# A re-analysis of the isolated black hole candidate
OGLE-2011-BLG-0462/MOA-2011-BLG-191

Casey Y. Lam Casey Y. Lam<EMAIL_ADDRESS>University of California,
Berkeley, Department of Astronomy, Berkeley, CA 94720  Jessica R. Lu
University of California, Berkeley, Department of Astronomy, Berkeley, CA
94720

###### Abstract

There are expected to be $\sim 10^{8}$ isolated black holes (BHs) in the Milky
Way. OGLE-2011-BLG-0462/MOA-2011-BLG-191 (OB110462) is the only such BH with a
mass measurement to date. However, its mass is disputed: Lam et al. (2022a, b)
measured a lower mass of $1.6-4.4M_{\odot}$, while Sahu et al. (2022); Mróz et
al. (2022) measured a higher mass of $5.8-8.7M_{\odot}$. We re-analyze
OB110462, including new data from the Hubble Space Telescope (HST) and re-
reduced Optical Gravitational Lensing Experiment (OGLE) photometry. We also
re-reduce and re-analyze the HST dataset with newly available software. We
find significantly different ($\sim 1$ mas) HST astrometry than Lam et al.
(2022a, b) in the de-magnified epochs due to the amount of positional bias
induced by a bright star $\sim$0.4 arcsec from OB110462. After modeling the
updated photometric and astrometric datasets, we find the lens of OB110462 is
a $6.0^{+1.2}_{-1.0}M_{\odot}$ BH. Future observations with the Nancy Grace
Roman Space Telescope, which will have an astrometric precision comparable or
better to HST but a field of view $100\times$ larger, will be able to measure
hundreds of isolated BH masses via microlensing. This will enable the
measurement of the BH mass distribution and improve understanding of massive
stellar evolution and BH formation channels.

##  1 Introduction

Although massive stars, the progenitors of black holes (BHs), are typically
born in binaries, the majority of the Milky Way’s $10^{7}-10^{9}$ BHs are
expected to be isolated (Fender et al., 2013; Wiktorowicz et al., 2019; Olejak
et al., 2020). Around 20-30% of O-stars are expected to merge and form a
single, even more massive star (Sana et al., 2012), and many of the remaining
binary systems are disrupted before, during, or after the formation of the BH
due to natal kicks or mass loss. Despite this, nearly all known Galactic BHs
are in binary systems (Corral-Santana et al., 2016; Thompson et al., 2019; El-
Badry et al., 2023a; Chakrabarti et al., 2022; El-Badry et al., 2023b). This
detection bias exists because, unlike BH binaries, isolated BHs do not have a
companion that can electromagnetically identify their presence, making them
particularly elusive. Detecting and characterizing isolated BHs is a critical
first step needed to understand the full Galactic BH population.

Gravitational lensing is the most practical way to detect isolated BHs, as the
observational signature depends only on the mass of the lens, and not its
luminosity. In particular, _microlensing_ , the regime of gravitational
lensing where the images are unresolved, provides a way to find and measure
the masses of dark objects. As a foreground lens (e.g. a BH) aligns in front
of a background source of light (e.g. a Bulge star), this causes the
background source to temporarily brighten and change its apparent position;
the transient brightening is called _photometric microlensing_ and the
transient change in position is called _astrometric microlensing_. The
combination of the photometric and astrometric signals can be used to measure
the mass, distance, and proper motion of the lens (Hog et al., 1995; Miyamoto
& Yoshii, 1995; Walker, 1995). For more details on astrometric microlensing,
see Dominik & Sahu (2000).

###  1.1 An isolated dark compact object found with microlensing

OGLE-2011-BLG-0462/MOA-2011-BLG-191 (hereafter OB110462) is the first
isolated, dark compact object to have its mass measured with astrometric
microlensing. It was identified as a microlensing event toward the Galactic
Bulge located at (17:51:40.19, -29:53:26.3) and has been observed both
photometrically and astrometrically in order to measure the lens’ mass.
However, the nature of OB110462’s lens is disputed. Sahu et al. (2022)
inferred the lens to be a $M_{L}=7.1\pm 1.3M_{\odot}$ dark object, making
OB110462 a firm BH detection similar in mass to other known Galactic BHs in
binary systems. Lam et al. (2022a, b) inferred a lower mass object, which
depending on the modeling, led to a $M_{L}=2.15^{+0.67}_{-0.54}M_{\odot}$ or
$M_{L}=3.79^{+0.62}_{-0.57}M_{\odot}$ dark object, implying a neutron star or
low-mass BH. Both groups analyzed slightly different subsets of high-cadence
ground-based photometry and high-resolution Hubble Space Telescope (HST)
astrometry; see Mróz et al. (2022) for a summary. In particular, despite
analyzing the same astrometric data, both groups derived different stellar
positions. In addition, they both found the microlensing parameters inferred
from the ground-based photometry were in tension with the parameters inferred
by their respective astrometric measurements.

Mróz et al. (2022) re-analyzed the ground-based OGLE photometry of OB110462.
They found systematics in the photometry resulting from imperfect image
subtraction due to variations in seeing. Updated modeling using the revised
data showed that the OGLE photometry could be self-consistently modeled with
the astrometry of Sahu et al. (2022), and Mróz et al. (2022) inferred a lens
mass of $M_{L}=7.88\pm 0.82M_{\odot}$.

Mereghetti et al. (2022) combined new and archival Chandra imaging to search
for X-ray emission from OB110462. No X-rays were detected, and based on the
detection upper limits, concluded OB110462 could be consistent with an
accreting isolated BH with low radiative efficiency. A neutron star moving
slowly or in a high density environment would be disfavored, but uncertainties
in the velocity and environment density, as well as the accretion efficiency,
did not allow stronger statements to be made on the nature of OB110462.

###  1.2 Rationale for re-analysis of OB110462

Since publication of the initial discovery and modeling papers of OB110462 in
July 2022, there have been several new developments. First, as mentioned in
§1.1, there are updated ground-based photometry data from OGLE. There are also
two additional HST data points for OB110462. OB110462 was one of 70 targets in
a HST snapshot program to image microlensing events (SNAP-16716; PI: K. Sahu);
observations of OB110462 were taken in May 2022. In addition, the second and
final epoch of a Cycle 29 program (GO-16760; PI: C. Lam) to obtain OB110462
astrometry was taken in September 2022. With regards to analysis tools, an
updated version of the software used in the extraction of the astrometry from
the HST data (hst1pass, Anderson, 2022) was released in July 2022.

The ability to find and characterize isolated BHs is necessary to understand
the evolution and death of massive stars. In turn, massive stars impact our
understanding of a wide range of astrophysical problems, from the high-mass
end of the stellar initial mass function, to chemical evolution, to galactic
feedback. Without understanding the properties of isolated BHs, these problems
cannot be solved. Thus, a re-analysis of OB110462 is a timely and worthwhile
pursuit.

The remainder of this paper is outlined as follows. §2 lists the new and
updated data used in this re-analysis of OB110462. §3 describes the new and
updated capabilities of the hst1pass software and their effect on the
measurement of source positions and magnitudes. §4 describes how these updated
measurements are used to derive an updated astrometric time series, and §5
describes how both the updated photometry and astrometry are fit with a
microlensing model. §6 presents the lens’ properties, compares them to
previous studies of OB110462, and shows that the choice of software
significantly affects the astrometry and in turn the lens mass. §7 discusses
OB110462 is the context of the known Galactic BH population and considers
future searches for BHs. §8 provides a summary and conclusions.

##  2 Observations

###  2.1 HST

11 epochs of HST observations of OB110462 were obtained between 2011 and 2022,
and are presented in Table 2.1. This includes all the data analyzed in Lam et
al. (2022a, b) (see Table 2 in Lam et al. (2022b)), with several additions.
First, all exposures taken on 2011-08-08 were included in this analysis
(compared to Lam et al. (2022a, b), who excluded 3 frames that had different
exposure times than the main dataset). The exposures from 2013-05-13 was also
included (compared to Lam et al. (2022a, b), who excluded them because there
were no other Spring epochs to calibrate reference frame alignment issues due
to parallax; however there is now a second Spring epoch that can be used to
perform this calibration). Finally, the new HST observations taken in 2022
have been included. Note that although the 2022-09-13 GO data was taken with
the UVIS2-2K2C-SUB subarray like the previous datasets, the 2022-05-29 SNAP
data was taken with the UVIS2-C1K1C-SUB subarray, which is a smaller subarray
(1k$\times$1k, vs. 2k$\times$2k).

Table 1: HST data analyzed

Epoch | PA | Filter | $T_{exp}$ | $N_{im}$
---|---|---|---|---
(UT) | (deg) |  | (sec) |
2011-08-08 | 270.0 | F606W | 75.0 | 3
2011-08-08 | 270.0 | F606W | 60.0 | 1
2011-08-08 | 270.0 | F814W | 75.0 | 3
2011-08-08 | 270.0 | F814W | 60.0 | 1
2011-08-08 | 270.0 | F814W | 120.0 | 1
2011-10-31 | 276.1 | F606W | 280.0 | 3
2011-10-31 | 276.1 | F814W | 200.0 | 4
2012-09-09 | 269.5 | F606W | 290.0 | 3
2012-09-09 | 269.5 | F814W | 190.0 | 4
2012-09-25 | 271.3 | F606W | 280.0 | 3
2012-09-25 | 271.3 | F814W | 200.0 | 4
2013-05-13 | 99.9 | F606W | 280.0 | 3
2013-05-13 | 99.9 | F814W | 200.0 | 4
2013-10-22 | 274.6 | F606W | 285.0 | 3
2013-10-22 | 274.6 | F814W | 285.0 | 4
2014-10-26 | 275.2 | F606W | 265.0 | 3
2014-10-26 | 275.2 | F814W | 265.0 | 4
2017-08-29 | 268.3 | F606W | 250.0 | 3
2017-08-29 | 268.3 | F814W | 250.0 | 4
2021-10-01 | 272.0 | F606W | 407.0 | 5
2021-10-01 | 272.0 | F814W | 307.0 | 6
2022-05-29 | 107.9 | F814W | 300.0 | 2
2022-09-13 | 269.9 | F606W | 407.0 | 5
2022-09-13 | 269.9 | F814W | 307.0 |

Note. — HST data. For each epoch, the position angle (PA), HST WFC3-UVIS
filter, exposure time $T_{exp}$, and number of images $N_{im}$ are listed.
_Bold text_ indicates data used in this re-analysis that was not used in Lam
et al. (2022a, b).

###  2.2 OGLE

As mentioned in §1, the ground-based OGLE photometry was re-reduced by Mróz et
al. (2022). Figure 1 shows the difference between the old and new lightcurve.
In addition, Mróz et al. (2022) found that data from the first half of 2010
was affected by systematics due to commissioning of a new camera, and removed
this data from their analysis. They also only modeled data through 2016 as
they found a potential systematic in old OGLE reductions around HJD = 2458000
(September 2017). For consistency, we also model the same subset of re-reduced
OGLE data, spanning HJD = 2455376 to 2457700 (roughly July 2010 - November
2016).

Figure 1:  Difference between the OGLE photometry used in Mróz et al. (2022)
as to that used in Lam et al. (2022a, b); Sahu et al. (2022). _Left:_ full
lightcurve. _Right:_ lightcurve zoomed into the first three years (photometric
peak year $\pm$ 1 year).

##  3 Updated astrometric reductions and analysis with hst1pass

The software package hst1pass extracts precise astrometry from HST WFC3-UVIS
imaging. It is described in Anderson & King (2006), and although it has been
updated and used in many publications over the years, never formally released
as Space Telescope Science Institue-supported software. Lam et al. (2022a, b)
and Sahu et al. (2022) used one of these unofficial hst1pass releases to
perform their astrometric analyses.

In July 2022 hst1pass was officially released (Anderson, 2022). Most notably,
this release included a tabular correction for charge transfer efficiency
(CTE, Anderson, 2021) that could be used instead of a pixel-based CTE
correction (Anderson et al., 2021), and new functionality to perform
artificial star injection and recovery simulations.

We note that the official release of hst1pass also comes with a new routine
called hst2collate to collate the starlists of the individual frames together
into a final starlist for that epoch. However, because this is a limited-use
early version of hst2collate, we find the flexibility of the existing software
routines xym2mat and xym2bar (Anderson & King, 2006) to be superior. Hence, we
do not use hst2collate in our analysis and do not discuss it further here.

###  3.1 CTE correction

Lam et al. (2022a, b) and Sahu et al. (2022) used the CTE-corrected flat-
fielded HST images (i.e. flc) for their analyses. The flc images were produced
using Version 2.0 of a pixel-based CTE correction algorithm (Anderson et al.,
2021). However, the pixel-based correction usually under-corrects the CTE
effect on photometry (Kuhn & Anderson, 2021).

The updated version of hst1pass includes a tabular correction that empirically
corrects for CTE based on the brightness of the source and the sky background,
which improves the extraction of photometry and astrometry (Anderson, 2021).

Both CTE correction methods alter the extracted source positions in the
detector $y$ direction, which is the parallel readout direction. At present,
CTE in the detector $x$ direction, which is the serial readout direction, is
not corrected. Although there is CTE in the serial readout direction, it is
negligibly small compared to CTE in the parallel readout direction (Anderson,
2014).

When performing the data reduction in this work, instead of reducing the CTE
pixel-corrected flc files, we instead reduce the flat-fielded data files (i.e.
flt) with hst1pass using the tabular CTE correction. Measured positions
between these two methods can differ by a tenth of a pixel and measured
brightness can differ by a tenth of a magnitude (Figure 2.1 in Appendix A). We
then proceed with the data reduction and intra-epoch alignment process
described in §4.1 of Lam et al. (2022b) using the tabular-CTE corrected
starlists.

###  3.2 Artificial star injection and recovery tests

OB110462 is located $\sim$10 pixels ($\sim$0.4 arcsec) away from an unrelated
neighbor star that is 3 magnitudes brighter. This neighbor star biases the
measurement of the flux and position of OB110462. Lam et al. (2022a, b) and
Sahu et al. (2022) took different approaches to calculate this bias.

Sahu et al. (2022) used 18 nearby isolated stars with color and magnitude
comparable to the neighbor to construct an “extended model” point spread
function (PSF). This extended model PSF was then subtracted from each exposure
to obtain an unbiased position and magnitude of OB110462. Using this method,
Sahu et al. (2022) found that the typical positional bias for OB110462 was
about 1.2 mas.

Lam et al. (2022a, b) performed injection and recovery tests to measure the
bias. They injected sources around an isolated star of similar brightness to
the neighbor, at the same azimuth, separation, and magnitude difference as the
neighbor-OB110462 pair. Using this method, Lam et al. (2022b) found a smaller
positional bias of around 0.3 and 0.5 mas for OB110462, in the F606W and F814W
filters, respectively.

In §3.2.1 we describe the new hst1pass software used to perform source
extraction for artificial stars and in §3.2.2, we present an updated and more
extensive star-planting analysis using the new hst1pass.

####  3.2.1 hst1pass vs. ks2 software

The version of hst1pass used by Lam et al. (2022a, b) did not have a method to
generate artificial stars in the images. Thus, Lam et al. (2022a, b) used a
different software package, called ks2, to perform the injection and recovery
tests. ks2 has not been formally released, but it is described in several
papers, e.g. Anderson et al. (2008); Bellini et al. (2018); Sabbi et al.
(2016).

Although both hst1pass and ks2 are used to extract precise astrometric
measurements from HST imaging, they work in slightly different manners. ks2
was specifically designed to find fainter sources than hst1pass. There are
also certain implementation differences across the two software packages. Of
relevance to the astrometry are the geometric distortion solutions used, and
the specifics of the PSF fitting. With regard to the distortion solution, ks2
has an internal geometric distortion solution that is slightly different from
the standard geometric distortion correction (STDGDC) files used by hst1pass.
With regard to the PSFs, the specific manner of fitting slightly differ, e.g.
the particulars of how outlier rejection is implemented.

In Lam et al. (2022a, b), hst1pass was used instead of ks2 for data reduction
because recovery depth was not an issue and hst1pass had been more robustly
used and tested on HST WFC3-UVIS data. To use hst1pass to obtain the
positions, and then use ks2 to calculate the bias in the position and flux of
OB110462 is formally inconsistent, since the two methods of source extraction
in these software differ. As a validation test, hst1pass and ks2 were used to
extract astrometry from the same epoch of HST observations and then the
differences between the resultant starlists were compared. No clear trends
were found to explain the differences. Since the bias correction is a relative
measurement and self-consistent across one software package, it was deemed a
reasonable approach to calculate the bias correction with ks2 and then apply
it to measurements made with hst1pass. However, now that the new version of
hst1pass has the ability to simulate artificial stars, the injection and
recovery analysis can be done in a fully self-consistent manner and this
assumption can be checked.

####  3.2.2 Updated methodology

Figure 2:  Spatial distribution of neighbor-like stars. The image shows a
$36\times 36$ arcsec area centered on OB110462. The black circle in the center
shows the location of OB110462’s bright neighbor. The four other colored
circles show the location of the neighbor-like stars.  Figure 3:  Location of
neighbor-like stars on an HST CMD. The black circle marks OB110462’s bright
neighbor. The four other colored circles mark the neighbor-like stars (colors
correspond to those in Figure 2). The black star marks OB110462 at baseline
(i.e. unmagnified).  Figure 4:  Measurement bias of a source in F814W (_left_)
and F606W (_right_) due to proximity to a bright star. The bias correction is
defined as the recovered minus injected value. The amount of positional bias
is shown decomposed into radial $r$ (_top row_) and azimuthal $\theta$
(_middle row_) components . The radial direction is defined by the
OB110462-neighbor star separation vector, and the azimuthal direction is
measured counterclockwise from the separation vector. The amount of magnitude
bias is shown in the _bottom row_. The colored points show the mean and
standard deviation of the measurement bias for the four different neighbor-
like stars; the colors correspond to those in Figure 2 and Figure 3. The black
points are the mean and standard deviation of the mean measurement bias of the
four neighbor-like stars. The positional and magnitude bias in F606W is
smaller than that in F814W by about a factor of two; this is not surprising
since at shorter wavelength the angular resolution is higher.  Table 2: Bias
correction derived from injection and recovery

F814W 2011-08-08 -0.21 $\pm$ 0.05 0.15 $\pm$ 0.07 0.26 -0.003 $\pm$ 0.001
2011-10-31 -0.73 $\pm$ 0.02 0.00 $\pm$ 0.07 0.73 -0.026 $\pm$ 0.002 2012-09-09
-1.39 $\pm$ 0.18 0.55 $\pm$ 0.17 1.49 -0.060 $\pm$ 0.015 2012-09-25 -1.50
$\pm$ 0.18 0.31 $\pm$ 0.13 1.53 -0.057 $\pm$ 0.003 2013-05-13 -1.72 $\pm$ 0.12
-0.47 $\pm$ 0.05 1.78 -0.052 $\pm$ 0.005 2013-10-22 -1.63 $\pm$ 0.08 0.05
$\pm$ 0.05 1.63 -0.053 $\pm$ 0.012 2014-10-26 -1.81 $\pm$ 0.10 0.11 $\pm$ 0.03
1.81 -0.053 $\pm$ 0.006 2017-08-29 -1.23 $\pm$ 0.13 0.53 $\pm$ 0.10 1.34
-0.053 $\pm$ 0.002 2021-10-01 -1.41 $\pm$ 0.05 0.15 $\pm$ 0.05 1.42 -0.039
$\pm$ 0.006 2022-05-29 -0.74 $\pm$ 0.06 -0.10 $\pm$ 0.07 0.74 -0.009 $\pm$
0.012 2022-09-13 -1.35 $\pm$ 0.07 0.27 $\pm$ 0.04 1.37 -0.054 $\pm$ 0.022

Note. — Bias correction derived from injection/recovery around a star of
comparable brightness at the same separation, azimuth, and magnitude
difference as OB110462 to its bright neighbor. The bias correction is defined
as the recovered minus the true injected value.

Here, we update the analysis performed in Lam et al. (2022b). We briefly
summarize the methodology here and only highlight new changes; see Appendix B
of Lam et al. (2022b) for full details.

We calculate the bias in the position and flux of OB110462 by injecting
artificial stars using hst1pass at the same azimuth and separation as OB110462
and its bright neighbor star, around “neighbor-like” stars. We then determine
whether these injected artificial stars are recovered, and if they are, how
different the recovered and injected positions and fluxes are.

The criteria for selecting nearby isolated neighbor-like stars are:

  * •

similar brightness to neighbor (within $\pm$ 0.5 mag in both F814W and F606),

  * •

similar color to neighbor (within $\pm$ 0.25 mag in F606W $-$ F814W),

  * •

nearby to neighbor (within $\pm$20 arcsec = 500 pix)

  * •

isolated from other stars (at least 0.4 arcsec = 10 pix away from any other
source detected by hst1pass in F814W).

There are four stars that fit all these criteria. Their positions in relation
to OB110462 are shown in Figure 2 and on a color-magnitude diagram (CMD) in
Figure 3.

45 artificial stars are injected in a $0.2\times 0.2$ pixel ($=8\times 8$ mas)
area adjacent to each of the four neighbor-like stars, with a magnitude so
that the artificial star has the same contrast with the neighbor-like star as
OB110462 to the neighbor. The results of the injection and recovery are shown
in Figures 4 and 12 (c.f. Figures 22 and 23 in Lam et al. (2022b)) and listed
in Table 3.2.2 (c.f. Table 16 in Lam et al. (2022b)).

##  4 Updated cross-epoch alignment

We take the starlists for epoch obtained in §3.1 and align them onto a common
reference frame, as described in §4.2 - 4.2.3 of Lam et al. (2022b). The
photometry is calibrated as described in §4.3 of Lam et al. (2022b); in short,
to obtain precise relative photometry, we calculate and apply a small
magnitude offset that assumes the reference stars are constant brightness. One
minor change in this work as compared to Lam et al. (2022b) is the value of
the additive error added in quadrature to the positional and magnitude
uncertainties. Appendix A of Lam et al. (2022b) describes an empirical
methodology to calculate the rescaling factor, which can vary epoch to epoch.
We instead simply use a constant additive error across all epochs; the value
of the additive error is chosen to make the alignment residuals follow the
expected $\chi^{2}$ distribution. We find an additive error of 0.25 mas and 12
milli-mag added to the F606W positions and magnitudes, and 0.10 mas and 8
milli-mag in F814W, produce acceptable $\chi^{2}$ distributions by eye (Figure
5, c.f. Figure 9 of Lam et al. (2022b).). Figures 6 and 7 show the 10
astrometric reference stars closest to OB110462.

The final result of the cross-epoch alignment is the HST photometric and
astrometric time series (Figures 8 and 9). The bias correction measured in
§3.2.2 is then applied to OB110462 to obtain its true positions and magnitudes
(Table 4). Note that in Table 4 the reported uncertainties do not include the
uncertainties in the transformation from a relative astrometric reference
frame to the absolute Gaia reference frame, which are 0.13 mas/yr and 0.11
mas/yr in RA and Dec, respectively.

Table 3: HST calibrated data

Note. — Relative positions and magnitudes of OB110462.

Figure 5:  _Top:_ Histogram of $\chi^{2}$ residual values to linear proper
motion fits (no parallax) of the reference stars. _Bottom:_ Histogram of
$\chi^{2}$ residual values to constant magnitude vs. time fits of the
reference stars. The _left_ column shows the reference stars for F606W; the
_right_ column shows the reference stars for F814W. $N$ denotes the number of
reference stars. Figure 6:  5 reference stars nearest to OB110462. The title
indicates the separation of the reference star from OB110462 in RA, Dec, and
total distance, and the magnitude of the star in F814W and F606W. In the
individual panels, data in the F606W (F814W) filters are shown in _blue_
(_red_). _1st column:_ Trajectory on sky. Note the scales across different
rows vary. The positions are relative to the $\Delta$RA and $\Delta$Dec
offsets, which are relative to OB110462. The best-fit linear trajectory is
shown in _gray_. Note these velocities are in a reference frame where the mean
velocity of reference stars is 0, and not the Gaia reference frame. _2nd
column:_ Residuals to the best-fit linear trajectory in RA. The dashed lines
are the $1\sigma$ uncertainties to the best-fit line. _3rd column:_ Same as
2nd column, but for Dec instead of RA. _4th column:_ Histogram of position
residuals to the best-fit linear trajectory, in units of sigma. The solid gray
is for RA; the outlined black is for Dec. _5th column:_ Residuals to the best-
fit constant magnitude. F814W and F606W are fit to the respective magnitudes
in the title; the dashed lines show the $1\sigma$ uncertainties to the best-
fit magnitude. _6th column:_ Residuals to the best-fit constant magnitude, in
units of sigma. The solid blue is for F606W; the outlined red is for F814W.
Figure 7:  Same as Figure 6, but for the next 5 reference stars nearest to
OB110462.

##  5 Modeling using updated data

Table 4: Posterior distributions of fit parameters

Note. — The columns list the median $\pm 1\sigma$ (68%) credible intervals,
maximum a posteriori (MAP) solution, and and maximum likelihood estimator
(MLE) solution for the microlensing model parameters.

Figure 8:  _Top panel_ : HST F814W _red triangles_ , HST F606W _black squares_
, and detrended OGLE lighturve _blue_ lightcurves, with the corresponding
maximum likelihood model (MLE, described in §5) plotted over the data. _Second
from top panel_ : Residuals to the MLE model. The Gaussian Process (GP) model
is plotted on top of the OGLE residual. _Second from bottom panel_ : Same as
top panel, but zoomed into the three most magnified years (2010-2012). _Bottom
panel_ : Same as second from top panel, but zoomed into the three most
magnified years (2010-2012). Figure 9:  OB110462 astrometry. _Left column, top
to bottom_ : RA vs. time with maximum likelihood (MLE) unlensed source motion
model subtracted; residuals to the MLE model for RA vs. time fit. HST F814W
astrometry data is shown in red; HST F606W astrometry data is shown in blue.
The MLE model is shown in black. Fifty random draws from the posterior
distribution are shown in light gray. _Middle column, top to bottom_ : Same as
left column, except Dec instead of RA. _Right panel_ : astrometry as seen on-
sky, in the barycentric frame.

Next, we fit the re-reduced OGLE data and updated HST data following the
procedure outlined in Lam et al. (2022a, b) for their “default weight” (DW)
fit. See §5 of Lam et al. (2022b) for details on the model parameters and
modeling framework.

Simultaneously fitting the photometry and astrometry is very time-intensive,
so to speed up the process, we first simultaneously fit the OGLE and HST
photometry, accounting for correlated noise while fitting the OGLE photometry
with a Gaussian Process. The joint photometric and astrometric geometric
parameters are $t_{0}$, $u_{0}$, $t_{E}$, $\pi_{E,E}$, and $\pi_{E,N}$. The
photometric-only parameters are $b_{SFF,O}$, $m_{base,O}$, $b_{SFF,H8}$,
$m_{base,H8}$, $b_{SFF,H6}$, and $m_{base,H6}$. The astrometric-only
parameters are $x_{S0,E}$, $x_{S0,N}$, $\pi_{S}$, $\log_{10}\theta_{E}$,
$\mu_{S,E}$, and $\mu_{S,N}$. The Gaussian process parameters are
$\log\sigma_{0,O}$, $\rho_{O}$, $\log\omega^{4}_{0,O}S_{0,O}$, and
$\log\omega_{0,O}$. See §3 of Lam et al. (2022a), and §5, §5.1 and Appendix F
of Lam et al. (2022b) for a full description of all these parameters.

We then take the posterior distributions for $t_{0}$, $u_{0}$, $t_{E}$,
$\pi_{E,E}$, and $\pi_{E,N}$ from the photometry fit, and use them as priors
when fitting the HST astrometry. Note the correlations between the 5
parameters are preserved when using them as a prior in the astrometry fit. Our
priors are listed in Table B in Appendix B.

##  6 Results

The best-fit photometry and astrometry models are shown in Figures 8 and 9,
and the posteriors are listed in Table 5. We find that that the lens of
OB110462 has a mass of $M_{L}=6.03^{+1.19}_{-1.04}M_{\odot}$, is at a distance
$D_{L}=1.72^{+0.32}_{-0.23}$ kpc, and has transverse velocity
$v_{T,L}=37.61^{+5.12}_{-5.13}$ km/s.

OB110462 cannot be a a high-mass star, and is thus a BH. In Lam et al. (2022a,
b), they rule out any possibility of a luminous lens for OB110462, for a lens
mass of $M_{L}=3.8M_{\odot}$ and source flux fractions $b_{SFF}=0.9$ and 0.94
in F814W and F606W, respectively. Here, we find $M_{L}=6.0M_{\odot}$ and
source flux fractions $b_{SFF}=0.98$ in both F814W and F606W. In both Lam et
al. (2022a, b) and this work, the lens is $D_{L}=1.7$ kpc away. In this work,
the mass is significantly higher and the source flux fractions are also
slightly higher. This means the constraint on a dark lens is much stronger–
given the lens is at some fixed distance, a higher mass star would be much
brighter, and a higher source flux fraction would mean there is less excess
flux that such a star could hide in. Thus, without any shadow of a doubt,
OB110462 is a BH.

In the next sections we compare these results to those of Lam et al. (2022a,
b), Sahu et al. (2022), and Mróz et al. (2022). Table 6.1 gives a short
summary of the differences in the data and models used to analyze OB110462
across these works. We first compare the inferred lens properties in §6.1,
evaluate the goodness-of-fits of the astrometric models in §6.2, then compare
the modeled astrometric time series in §6.3 to understand the reasons for the
differences in the inferred lens properties.

###  6.1 Comparison of inferred microlensing parameters

We compare the inferred lens mass, distance, transverse velocity, and proper
motion of OB110462 to Lam et al. (2022a, b), Sahu et al. (2022), and Mróz et
al. (2022) in Table 6.1 and Figure 10. Mróz et al. (2022) provides two sets of
$D_{L}$ and $v_{T,L}$ depending on the source distance assumed. The two
solutions are consistent with each other to $1\sigma$; here we only compare to
their results using $D_{S}=8.8\pm 1.4$ kpc. In addition, Sahu et al. (2022) do
not report their uncertainties on the lens transverse velocity; we estimate it
to be $\sim 5.5$ km/s based on the reported lens proper motion and distance
uncertainties.

The lens mass of OB110462 inferred in this work
$M_{L}=6.03^{+1.19}_{-1.04}M_{\odot}$ is consistent with the measurement of
Sahu et al. (2022) $M_{L}=7.1\pm 1.3M_{\odot}$ to 1$\sigma$, and consistent
with the Lam et al. (2022a, b) DW model $M_{L}=3.79^{+0.62}_{-0.57}M_{\odot}$
and Mróz et al. (2022) $M_{L}=7.88\pm 0.82M_{\odot}$ measurement to 2$\sigma$
(in different directions). Our uncertainties are likely larger due to using
wider priors for the astrometry; Mróz et al. (2022) state they use uniform
priors in their modeling, but do not state the support.

The lens distance $D_{L}=1.72^{+0.32}_{-0.23}$ kpc inferred in this work is
consistent with the measurements of Lam et al. (2022a, b) DW model
$D_{L}=1.67^{+0.26}_{-0.20}$ kpc, Sahu et al. (2022) $D_{L}=1.58\pm 0.18$ kpc,
and Mróz et al. (2022) $1.62\pm 0.15$ kpc to 1$\sigma$.

The lens transverse velocity $v_{T,L}=37.61^{+5.12}_{-5.13}$ km/s inferred in
this work is consistent with Mróz et al. (2022) $v_{T,L}=43.4\pm 3.8$ km/s and
Sahu et al. (2022) $v_{T,L}\sim 45\pm 5.5$ km/s to 1$\sigma$. It is consistent
with the Lam et al. (2022a, b) DW model $v_{T,L}=23.95^{+2.95}_{-2.95}$ km/s
to 2$\sigma$.

The lens proper motion
$(\mu_{L,E},\mu_{L,N})=(-3.80^{+0.48}_{-0.55},2.60^{+0.83}_{-0.80})$ mas/yr
inferred in this work is consistent with Mróz et al. (2022)
$(\mu_{L,E},\mu_{L,N})=(-4.48\pm 0.39,3.29\pm 0.5)$ mas/yr and Sahu et al.
(2022) $(\mu_{L,E},\mu_{L,N})=(-4.36\pm 0.22,3.06\pm 0.66)$ mas/yr to
1$\sigma$ in RA and Dec. The proper motion inferred by the Lam et al. (2022a,
b) DW model
$(\mu_{L,E},\mu_{L,N})=(-2.64^{+0.18}_{-0.24},1.46^{+0.63}_{-0.71})$ mas/yr is
discrepant to this work $\sim 3\sigma$ in RA, and consistent to this work
$\sim 2\sigma$ in Dec (the measurements in Dec have larger uncertainties than
in RA).

The direction of the lens-source relative proper motion $\varphi$ is not a
property of the lens itself, but was an important point of comparison across
previous work so we consider it here. It is defined in Sahu et al. (2022) as
the position angle of the lens-source relative proper motion in equatorial
coordinates. In this work, we find $\varphi=343.75^{+4.80}_{-3.95}$ deg,
consistent with the measurements of Mróz et al. (2022) $\varphi=342.5\pm 4.9$
deg and Sahu et al. (2022) $\varphi=342.3\pm 3.0$ deg to $1\sigma$. The value
of Lam et al. (2022a, b) from the DW model $\varphi=355.47^{+2.66}_{-2.11}$ is
discrepant at $>2\sigma$ from this work.

The photometric and astrometric measurements independently constrain
$\varphi$. Modeling the re-reduced OGLE photometry alone yields
$\varphi=345.1\pm 3.7$ deg (Mróz et al., 2022). Modeling the re-reduced HST
astrometry presented here alone measures $\varphi=326.9^{+11.5}_{-11.0}$ deg.
The updated photometry and astrometry are consistent with each other to
$<2\sigma$.

In general, the properties of the lens of OB110462 inferred in this work are
somewhat discrepant with the Lam et al. (2022a, b) DW model, and in reasonable
agreement with Sahu et al. (2022) and Mróz et al. (2022). We also note that
they are inconsistent with the Lam et al. (2022a, b) EW model; the
measurements of $M_{L}=2.15^{+0.67}_{-0.54}M_{\odot}$,
$D_{L}=0.92^{+0.38}_{-0.22}$ kpc, $v_{T,L}=7.26^{+4.88}_{-4.88}$ km/s,
$(\mu_{L,E},\mu_{L,N})=(-0.69^{+0.91}_{-0.94},1.53^{+1.21}_{-1.12})$, and
$\varphi=18.08^{+8.60}_{-8.31}$ are discrepant $>3\sigma$ in mass, lens-source
relative proper motion, and lens proper motion in RA and $>4\sigma$ in
transverse velocity with this work.

Figure 10:  Comparison of lens mass $M_{L}$, distance $D_{L}$, transverse
velocity $v_{T,L}$, proper motion vector $(\mu_{L,E}$, $\mu_{L,N}$), and lens-
source relative proper motion direction $\varphi$ inferred from various
studies of OB110462.  Table 5: Summary of data and models in different
analyses

Note. — _OGLE data_ : “Original” is the original OGLE reduction of OB110462
used in Lam et al. (2022a, b); Sahu et al. (2022); “Updated” is the OGLE re-
reduction of OB110462 presented in Mróz et al. (2022). _HST data_ : the
timespan listed indicates the years of HST data used. _Likelihood_ : “DW” is
the likelihood used in performing the model fit that weights each data point
equally; “EW” is the likelihood that weights the OGLE photometry, HST
photometry, and HST astrometry equally.

Table 6: Comparison of lens properties

Note. — Comparison of lens mass $M_{L}$, distance $D_{L}$, transverse velocity
$v_{T,L}$, proper motion vector $(\mu_{L,E}$, $\mu_{L,N}$), and lens-source
relative proper motion direction $\varphi$ inferred from various studies of
OB110462.

###  6.2 Goodness-of-fits

Figure 11:  CDF of normalized residuals for astrometric models and data, as
compared to the CDF of a standard normal distribution. The CDF of normalized
residuals for the astrometry in RA (Dec) is shown in the blue solid (orange
dotted) line. The CDF of a standard normal distribution is shown as the black
curve.

Next, we consider the goodness-of-fits of the astrometric model to the data.
As a reminder, the data and models used for each work are summarized in Table
6.1. Figure 11 shows the CDF of the normalized residuals of the astrometric
data and models against the CDF of a standard normal distribution. For all the
data sets except Sahu et al. (2022), the astrometry in F606W and F814W are
separate data points; Sahu et al. (2022) averages astrometry across both F606W
and F814W filters to obtain a single position. The residuals in this work are
in good agreement with that of a standard normal, as are the residuals in Mróz
et al. (2022) and the Lam et al. (2022a, b) EW model. The residuals in Lam et
al. (2022a, b) are somewhat larger than expected in RA due to the model being
a poor fit to the data. The residuals in Sahu et al. (2022) are smaller than
expected, possibly indicative of underestimated uncertainties.

To be more quantitative, we also perform an Anderson-Darling (AD) test to
check whether the distribution of normalized residuals is consistent with a
standard normal distribution. Table 6.2 lists the the AD test $S$-statistic,
for the model fits in the RA and Dec. The critical values for significance
levels of 5% and 1% are 0.709 and 0.984, respectively. Thus, all models
presented are consistent with being drawn from a standard normal distribution.

Table 7: AD test statistic

###  6.3 Bias correction method

Figure 12:  Comparison of the bias correction measured in this work to those
in Lam et al. (2022b) and Sahu et al. (2022). The _left (right) column_ shows
the bias in F814W (F606W). The _top (bottom) row_ shows the astrometric
(photometric) bias, where the bias is defined as the recovered minus injected
value. The bias measured in this work performing injection and recovery tests
with hst1pass is shown in _black circles_. The bias measured in Lam et al.
(2022b) performing injection and recovery tests with ks2 is shown as _light
blue squares_. The astrometric bias reported by Sahu et al. (2022) performing
PSF subtraction is shown as the _dashed gray line_. Sahu et al. (2022) report
a single value for the astrometric bias; we show this same value in both the
F814W and F606W panels.

We find the main difference in the astrometry between this work and Lam et al.
(2022a, b) stems from the bias correction. The other changes in the
astrometric analysis did not significantly change the astrometry; we show this
along with more detailed comparisons in Appendix C.

We compare the measured photometric and astrometric bias corrections due to
the bright neighbor star in Figure 12. Using hst1pass, we find the average
positional bias in non-magnified epochs is around 1.6 mas in F814W and 0.6 mas
in F606W (also see Figure 4, Table 3.2.2). This is two to three times larger
than the bias found by Lam et al. (2022b) using ks2 (c.f. Figures 22 and 23 in
Lam et al. (2022b), the “Neighbor-like” columns). This suggests that PSF
fitting in ks2 is more precise than hst1pass.

Similar to Lam et al. (2022b), there is minimal positional and magnitude bias
in the first epoch, where OB110462 and the neighbor are of roughly equal
brightness. In the third epoch onwards, the bias becomes non-negligible when
OB110462 is much fainter, and the bias is primarily in the radial direction in
F814W, and more mixed between radial and azimuthal in F606W. We find that the
magnitude bias to be also larger in hst1pass than in ks2. The average bias in
non-magnified epochs is around 15 mmag in F814, and about 5 mmag in F606W when
using ks2, as compared to around 50 mmag in F814W and 20 mmag in F606W when
using hst1pass. This again suggests that PSF fitting in ks2 is more precise
than hst1pass.

The hst1pass positional bias, when averaged across the two filters, is
comparable to the bias of 1.2 mas111Sahu et al. (2022) does not specify the
bias as a function of filter; we assume that their stated bias of 1.2 mas is
the average of F606W and F814W. found by Sahu et al. (2022). This is true,
even though different sets of stars were used to compute the bias (18 vs. 4)
as well as different methods (PSF subtraction vs. artificial star planting
tests).

The results imply combining relative measurements across ks2 and hst1pass is
not valid. Although the majority of the underlying source extraction
algorithms are identical, the particulars of PSF fitting are different enough
to significantly alter the measured positions. The two software source
extraction methods cannot be combined together in a self-consistent manner.
Thus, the positional bias as calculated in Lam et al. (2022a, b) was too
small, and resulted in an incorrect set of astrometric measurements.

In addition, §4.2.5 of Lam et al. (2022b) noted an “astrometric color offset”
between the F814W and F606W positions of OB110462 and another microlensing
event called OB110037. For OB110462, the astrometry across the F814W and F606W
filters were offset from each other by about 0.5 mas. This color difference
was tentatively attributed to binarity. For OB110462, this color difference
can mainly be attributed to the bias correction (Figure 13). However, the
color difference in OB110037 and other stars are still unexplained.

Figure 13:  Comparison of color differences in OB110462 astrometry. The _top
row_ shows the difference in F606W and F814W position for OB110462 vs. time
from Lam et al. (2022a, b). The bias correction calculated in Lam et al.
(2022b) has been applied, but the constant positional offset between F814W and
F606W is _not_ included. The _dashed line_ is the “empirical color offset”
applied by Lam et al. (2022b) to get the data in the two filters to better
agree with each other. There is a significant difference between the positions
measured between the two filters, especially in RA. The _bottom row_ shows the
difference in F606W and F814W positions for OB110462 vs. time as derived in
this work. The color difference has largely disappeared from the RA, and also
slightly decreased in Dec.

##  7 Discussion

###  7.1 OB110462 in the context of the Galactic BH population

Figure 14:  OB110462 in the context of the observed Galactic BH population.
_Left:_ Dynamically confirmed BHs, coded by their type (isolated, detached
field binary, detached globular cluster binary, low-mass X-ray binary, high-
mass X-ray binary). _Right:_ Histogram of Galactic BH mass measurements, coded
by type. Note that not all BHs in the left panel are included in the right
panel, as some only have lower limits on the mass. Note the histogram does not
account for uncertainties on measurements.  Figure 15:  Microlensing parallax
$\pi_{E}$ vs. Einstein crossing time $t_{E}$ (_left_) and maximum astrometric
shift $\delta_{c,max}$ (_right_) of 5 BH microlensing candidates analyzed in
Lam et al. (2022a, b). It is the same as Figure 6 of Lam et al. (2022a),
except the contours for OB110462 have been updated with the values inferred in
this work. MB09260, MB10364, OB110037, and OB110310 are low mass
($<4M_{\odot}$) lenses and not BHs. The points are simulated microlensing
events from the PopSyCLE microlensing simulation (Lam et al., 2020). Compared
to the simulated BH population, OB110462 has a relatively large astrometric
shift, large microlensing parallax, and long Einstein crossing time.

Before 2019, all information about Galactic BHs came from X-ray binary
systems, mainly low-mass X-ray binaries (LMXBs). The observed population of
BHs in LMXBs have masses tightly centered around $\sim 8M_{\odot}$ (Özel et
al., 2010).

A portrait of the complete Galactic BH population is finally emerging (Figure
14; X-ray sources were compiled across catalogs from Aaron Geller
(Northwestern)222https://github.com/ageller/LIGO-Virgo-Mass-
Plot_v2.0/blob/main/src/data/EMdata.json, Grzegorz Wiktorowicz and Chris
Belczynski333https://stellarcollapse.org/sites/default/files/table.pdf, and
BlackCAT444https://www.astro.puc.cl/BlackCAT/transients.php (Corral-Santana et
al., 2016)). Over the last 5 years, our knowledge of other types of Galactic
BH systems has grown. The first BH in a non-interacting system was found to
have a somewhat surprising low mass of 3$M_{\odot}$, in the “lower mass gap”
where neutron stars and BHs had not previously been observed
electromagnetically (Thompson et al., 2019). Since then, two more BHs in non-
interacting binary systems have been found, with masses of $9-10M_{\odot}$
(El-Badry et al., 2023a; Chakrabarti et al., 2022; El-Badry et al., 2023b).
This is somewhat higher than the average observed BH mass in a LMXB, but falls
within the typical mass range.

Now, OB110462 is the first isolated BH to have its mass measured. At
$6M_{\odot}$, it is slightly lower than the average observed BH mass in an
LMXB, although still falling within the typical mass range.

The selection effects that affect observations of BHs are important to
consider if we are to understand the underlying population and BH mass
function. For example, observational selection effects may cause more massive
BHs in LMXBs to be undetected (Jonker et al., 2021). For BHs in detached
binaries, the picture is more muddled: there is tentative evidence of a dearth
of BHs below $8M_{\odot}$ (El-Badry et al., 2023b) as detected from Gaia, but
ground-based RV surveys seem to not be finding these more massive BHs and have
only found a single low-mass BH despite an observational bias toward higher
masses (Thompson et al., 2019). For isolated BHs, there is also a selection
bias towards more massive BHs as they have larger lensing cross sections;
however, if there are many more low-mass BHs, those will dominate the observed
sample. With one detection, no strong conclusions can be made, but there
should be at least as many $6-8M_{\odot}$ BHs as $10M_{\odot}$ BHs in the
Galactic BH population.

Figure 15 compares OB110462 to a simulated population of Galactic microlensing
events. OB110462 is somewhat unusual compared to typical microlensed BHs—it
has a relatively large astrometric shift for its mass, as well as a large
microlensing parallax for a BH. This is due to OB110462 being a nearby ($<2$
kpc) lens; its large astrometric shift and microlensing parallax facilitated
its detection and characterization.

Now that detections of BHs in various types of systems have been made,
understanding their selection effects are needed to quantify their population
properties. This is the next research frontier that will enable us to
understand the Galactic BH population as a whole.

###  7.2 Origin and formation scenarios for OB110462

With only a single isolated BH detection, tight constraints cannot be placed
yet on their origins or specific formation scenarios. In addition, OB110462
does not have full 6-D kinematic information available, since microlensing
does not measure the lens radial velocity. However, recent works examine
possible situations that are consistent with observations of OB110462 and lay
the groundwork for future studies.

Using statistical arguments and assuming OB110462 originated from a single
star, Andrews & Kalogera (2022) finds that OB110462 is kinematically
consistent with the Galactic thick disk. Given the mass, distance, and
transverse velocity found in this work, if OB110462 was born in the thick
disk, natal kicks up to 100 km/s would be consistent with its current
velocity. On the other hand, if OB110462 formed in the thin disk and received
a kick to a thick disk-like orbit, the kick would have had to be around 50-100
km/s.

Vigna-Gómez & Ramirez-Ruiz (2023) study the origins of isolated BHs. They find
that the majority of BHs with masses $<10M_{\odot}$ originated from binary
systems, while the majority of BHs with masses $>10M_{\odot}$ originated as
single stars. This would imply that although OB110462 is now an isolated BH,
it likely originated in a stellar binary system.

With additional mass measurements of isolated BHs, these studies and their
extensions will be able to place increasingly tight constraints on the
formation channels and origins of isolated BHs. By performing targeted
astrometric follow-up with existing facilities, it is possible to build the
sample of isolated BHs to a few ($\lesssim 10$) over the next 5 to 10 years.

###  7.3 Towards a large sample of isolated BHs

Lam et al. (2022a, b) analyzed a sample of 5 archival BH microlensing
candidates, which included OB110462. The other four candidates were not BHs.
They found that after accounting for selection effects, this single BH
detection was consistent with $2\times 10^{8}$ isolated Galactic BHs. Although
the results were consistent, they were not highly constraining due to the
small sample size. The Nancy Grace Roman Space Telescope (Roman), NASA’s next
flagship mission, presents the opportunity to find and characterize hundreds
of isolated BHs with astrometric microlensing (Lam et al., 2020), which will
expand the sample size and enable stringent constraints on the Galactic BH
population.

Roman will conduct several wide-field infrared surveys to answer questions
about dark energy and dark matter, and find exoplanets (Spergel et al., 2015).
One of the surveys, the Galactic Bulge Time Domain Survey, nominally plans to
observe $\sim 2$ deg2 of the Galactic Bulge, finding several tens of thousands
of microlensing events and a thousand exoplanets (Penny et al., 2019; Johnson
et al., 2020). This survey also provides an excellent opportunity to find
isolated BHs with microlensing.

OB110462, in addition to being a proof-of-concept of the method, raises
several technical issues that have not been previously considered, and should
be examined in preparation for using microlensing to find BHs with Roman.

Roman’s mirror is the same size as Hubble (a diameter of 2.4 m), but will be
observing at longer wavelengths than the optical bands used for this and most
other microlensing work. Thus the bias due to a nearby star in an event like
OB110462 would be larger.

Roman will also be similarly or more undersampled than HST. Roman’s pixel
scale is 110 mas/pix; the main filter for the Galactic Bulge Time Domain
Survey is nominally F146, a wide filter centered at 1.46 micron corresponding
to an angular resolution of $\theta_{res}=153$ mas at the diffraction limit.
For comparison, the HST WFC3-UVIS pixel scale is 40 mas/pix; the resolution at
the effective wavelength of F814W = 814 nm (F606W = 606 nm) is 85 (64) mas.
Thus, these PSF reconstruction and modeling methods currently required to
achieve precise astrometry with HST will also be necessary for Roman.

In addition, the density of sources will be much higher in the infrared than
in the (red) optical. Roman will observe hundreds of millions of stars down to
25th magnitude in F146, exacerbating this issue. In addition, if Roman chooses
to observe fields closer to the Galactic plane towards $b=0$ deg, the density
of sources will also increase. Finally, Roman’s field of view will be 200
times that of Hubble’s in the infrared. Events like OB110462, where a faint
source of interest is near a bright one, will be common and not necessarily an
ignorable edge case. A more automated or generalized way to correct the
photometry and astrometry in this situation will be needed.

This also will have impact on the assumptions made in astrometry measurements.
This work as well as previous studies (Lu et al., 2016; Sahu et al., 2022; Lam
et al., 2022a, b) assume that the astrometric shift measured is completely
unblended; i.e. the lens is dark, there are no unrelated neighboring stars,
and the only light that makes it to the telescope aperture is from the images
of the source. However, for all the reasons mentioned above, this assumption
will not necessarily hold for Roman. Despite the resolution gain from going to
space, there is expected to be a non-neglible fraction of blended microlensing
events (Penny et al., 2019). For dark lenses like BHs, the concern of blending
would be from unrelated neighbor stars falling in Roman’s aperture. This extra
flux would dilute the astrometric signal. In addition, if the proper motion of
the neighbor(s) was comparable to the lensing, it could affect not only the
magnitude, but the shape of the astrometric shift.

Other considerations are how to best perform relative astrometry over such
large fields of view. The details of detector-to-detector calibration issues
will be important. Design tradeoffs in terms of observing strategy in order to
attain sufficient astrometric precision should also be studied. All the
statements made here are qualitative, but these issues deserves further
quantitative study.

##  8 Conclusion

We reanalyze OB110462, a microlensing event due to a dark, compact-object
lens. The astrometry we measure is significantly different from Lam et al.
(2022a, b); the discrepancy is caused by a difference in the measured bias
correction from a neighboring bright star. By performing both the astrometric
source extraction and bias correction measurement with the new version of
hst1pass in a self-consistent manner, we find our astrometry for OB110462
between 2011 and 2017 to be consistent with Sahu et al. (2022). Modeling the
updated HST photometry and astrometry along with the re-reduced OGLE
photometry, we find OB110462 to be a BH with a mass of
$6.03^{+1.19}_{-1.04}M_{\odot}$, consistent with the measurement of Sahu et
al. (2022). Thus, it appears so far that the masses of isolated Galactic BHs
are similar to those in binary systems. With the Roman Space Telescope, many
more isolated BH systems can be characterized, and ultimately enable the
measurement of the Galactic BH mass function.

C.Y.L. and J.R.L. acknowledge support from the National Science Foundation
under grant No. 1909641 and the Heising-Simons Foundation under grant No.
2022- 3542. C.Y.L. also acknowledges support from NASA FINESST grant No.
80NSSC21K2043 and a research grant from the H2H8 Foundation.

We thank the anonymous referee and Dan Weisz for helpful comments that
improved the manuscript. We thank Kailash Sahu, Howard Bond, and Jay Anderson
for proposing and taking the HST SNAP observation used in this work.

This research has made use of NASA’s Astrophysics Data System. Some of the
data presented in this paper were obtained from the Mikulski Archive for Space
Telescopes (MAST) at the Space Telescope Science Institute. The specific
observations analyzed can be accessed via https://doi.org/10.17909/gpw0-w659
(catalog 10.17909/gpw0-w659).

##  Appendix A Comparison of pixel-based vs. tabular CTE correction

Here we compare the differences between the tabular and pixel-based CTE
correction. The differences alone do not indicate which type of correction is
better or worse, but allows us to quantify the differences in the resultant
astrometry.

Specifically, the comparison is made between the starlists in Lam et al.
(2022a, b) created using the pixel-based CTE correction (i.e. the old version
of hst1pass run on the flc files) as compared to the starlists in this work
created using the tabular CTE correction (i.e. the new version of hst1pass,
run on the flt files). Several frames spanning 10 years of the OB110462
dataset are shown in Figure 2.1.

As a function of $y$ detector position, there are only differences between the
$y$ position and instrumental magnitude; the differences are symmetric in $x$
detector position. This is expected, since CTE corrections only change the $y$
detector position and magnitude of the sources. The scatter in x-position is
likely due to minor differences between the versions of hst1pass and
differences in the choices of certain runtime parameters between the old and
new reductions. The scatter in $x$ position also allows us to see how much of
the trend in $y$ position is due to scatter, vs. the CTE correction methods
themselves. As time goes on, the scatter increases greatly in the $x$ and $y$
position differences. The scatter also seems to increase with time for
magnitude difference, although it is mostly constant after 2013 or so.

As a function of magnitude, the differences are again symmetric about $x$
position, but not for $y$ position and magnitude. In particular, for the
differences in $y$ position before 2013, the bright stars with $m<-10$ are not
affected, but after 2013 there is magnitude-dependent structure.

Figure 16:  Comparison of measured positions and magnitudes using new hst1pass
and tabular CTE correction vs. old hst1pass and pixel-based CTE correction.
These comparisons are performed on an single exposure, i.e. on the individual
flc.xym or flt.XYM files. The y-axis is the difference between the new tabular
$-$ old pixel based correction for $x$ detector position, $y$ detector
position, and instrumental magnitude. The top (bottom) plot shows the
differences as a function of $y$ detector position (instrumental magnitude).

##  Appendix B Priors

The priors for the fits are summarized in Table B. $\mathcal{N}(\mu,\sigma)$
denotes a normal distribution with mean $\mu$ and standard deviation $\sigma$.
$\mathcal{N}_{T}(\mu,\sigma,l_{\sigma},u_{\sigma})$ denotes a normal
distribution with a low end truncation at $\mu+\sigma l_{\sigma}$ and a high
end truncation at $\mu+\sigma u_{\sigma}$. $\mathcal{U}(a,b)$ denotes a
uniform distribution from $a$ to $b$. $\Gamma^{-1}(\alpha,\beta)$ is the
inverse gamma distribution (Equation G1 in Lam et al. (2022b)).

The choice of priors is nearly identical to those in Lam et al. (2022a, b);
see Appendix G of Lam et al. (2022b) for details. The only difference is for
the photometric priors $b_{SFF}$ and $m_{base}$. We changed the priors on
$m_{base}$ as some of the photometry was recalibrated and the baseline
magnitudes were now different. We also made the priors on $b_{SFF}$ tighter;
the blending is well constrained from previous work (Lam et al., 2022a; Sahu
et al., 2022; Mróz et al., 2022). We did allow for some spread, but consider
it well established that the OGLE lightcurve is highly blended ($b_{SFF}$
close to 0) and the HST lightcurves are less or unaffected by blending
($b_{SFF}$ close to 1).

Table 8: Priors

##  Appendix C Detailed comparison to previous work

###  C.1 CTE correction method

First, we compare the pre-bias corrected astrometry presented in this work to
that of Lam et al. (2022a, b). The differences are the number of epochs of
data used, the version of the hst1pass software, the values of the additive
error, and the method of CTE correction. Of these differences, the method of
CTE correction has the largest impact on the resultant astrometry.

We find no significant difference between the two astrometric time series
after cross-epoch alignment and before implementing the bias correction. While
the different CTE corrections can change the measured positions of stars by
several mas within a single epoch (Figure 2.1), these differences are
effectively removed by the cross-epoch alignment when the starlists are
transformed into a common reference frame using first and second-order
polynomials.

Thus, the specific choice of CTE correction method, tabular or pixel-based,
does not produce any significant change in the astrometric time series
derived. It is only important that CTE be corrected in some manner; in
previous work we find that if no CTE correction of any kind is applied, there
are systematics in the astrometry that cannot be removed by the first or
second order polynomial transformations.

###  C.2 Bias-corrected astrometric time series

Figure 17 directly compares the astrometry of OB110462 in Sahu et al. (2022)
(their Figures 16 and 18), Lam et al. (2022a) (their Figure 2), and this work.
We emphasize that no microlensing model is being fit in Figure 17. Only a
constant proper motion of (RA, Dec) = (-2.263, -3.597) mas/yr has been
subtracted from the positions for easier visualization. The subtracted proper
motion was chosen to match the source proper motion inferred in Sahu et al.
(2022) for an “apples-to-apples” comparison of the astrometry.

There is a clear discrepancy in the deflection in RA between the astrometry of
Sahu et al. (2022) and Lam et al. (2022a, b) (_top left panel_) across the
magnified epochs (2011) vs. the non-magnified epochs (after 2011). There is
some minor discrepancy in Dec (_top right panel_), but the measurements are
generally within 1-2$\sigma$ of each other. The new astrometry, including the
updated bias correction, presented in this work is now consistent with the
measurements of Sahu et al. (2022) between 2011 and 2017. In both RA (_bottom
left panel_) and Dec (_bottom right panel_ , the magnified and non-magnified
astrometry are now all within $1\sigma$ of each other.

The new 2021-2022 measurements provide a more accurate source proper motion in
baseline than in previous work. From Figure 17, the source proper motion from
our updated astrometric reductions appears to be different from that of Sahu
et al. (2022). For $t\gg t_{0}$, the slope of the proper motion-removed
measurements should asymptotically approach 0. This is not the trend seen in
the new 2021-2022 epochs, implying the source proper motion is different from
the value inferred by Sahu et al. (2022). Although the source proper motion is
not particularly interesting in and of itself, its inferred value affects the
measured astrometric shift, which in turn affects the measured lens mass.

We consider whether this different proper motion could be the result of
systematics in the analysis of later epochs. No systematic errors of 1-2 mas
in the 2021-2022 astrometry were detected in the reference stars (Figures 6
and 7).

The source proper motion inferred from the fit are presented in Table 5. The
uncertainties in the transformation from a relative astrometric reference
frame to the absolute Gaia reference frame in which the proper motions are
reported are 0.13 mas/yr and 0.11 mas/yr in RA and Dec, respectively. Sahu et
al. (2022) do not state their uncertainties moving from their relative to
absolute Gaia astrometric frame, but assuming their systematic uncertainties
are comparable to ours, the source proper motion inferred in this work of
(-2.02, -3.45) mas/yr and Sahu et al. (2022) of (-2.263, -3.597) mas/yr are
consistent within $1-2\sigma$. Thus, the source proper motions across the
model fits are consistent with each other. However, there appears to be a hint
of correlated residuals in the later epochs from 2014-2022 in RA (Figure 9).

Additional observations in the future can establish the unlensed source proper
motion and confirm or reject any potential discrepancy.

Figure 17:  Comparison of HST astrometry derived by Sahu et al. (2022), Lam et
al. (2022a, b), and this work. The _top (bottom) row_ shows the astrometry in
RA (Dec) vs. time, with the source proper motion reported in Sahu et al.
(2022) (RA, Dec) = (-2.263, -3.597) mas/yr subtracted in order to more easily
compare the astrometric deflections. The _left (right) column_ compares the
astrometry in Sahu et al. (2022) to Lam et al. (2022a, b) (this work). The
positions of Sahu et al. (2022) _(black squares)_ average all F814W and F606W
measurements. Lam et al. (2022a, b) and this work keep the F814W and F606W
sets separate; F814W (F606W) positions are shown in red (blue) circles.

###  C.3 Bias-corrected photometric time series

Figure 18:  Comparison of HST photometry derived by Sahu et al. (2022), Lam et
al. (2022a, b), and this work. The _top (bottom) row_ compares the differences
in the photometry in Sahu et al. (2022) to Lam et al. (2022a, b) (this work);
red (blue) points denote data in the F814W (F606W) filter.

Figure 18 compares the HST photometry of OB110462 derived by Sahu et al.
(2022) (their Figure 13), Lam et al. (2022b) (their Figures 7 and 8), and this
work. The calibration of the photometry across the three datasets slightly
differs. To calculate the difference between the zeropoint in the photometry
of Lam et al. (2022a) and this work as compared to Sahu et al. (2022), we
calculate the average magnitude difference between those datasets and the Sahu
et al. (2022) photometry for the 7 common epochs between 2011 to 2017. Then to
put the photometry all onto comparable footing, this constant zeropoint
difference is subtracted from the photometry in Lam et al. (2022a) and this
work.

There are minor differences between the photometry in Lam et al. (2022a, b)
and Sahu et al. (2022) in the early (2011-2012) epochs. With the updated bias
correction in the photometry we derive in this work, we find the same
photometry as measured by Sahu et al. (2022) within the uncertainties.

Although the astrometric bias is the main reason for the updated mass result,
Figure 18 illustrates the photometric bias should also not be neglected. In
particular, for Roman, with its extremely precise photometry, accounting for
such systematics will be important.

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# Optical levitation of high purity nanodiamonds in vacuum without heating

A. C. Frangeskou<EMAIL_ADDRESS>Department of Physics,
University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom  A.
T. M. A. Rahman  Department of Physics, University of Warwick, Gibbet Hill
Road, Coventry CV4 7AL, United Kingdom  Department of Physics and Astronomy,
University College London, Gower Street, London WC1E 6BT, United Kingdom  L.
Gines  S. Mandal  O. A. Williams  School of Physics and Astronomy, Cardiff
University, The Parade, Cardiff CF24 3AA, United Kingdom  P. F. Barker
Department of Physics and Astronomy, University College London, Gower Street,
London WC1E 6BT, United Kingdom  G. W. Morley<EMAIL_ADDRESS>Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4
7AL, United Kingdom

###### Abstract

Levitated nanodiamonds containing nitrogen vacancy centres in high vacuum are
a potential test bed for numerous phenomena in fundamental physics. However,
experiments so far have been limited to low vacuum due to heating arising from
optical absorption of the trapping laser. We show that milling pure diamond
creates nanodiamonds that do not heat up as the optical intensity is raised
above 700 GW/m2 below 5 mbar of pressure. This advance now means that the
level of attainable vacuum for nanodiamonds in optical dipole traps is no
longer temperature limited.

###### pacs:

42.50.Wk, 65.80.-g, 01.55.+b, 05.40.Jc

Optically levitated nanodiamonds containing nitrogen vacancy (NV-) centres
have been proposed as probes of quantum gravity Albrecht  _et al._ (2014),
mesoscopic wavefunction collapse Scala  _et al._ (2013); Yin  _et al._ (2013);
Wan  _et al._ (2015, ), phonon mediated spin coupling Albrecht  _et al._
(2013), and the direct detection of dark matter Riedel (2013). The NV- centre
is a point defect in diamond that has a single electron spin which has long
coherence times at room temperature and can be both polarized and read out
optically Wrachtrup and Jelezko (2006); Doherty  _et al._ (2013). Nanodiamonds
containing NV- centres have been trapped at atmospheric pressure using ion
traps Kuhlicke  _et al._ (2014); Delord  _et al._ (2016), and a magneto-
gravitational trap has allowed nanodiamond clusters to be held below $1\times
10^{-2}$ mbar Hsu  _et al._ (2016). However, this design requires permanent
magnets for the trapping, which is incompatible with the trap-and-release
experiments that reach large distance spatial superpositions of the centre-of-
mass as required for some of the fundamental physics experiments mentioned
above Albrecht  _et al._ (2014); Wan  _et al._ ; Riedel . Progress with
nanodiamonds levitated in optical dipole traps, where trap-and-release is
possible, includes the detection of NV- fluorescence Neukirch  _et al._
(2013), optically detected magnetic resonance Neukirch  _et al._ (2015); Hoang
_et al._ (2016a), and the observation of rotational vibration exceeding 1 MHz
Hoang  _et al._ (2016b).

A key requirement of the aforementioned proposals is that the nanodiamonds are
levitated in high vacuum to prevent motional decoherence arising from gas
collisions. However, nanodiamond has been reported to heat up and eventually
burn or graphitise below 10-40 mbar due to the absorption of trapping light by
defects and impurities prevalent in the commercially available nanodiamonds
used thus far Neukirch  _et al._ (2015); Hoang  _et al._ (2016a); Rahman  _et
al._ (2016). Furthermore, heating has been shown to be detrimental to the
fluorescence intensity of the NV- centre, which is necessary for the optical
read out of the spin state Toyli  _et al._ (2012). With non-levitated
nanodiamonds, reducing the electron spin concentration has been shown to
drastically improve spin coherence times of NV- centres in nanodiamond Knowles
_et al._ (2014); Trusheim  _et al._ (2013). Since the source of absorption of
near-infrared light by diamond is known to be extrinsic Dyer  _et al._ (1965);
Bennett  _et al._ (2014); Webster  _et al._ (2015), improving the purity of
the material used in optomechanics experiments involving nanodiamond to avoid
heating and improve spin coherence is of vital importance to making the
proposals in Albrecht  _et al._ (2014, 2013); Scala  _et al._ (2013); Yin  _et
al._ (2013); Wan  _et al._ (2015, ); Riedel (2013) viable. We also note that
heating is a problem more generally in optical trapping and not unique to
nanodiamond Peterman  _et al._ (2003); Millen  _et al._ (2014).

Here we report on levitated nanodiamonds milled from pure low nitrogen
chemical vapour deposition (CVD) grown bulk diamond, and whilst methods such
as reactive ion etching of diamond to form nano-pillars are known to produce
superior quality nanodiamonds compared to milling Trusheim  _et al._ (2013),
milling is the only technique we are aware of that produces large enough
quantities of nanodiamonds for the common nanoparticle injection methods
employed in optomechanics. A previous study had shown that commercial
nanodiamonds (Adámas Nanotechnologies) can reach temperatures in excess of 800
K at 20 mbar, enough to burn or graphitise the nanodiamond Rahman  _et al._
(2016). This study had suggested that the source of heating was absorption of
the trapping light by amorphous carbon on the surface of the nanodiamond and
nitrogen defects in the diamond. Our milled CVD nanodiamonds remain at room
temperature at lower pressures than the pressures attainable with commercially
available material. These results confirm that nitrogen impurities are the
dominant source of unwanted absorption and heating in commercially available
nanodiamonds. We observe nanodiamonds to be suddenly ejected from the trap
below 4 mbar (typically at $\sim$ 1 mbar), which we attribute to previously
observed trap instabilities at intermediate vacuum that can be overcome with
damping of the centre-of-mass motion Ranjit  _et al._ (2015); Mestres  _et
al._ (2015); Gieseler  _et al._ (2012).

An optical dipole trap (optical tweezers) is formed by a focussed laser beam.
A sub-wavelength sized dielectric bead satisfies the Rayleigh scattering
criterion and can be approximated as a point dipole, and in the limit of small
oscillations Gieseler  _et al._ (2013) and the paraxial approximation Novotny
and Hecht (2012), the trap potential is harmonic with a spring constant

$k_{trap}=4\pi^{3}\frac{\alpha
P}{c\varepsilon_{0}}\frac{\text{(N.A.)}^{4}}{\lambda^{4}},$ (1)

where $\alpha$ is the polarisability of the bead, $P$ is the optical power,
N.A. is the numerical aperture of the trapping lens, $c$ is the speed of
light, $\varepsilon_{0}$ is the vacuum permittivity, and $\lambda$ is the
trapping laser wavelength Gieseler  _et al._ (2012). Whilst the trap forms a
harmonic potential, collisions with the surrounding gas induce Brownian
dynamics Millen  _et al._ (2014) and the motion of the bead is therefore
governed by

$m\ddot{x}(t)+m\Gamma_{0}\dot{x}(t)+m\omega_{0}^{2}x(t)=f_{B}(t),$ (2)

where $x(t)$ is the time dependent position along the $x$ axis (transverse to
the optical axis), $m$ is the mass, $\Gamma_{0}$ is the damping rate,
$\omega_{0}=\sqrt{k_{trap}/m}$, and $f_{B}(t)$ is a Gaussian random force with
$\langle f_{B}(t)\rangle=0$ and $\langle
f_{B}(t)f_{B}(t-t^{\prime})\rangle=2m\Gamma_{0}k_{B}T_{cm}\delta(t-t^{\prime})$,
where $T_{cm}$ is the centre-of-mass temperature. Similar equations may be
written for the $y$ (transverse) and $z$ (along the optical axis) directions.
It can then be shown that the power spectral density is

$S_{x}(\omega)=\frac{2k_{B}T_{cm}}{m}\frac{\Gamma_{0}}{(\omega^{2}-\omega_{0}^{2})^{2}+\omega^{2}\Gamma_{0}^{2}}.$
(3)

We fit the experimental data with equation 3, from which we may extract the
beads centre-of-mass temperature, damping rate, size, and mechanical
frequency.

The centre-of-mass and internal temperatures are linked by
$T_{cm}=(T_{imp}\Gamma_{imp}+T_{em}\Gamma_{em})/(\Gamma_{imp}+\Gamma_{em})$,
where the subscripts $imp$ and $em$ denote the temperature and damping
coefficients of impinging and emerging gas molecules, respectively. Gas
molecules thermalise with the bead, and the temperature of the bead is then
$\alpha_{g}T=T_{em}$, where $0\leq\alpha_{g}\leq 1$ is the thermal
accommodation coefficient which determines the degree of thermalisation Millen
_et al._ (2014). By measuring the centre-of-mass temperature of levitated
Rayleigh beads as a function of trapping power, $P$, one may deduce whether
the bead is heating, cooling, or at room temperature. We show that the centre-
of-mass temperature of the majority of nanodiamonds investigated in this study
showed no dependence on power, and were therefore still at room temperature at
$\sim$4 mbar.

Approximately 150 mg of single crystal CVD bulk diamonds (Element Six
145-500-0274-01) were converted into nanodiamonds using silicon nitride ball
milling. The nanodiamonds were cleaned with phosphoric acid at 180∘C and
sodium hydroxide at 150∘C to remove the milling material, followed by a 5 hour
600∘C air anneal. Raman spectroscopy revealed no detectable contamination of
silicon nitride (supplementary information). The concentration of single
substitutional nitrogen defects (N${}_{\text{s}}^{0}$) in 20 bulk samples of
the same material were measured to vary from 95 ppb to 162 ppb using
quantitative electron paramagnetic resonance Eaton, G.R.  _et al._ (2010)
(supplementary information). This signifies an increase in purity of
approximately three orders of magnitude compared to the 150 ppm high-pressure
high-temperature (HPHT) synthesised starting material used to make the
nanodiamonds used for previous work Kuhlicke  _et al._ (2014); Neukirch  _et
al._ (2013, 2015); Hoang  _et al._ (2016a); Rahman  _et al._ (2016); Hoang
_et al._ (2016b); Hsu  _et al._ (2016).

The optical dipole trap shown in Fig. 1 (a) was formed by focussing a single
longitudinal mode 1064 nm Nd:YAG laser (Elforlight I4-700) with a microscope
objective (Nikon N.A. 0.95) housed inside a vacuum chamber. The power of the
laser was controlled with a half-wave plate and polarising beam-splitter,
which transmits horizontally polarised light to the objective lens. Un-
scattered and scattered light from the trapped nanodiamond was collimated by
an aspheric lens and sent to an InGaAs balanced detector which monitors the
$x$ motion in an interferometric scheme described in Gieseler  _et al._
(2012).

Figure 1:  (a) Schematic of the experimental setup. A half-wave plate
($\lambda/2$) and polarising beamsplitter (PBS) control the power of the
laser. A microscope objective housed inside a vacuum chamber focusses the beam
to form the dipole trap. Un-scattered and scattered light from trapped
nanodiamonds is collimated by an aspheric lens and sent to a balanced detector
using one half-mirror and one full mirror. A CMOS camera monitors scattering
intensity from above the vacuum chamber. (b) Photograph of a levitated
nanodiamond (white spot, left) trapped at the focus of the objective (right).

The nanodiamonds were suspended in pure methanol and sonicated prior to
trapping (SEM image in Fig. 2 (b)). Nanodiamonds were trapped by dispersing
them into the vacuum chamber at atmospheric pressure using a nebuliser.
Constancy of mass is a requirement of the power spectral density analysis,
therefore nanodiamonds were first taken to $\sim$3-4 mbar using the maximum
available trapping power to remove surface contaminants Rahman  _et al._
(2016), and then brought back to atmospheric pressure after a minimum of one
hour at vacuum. Scattered light from the nanodiamond was monitored with a CMOS
camera above the vacuum chamber to ensure the size remained constant across
all centre-of-mass measurements. Figure 2 (a) shows that the scattering
intensity falls by 10-20$\%$ after the initial evacuation due to the removal
of surface contaminants, and then remain constant on the second evacuation
when measurements were made Rahman  _et al._ (2016).

Figure 2: (a) Scattering intensity of a single nanodiamond as pressure is
reduced from atmospheric to 4 mbar (red triangles). The chamber is then leaked
back to atmospheric pressure and evacuated for a second time (blue squares).
Minimum trapping power is used on the second evacuation, with the scattering
intensity recorded after measurements to verify that the size of the
nanodiamond has not changed during power variation. (b) SEM image of the
nanodiamonds after ultra-sonication.

The live position signal from the balanced detector was recorded with a high
resolution PC oscilloscope. After normalising by the power, the data is
Fourier transformed to reveal the power spectral density and fit with
$S_{x}(\omega)$ as shown in Fig. 3 (a). From the fit, $A=2C^{2}k_{B}T_{cm}/m$
was extracted for different powers at a fixed pressure of 4 mbar, where $C$ is
a calibration constant for converting from the detector signal in volts to
meters. By demonstrating that $A$ does not depend on power for a fixed mass,
we confirm that the nanodiamonds remained at room temperature as the trapping
power was more than doubled up to approximately 300 mW, corresponding to an
intensity at the focus of $\sim$750 GW/m2. The centre-of-mass temperature was
measured as a function of power rather than pressure to take advantage of the
increased signal-to-noise at low pressure. The centre-of-mass temperature was
inferred from $A$ by measuring $A$ at a higher pressure $p_{1}$ where the
nanodiamond is confirmed to be at room temperature, and then
$T_{cm}=T_{0}(A_{p_{1}}/A_{p_{2}})$, where $T_{0}=298$ K, and $A_{p_{1}}$ and
$A_{p_{2}}$ are the values of $A$ at low pressure $p_{2}$ and high pressure
respectively.

Representative data of the nanodiamonds studied in this article are presented
in Fig. 3. We demonstrate that the centre-of-mass temperature does not depend
on the trapping laser power. This shows that by absorbing less trapping light,
the nanodiamonds are able to dissipate their excess heat. Furthermore, we were
able to keep most nanodiamonds trapped at $\sim$ 2.5-4 mbar for over a week
with the maximum available trapping power.

Figure 3: (a) Experimentally measured power spectral densities (dots) of a
single trapped nanodiamond at 4 mbar as trapping power is increased from 130
to 290 mW (left to right), fit with equation 3 (solid lines). As expected from
equation 1, the trap frequency increases as $\omega_{0}\sim\sqrt{P}$. From the
fit, we determine the radius of the nanodiamond to be $r\approx 18$nm
(equivalent sphere). (b) Corresponding centre-of-mass temperatures of the same
nanodiamond, showing no dependence on the trapping laser power.

The upper-bound absorption coefficient at 1064 nm of the parent material used
to make our nanodiamonds is 0.03 cm-1 Bennett  _et al._ (2014); Bennett ,
corresponding to an absorption index (i.e. the imaginary part of the complex
refractive index) of $2.5\times 10^{-7}$ i.e., comparable to the known
absorption index of pure silica Kitamura  _et al._ (2007), which has already
been optically trapped at high vacuum in numerous studies Li  _et al._ (2011);
Gieseler  _et al._ (2012); Ranjit  _et al._ (2015); Mestres  _et al._ (2015);
Jain  _et al._ (2016); Vovrosh  _et al._ (2016). Therefore, if the motion of
the nanodiamond can be damped through the period of intermediate vacuum, it
may be possible to take the nanodiamonds to high vacuum where proposals
Albrecht  _et al._ (2014, 2013); Scala  _et al._ (2013); Yin  _et al._ (2013);
Wan  _et al._ (2015, ); Riedel (2013) could be realised. We also note that
single crystal CVD diamond with over 2 orders of magnitude greater purity than
the nanodiamonds used here is commercially available.

In order to predict an upper-bound to the pressure our nanodiamonds can reach,
we utilise a thermodynamic model based on two heat dissipation mechanisms: gas
cooling and black-body radiation. Equation 4 models a sub-wavelength sphere
with an absorption cross-section related to the complex permittivity of the
sphere, a cooling rate due to gas molecule collisions, and both the absorption
and dissipation of black-body radiation (see supplementary information). The
excess heat remaining in the sphere can be expressed as Bohren and Huffman
(1983); Chang  _et al._ (2010)

$\displaystyle
C_{V}V(T-T_{0})=\underbrace{3IkV\left(\text{Im}\frac{\epsilon-1}{\epsilon+2}\right)}_{\text{Absorption}}$
(4) $\displaystyle-\underbrace{6\alpha_{g}\pi
r^{2}\bar{v}N_{0}\frac{p}{p_{0}}k_{B}\left(T-T_{0}\right)}_{\text{Gas}}$
$\displaystyle-\underbrace{\frac{72\zeta(5)V}{\pi^{2}c^{3}\hbar^{4}}\left(\text{Im}\frac{\epsilon_{bb}-1}{\epsilon_{bb}+2}\right)k_{B}^{5}T^{5}}_{\text{Black-
body}},$

where $C_{V}$ is the volumetric heat capacity, $I$ is the trapping laser
intensity, $k=2\pi/\lambda$, $V$ is the nanoparticle volume, $\epsilon$ is the
complex permittivity, $\alpha_{g}$ is the thermal accommodation coefficient,
$r$ is the nanoparticle radius, $\bar{v}$ is the mean gas velocity, $N_{0}$ is
the number of gas molecules at atmospheric pressure, $p$ is the gas pressure,
$p_{0}$ is atmospheric pressure, $\zeta(5)$ is the Reimann zeta function, and
$\hbar$ is Planck’s constant divided by $2\pi$. It is assumed that
$\epsilon_{bb}\approx\epsilon$ Chang  _et al._ (2010).

We model the steady state temperatures as a function of pressure of 20 nm
radius nanodiamonds using upper-bounds of the absorption coefficients measured
in Bennett  _et al._ (2014); Webster  _et al._ (2015) by laser calorimetry.
Assuming a linear relationship between defect concentration and absorption
coefficient Morelli and Uher (1993), we may also model diamonds for which the
absorption coefficient is below the detection limit of laser calorimetry
(0.001 cm-1 for a 1 mm thick sample Webster  _et al._ (2015)). Figure 4 shows
the predicted temperature as a function of pressure for nanodiamonds of
various absorption coefficients.

Figure 4:  Modelled upper-bounds of temperature as a function of pressure for
$r=20$ nm nanodiamonds and trapping laser intensity of $I=60$ GW/m2 Hoang  _et
al._ (2016a). The absorption coefficient of the Adámas Nanotechnologies
nanodiamonds (dot-dash line) used in Kuhlicke  _et al._ (2014); Neukirch  _et
al._ (2013, 2015); Hoang  _et al._ (2016a); Rahman  _et al._ (2016); Hoang
_et al._ (2016b); Hsu  _et al._ (2016) has been set to $30$ cm-1 based on 150
ppm N${}_{\text{s}}^{0}$. Standard grade (solid line) corresponds to the bulk
diamond grade used to make the nanodiamonds used in this paper, with an upper-
bound absorption coefficient $0.03$ cm-1. Low absorption grade (dashed line)
has $0.003$ cm-1 Bennett  _et al._ (2014); Webster  _et al._ (2015).
Electronic grade (dotted line) has a predicted absorption coefficient of
$4.5\times 10^{-5}$ cm-1.

No significant heating is expected from 1 to 0.001 bar, as confirmed by our
experimental results. At atmospheric pressure, nanodiamond graphitises at
$\sim$ 940 - 1070 K Chen  _et al._ (1999); Xu  _et al._ (2002). Taking 940 K
as a cut-off, we estimate an attainable pressure of at least ${10^{-3}}$ mbar
for the nanodiamonds used in this article, or ${10^{-4}}$ mbar for 20-30 ppb
N${}_{\text{s}}^{0}$ low absorption grade material. To reach the pressures
required for proposals Albrecht  _et al._ (2014, 2013); Scala  _et al._
(2013); Yin  _et al._ (2013); Wan  _et al._ (2015, ); Riedel (2013),
electronic grade diamond containing less than 5 ppb N${}_{\text{s}}^{0}$ may
be required, which is available commercially.

The actual attainable pressures are likely to be lower than this, as we have
used the upper-bounds of the absorption coefficients. It is also likely that
the absorption will differ at the nanoscale compared to the bulk. For an
average particle radius of 25 nm, as is the case in this study, there are
likely to be only 1 or 2 N${}_{\text{s}}^{0}$ defects per nanodiamond if the
concentration of N${}_{\text{s}}^{0}$ is 100-160 ppb. In this regime, the
surface is probably a more significant source of absorption than the bulk. We
also note that high vacuum is commonly used in high-temperature annealing of
diamond to prevent the onset of graphitisation, which may further extend the
level of attainable vacuum Chu  _et al._ (2014). Since nanodiamonds are non-
spherical (Fig. 2 (b)), they have a higher surface area-to-volume ratio, which
assists gas cooling. It should also be noted that a previous study has shown
that the spin coherence lifetime of the NV- centre ($T_{2}^{*}$) can be over 1
$\mu$s even for temperatures above 600 K Toyli  _et al._ (2012).

Although we were unable to measure the temperature at the loss pressure, the
simulation shows that heating would be limited to 10 K above room temperature
at 1 mbar at most (using the highest intensities and bead radius), which would
suggest a different loss mechanism to the one proposed in Ranjit  _et al._
(2015) involving radiometric forces arising from temperature gradients across
3 $\mu$m silica spheres. Given diamonds large thermal conductivity,
temperature gradients are also unlikely to be present. Rather, the smaller
damping coefficient at lower pressure, combined with nonconservative
scattering forces Roichman  _et al._ (2008); Wu  _et al._ (2009), also
proposed in Ranjit  _et al._ (2015), and shot-noise from the gas, are the more
probable culprits in this trapping regime.

In conclusion, we have milled pure CVD diamonds into nanodiamonds and measured
their resulting centre-of-mass motion in an optical dipole trap, from which we
infer that the nanodiamonds do not heat up at a few mbar for the first time.
We attribute the previously observed heating in commercial nanodiamonds
predominantly to nitrogen defects within the diamond. We have therefore
demonstrated a route to levitating nanodiamonds in high vacuum and have set
upper-bounds on the pressure they can sustain by modelling the temperature of
various types of nanodiamonds as a function of pressure. This advance makes
experiments in dark matter detection, quantum gravity, phonon mediated
coupling of electron spins, and matter-wave interferometry viable.

We acknowledge the EPSRC grants EP/J014664/1 and EP/J500045/1, and the
European Union Seventh Framework Programme (FP7/2007-2013) under grant
agreement no. 618078. G.W.M. is supported by the Royal Society. We are
grateful to James Millen for comments that improved the manuscript, and to Ben
Breeze for assistance with EPR.

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