the Heliocentric reference
frame, $\bm{\mu}_{\mathrm{rel,H}}$, is given by

$\displaystyle\bm{\mu}_{\mathrm{rel,H}}=\bm{\mu}_{\mathrm{rel,G}}+\frac{\bm{v}_{\oplus\mathrm{N,E}}}{\mathrm{AU}}\pi_{\mathrm{rel}}\
,$ (3)

(Dong et al., 2007; Bhattacharya et al., 2018), where the relative parallax is
given by $\pi_{\rm rel}=\mathrm{AU}/D_{L}-\mathrm{AU}/D_{S}$. This
transformation from $\bm{\mu}_{\mathrm{rel,G}}$ to $\bm{\mu}_{\mathrm{rel,H}}$
cannot be computed with light curve parameters only as $\pi_{\mathrm{rel}}$
depends on the source and lens distances. Therefore, we must invoke a Galactic
model to properly sample the source distance, $D_{S}$, values. We have used
the Galactic model from Bennett et al. (2014) in our analysis. Our results are
largely independent of the choice of Galactic model for this Bayesian
analysis. The only poorly understood prior is the host mass dependence of the
planet hosting probability, since we are considering a lens that is known to
host a planet. For simplicity, we assume that all white dwarfs are equally
likely to host a planet. Once the source distance is selected, the lens
distance can be determined from light curve parameters,

$D_{L}={{\rm AU}\over\pi_{\rm E}\theta_{\rm E}+{\rm AU}/D_{S}}\ .$ (4)

Extended Data Fig. 3: White Dwarf Mass-Luminosity distribution derived a
sample of 130 white dwarfs from a homogeneous and complete sample of white
dwarfs within 20pc of the Sun Giammichele et al. (2012). Two unresolved
double-white dwarfs (DWD), eight unresolved DWD candidates and one unresolved
binary white dwarf with a main-sequence companion have been removed from this
sample (Toonen et al., 2017). 14 stars with distances $>20\,$pc have also been
removed. The white dots indicate the masses and $V$ band magnitudes of the
white dwarfs in this sample, and the color distribution indicates the smooth
Gaussian multivariate kernel-density distribution that we have used in our
analysis. Extended Data Fig. 4: Cumulative $\Delta\chi^{2}$ comparing a static
light curve model with that including parallax and orbital motion. The bulk of
the signal comes following the light curve peak. The parallax plus orbital
motion comes primarily from the MOA data ($\Delta\chi^{2}=45$) and the SAAO
data ($\Delta\chi^{2}=9.0$). Extended Data Fig. 5: Predictions of the
microlens parallax vector $\bm{\pi}_{E}$ and the corresponding predicted
relative lens-source proper motion $\bm{\mu}_{rel}$ for a main sequence and
white dwarf lens. Based on a Markov-Chain Monte-Carlo (MCMC) analysis using
Galactic model priors as in Bennett et al. (2014), the upper panels (a) and
(b) show the unweighted predicted components of $(\pi_{EN},\pi_{EE})$ and
$(\mu_{rel,HN},\mu_{rel,HE})$. The middle panels (c) and (d) show the weighted
predictions for a main-sequence lens. The lower panels (e) and (f) show the
weighted predictions for a white-dwarf lens. The three shades of blue from
dark to light denote probabilities of of 0.393, 0.865, 0.989. When integrating
over all parameters the limit of the 0.393 contour corresponds to the
1$\sigma$ distribution of any chosen parameter. Extended Data Fig. 6: H-band
adaptive optics imaging from the KECK observatory, with contours showing the
predicted position of a white dwarf lens (analogous to Fig. 1) (a) A crop of a
narrow-camera H-band image obtained with the NIRC2 imager in 2015 centered on
MOA 2010-BLG-477 with an 8 arcsec field of view. (b) A 0.36 arcsec zoom of the
same image.The bright object in the center is the source. To the north-east
(the upper left) is an unrelated $H=18.52\pm 0.05$ star $123\,$mas from the
source (star 123NE. (c) The field in 2018. The contours indicate the likely
positions of the white dwarf host (probability of 0.393, 0.865, 0.989 from
light to dark blue) using constraints from microlensing parallax and lens-
source relative proper motion.

While $\pi_{\rm E}$ is only partially constrained by the light curve data, it
is precisely determined for each model in the Markov Chain, so equation 4 can
be used for each Markov Chain model. The Galactic model also provides weights
for the density of stars at the distance to the lens, $D_{L}$, and by the
probability of a star with a mass equal to the lens mass. This implicitly
includes the assumption that all stars are equally likely to host the planet
of the measured mass ratio and projected separation.
Extended Data Fig. 5 shows how the Galactic model affects the distributions of
the $\bm{\pi}_{\mathrm{E}}$ and $\bm{\mu}_{\mathrm{rel,H}}$ vectors. Panels 5a
and 5b show the $\bm{\pi}_{\mathrm{E}}$ distribution from the light curve
models in (Bachelet et al., 2012) and the implied $\bm{\mu}_{\mathrm{rel,H}}$
distribution, with the help of the $D_{S}$ distribution from the (Bennett et
al., 2014) Galactic model. The component of the $\bm{\pi}_{\mathrm{E}}$
parallel to the direction of the Earth’s acceleration during the event is
tightly constrained. This component is close to the East direction. Conversely
the perpendicular component, which is largely in the North-South direction is
very weakly constrained. Panels c and d of Extended Data Fig. 5 show the
result when we apply the complete Galactic model including the mass function
for main sequence stars. This removes the light curve models with small
$\pi_{\mathrm{E}}$ values, and therefore, large masses. It was these
low-$\pi_{\mathrm{E}}$, high-mass light curve models that allowed the
$\bm{\pi}_{\mathrm{E}}$ vector to point in any direction. This was responsible
for the ring distribution of $\bm{\mu}_{\mathrm{rel,H}}$, but with the high
mass lens systems excluded, this ring is broken into two arcs to the North and
South, with low-mass lens systems only allowed in the Northern arc. Our Keck
observations have ruled out any main sequence stellar lenses in these arcs.
Our model does assume that stars with masses above $1.1\,M_{\odot}$ have left
the main sequence, but such stars would also be brighter than the source star,
which are clearly ruled out over the full ring in Extended Data Fig. 5b ring.

With main sequence hosts ruled out, we can now turn our attention to white
dwarf host stars for the MOA-2010-BLG-477Lb exoplanet. We can repeat the same
calculation that we did for main sequence sources. This requires a mass-
luminosity distribution for white dwarfs. We construct such a distribution
from the 20 pc sample as in (Giammichele et al., 2012), excluding unresolved
binary white dwarfs because these are likely to have unreliable parameters.
The resulting white dwarf mass-luminosity relation, constructed using a
multivariate Gaussian kernel density-estimation is shown in Extended Data Fig.
3. The results of a repeat of the Bayesian analysis with the main sequence
mass function replaced by our white dwarf mass function, from Extended Data
Fig. 3, is shown in Extended Data Fig. 5 panels e and f. The results are quite
similar to the main sequence case, but the exclusion of
low-$\pi_{\mathrm{E}}$, high-mass light curve models is now somewhat stronger
because the white dwarf mass function is strongly peaked around $M_{L}\sim
0.57-0.58M_{\odot}$. This Bayesian analysis is also used to produce the lens
system properties presented in Table 1. The contours from Extended Data Fig.
5d are reproduced in Extended Data Fig. 6c, where they replace the contours
for the main sequence stars shown in Fig. 1c.

### Detection Limits and Point Spread Function Deconvolution

We determine the detection threshold of our Keck images by estimating the flux
of a point source and evaluating the normalized cross-product with the point-
spread function (PSF). We obtain this quantity for all points in the
subtracted image and construct a map of the amplitudes. In order to obtain an
estimate of the fluctuations due to the noise we calculate the standard
deviation in this map. We consider that these fluctuations are significant at
the 3$\sigma$ level and that this 3$\sigma$ level is our detection limit. We
then convert this 3$\sigma$ limit to magnitude and obtain a minimum limit of
detection of $H~{}\simeq$ 21.1.

While the relative proper motion indicates that the lens is distinct from the
PSF of the source star, we then perform an analysis to determine if the star
at the position of the source is consistent with a single object or if there
is evidence of a two component system. To do this we make a numerical estimate
of the PSF by stacking the brightest stars in the neighborhood of our target.
The accurate position of each PSF is estimated by iterative Gaussian weighted
centering. The PSFs are then interpolated, re-centered and stacked, and median
solution obtained. In our H-band images the star of interest is quite close to
the star to the north-east, which means it may receive some contribution from
the PSF wings of its neighbour. To subtract any neighbour contributions we
reconstruct the wings of the PSF of the more distant star as a single function
of distance and subtract this weak contribution from the image. Then the best
solution obtained is subtracted, leaving only the stars of interest. The
result of this subtraction and the residuals are presented in Extended Data
Fig. 7.

Extended Data Fig. 7: Keck Point Spread Function (PSF) fit and residuals. (a)
Keck/NIRC2 H-band image from 2018. (b) Residuals after fitting the PSF using
multiple stars in the neighbourhood of the source. Both the object to the
north-east (upper left in panel A) and the object at source position (lower
right) are subtracted using this PSF fit. There is no structure or indication
of a double star in either of the two objects. (c) The residuals from panel B
normalized to the Poisson noise. (d) Panel A but subtracting the fitted PSF
from the unrelated companion.

When reconstructing a single star a common problem is to determine whether
this star is a single PSF or a very close system of multiple PSFs. If the
single PSF fit results in significant residuals, for example, it is clear that
the fit of a binary system should be attempted. In this case constraints on
the PSF can be used to find upper limits on allowed separations that are
compatible with the data and noise. Our method to achieve this and to overcome
the degeneracy created when the PSF components are very close are described
below.

The image as shown in Extended Data Fig. 7, $I_{C}$ is the result of the
convolution of the object data, $I_{0}$ with the PSF, $\phi$. Let’s consider
the case where the un-convolved data $I_{0}$ is very narrow with respect to
the PSF, which is typical in cases of two close-together stars. In this
instance, we can write:

$I_{C}(x,y)=\int\phi(x-u,y-v)I_{0}(u,v)dudv$ (5)

Considering that the variations are small with respect to the variations of
$\phi,$ we can write $\phi$ as a local expansion in the local variables
$(u,v)$:

$I_{C}(x,y)\simeq m_{0}\phi(x,y)-m_{1}\frac{\partial\phi}{\partial
x}-m_{2}\frac{\partial\phi}{\partial y}+m_{3}\frac{\partial^{2}\phi}{\partial
x^{2}}+m_{4}\frac{\partial^{2}\phi}{\partial x\partial
y}+m_{5}\frac{\partial^{2}\phi}{\partial y^{2}}$ (6)

where:

$\begin{cases}m_{0}=\int I_{0}(u,v)dudv\\\ m_{1}=\int I_{0}(u,v)u\;dudv\\\
m_{2}=\int I_{0}(u,v)v\;dudv\\\ m_{3}=\frac{1}{2}\int I_{0}(u,v)u^{2}dudv\\\
m_{4}=\int I_{0}(u,v)uv\;dudv\\\ m_{5}=\frac{1}{2}\int I_{0}(u,v)v^{2}dudv\\\
\end{cases}$ (7)

The $m_{1}$ and $m_{2}$ coefficients represent the degrees of freedom related
to the centered of the function $I_{0}$. To eliminate these degrees of freedom
we make the center of flux of $I_{0}$ coincide with the origin of the
coordinate system. In this case, $m_{1}=0$ and $m_{2}=0$. As such we are left
with an expansion with 4 basis functions, the PSF and its 3 second order
derivatives. The moments of the function $I_{0}$ are simply the 3 coefficients
$\left(m_{3},m_{4},m_{5}\right)$ normalized by the total flux (Note that
provided that the PSF is normalized the total flux and the coefficient $m_{0}$
should be very similar).

We use the numerical model of the PSF to reconstruct the derivatives up to the
second order. The derivatives are obtained by shifting the PSF model and
taking the difference with the original PSF. The value of the shift is small
with respect to the size of the PSF grid. We choose a value of 0.01, but also
tested 0.1 and 0.001 and no significant changes were observed. In creating the
PSF model we take an area around the object large enough to include the PSF
wings but small enough to avoid including another object.

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## Data availability

The KECK Observatory data used in this study are freely available on the Keck
Observatory Archive (https://koa.ipac.caltech.edu/cgi-bin/KOA/nph-KOAlogin).
Data from the VISTA Variables in the Via Lactea (VVV) survey are available on
the European Southern Observatory archive
(http://archive.eso.org/wdb/wdb/adp/phase3_main/form?phase3_collection=VVV&release_tag=6).
Data used to model the light curve are available from the authors on request.

## Code availability

The KECK pipeline is available on Github
(https://github.com/blackmanjw/KeckPipeline). The Bayesian analysis code of
D.P. Bennett uses routines from (Press et al., 1992) which are subject to
restricted availability.

## Acknowledgements

Data presented in this work was obtained at the W. M. Keck Observatory from
telescope time allocated to the National Aeronautics and Space Administration
through the agencies scientific partnership with the California Institute of
Technology and the University of California. Funding: This work was supported
by the University of Tasmania through the UTAS Foundation, ARC grant
DP200101909 and the endowed Warren Chair in Astronomy. We acknowledge the
support of ANR COLD WORLDS (ANR-18-CE31-0002) at the Institut d’astrophysique
de Paris and the Laboratoire d’astrophysique de Bordeaux. D.P.B., A.B., N.K.,
C.R. and S.K.T. were supported by NASA through grant NASA-80NSSC18K0274 and by
NASA award number 80GSFC17M0002. E.B. gratefully acknowledges support from
NASA grant 80NSSC19K0291. Work by N.K. is supported by JSPS KAKENHI Grant
Number JP18J00897. C.D. acknowledges financial support from the State Agency
for Research of the Spanish MCIU through the “Center of Excellence Severo
Ochoa" award to the Instituto de Astrofísica de Andalucía (SEV-2017-0709), and
the Group project Ref. PID2019-110689RB-I00/AEI/10.13039/501100011033. D.V.
gratefully acknowledges the support of the STFC via an Ernest Rutherford
Fellowship (grant ST/P003850/1).

## Author contributions

J.W.B. led the photometric and formal analysis and wrote the manuscript.
J.W.B, V.B. and J.-P.B. took and reduced the photometric data using a pipeline
written by J.W.B. and A.V. with contributions from J.-B.M. for magnitude
calibrations. D.P.B., J.-P.B and A.A.C discussed conceptual and analysis
approaches. D.P.B. was the PI of the KECK telescope proposal, led the planning
of the observations, and conducted the light curve modeling and Bayesian
analyses. C.D. provided insight into single and double white dwarf planetary
systems. A.B. and E.B. assisted with proper motion calculations. A.B. and N.K
assisted with observing on Keck. C.R. calculated the parallax, proper motion
and lens prediction contours. C.A. worked on the PSF analysis and the
determining of detection limits. D.P.B., J.-P.B., A.A.C., C.D., S.T. and D.V.
contributed to the review and editing of the manuscript.

## Competing interests

The authors have no competing interests to declare.

Correspondence and requests for materials should be addressed to Joshua
Blackman (joshua.blackman@utas.edu.au).