figure table

# Reconstruction of the NuSTAR point spread function using single-laser
metrology

Hannah P. Earnshaw  Cahill Center for Astronomy and Astrophysics, California
Institute of Technology, Pasadena, CA 91125, USA  Kristin K. Madsen  Cahill
Center for Astronomy and Astrophysics, California Institute of Technology,
Pasadena, CA 91125, USA  Goddard Space Flight Center, Greenbelt, MD 20771, USA
Karl Forster  Cahill Center for Astronomy and Astrophysics, California
Institute of Technology, Pasadena, CA 91125, USA  Brian W. Grefenstette
Cahill Center for Astronomy and Astrophysics, California Institute of
Technology, Pasadena, CA 91125, USA  Murray Brightman  Cahill Center for
Astronomy and Astrophysics, California Institute of Technology, Pasadena, CA
91125, USA  Andreas Zoglauer  Space Sciences Laboratory, University of
California, Berkeley, CA 94720, USA  Fiona Harrison  Cahill Center for
Astronomy and Astrophysics, California Institute of Technology, Pasadena, CA
91125, USA

###### Abstract

This paper describes a method by which the metrology system of the Nuclear
Spectroscopic Telescope Array (NuSTAR) X-ray space observatory, which uses two
lasers to characterize the relative motion of the optics and focal plane
benches, can be approximated should one laser fail. The two benches are
separated by a ten-meter-long rigid mast that undergoes small amounts of
thermal flexing which need to be compensated for in order to produce a non-
blurred image. We analyze the trends of mast motion by archival observation
parameters in order to discover whether the mast motion in future observations
can be predicted. We find that, by using the solar aspect angle (SAA),
observation date, and orbital phase, we can simulate the motion of one laser
by translating the track produced by the other and applying modifications to
the resulting mast aspect solution, allowing the reconstruction of a minimally
distorted point spread function in most cases. We will implement the
generation of simulated mast files alongside the usual NuSTAR data reduction
pipeline for contingency purposes. This work has implications for reducing the
risk of implementing laser metrology systems on future missions that use
deployable masts to achieve the long focal lengths required in high-energy
astronomy by mitigating the impact of a metrology laser failure in the
extended phase of a mission.

###### keywords:

Laser metrology – Space telescopes – Point spread functions – X-rays

*Hannah Earnshaw<EMAIL_ADDRESS>
##  1 Introduction

The Nuclear Spectroscopic Telescope Array (NuSTAR) is a NASA Small Explorer
mission[1] and the first space-based telescope to be able to focus hard X-rays
(3–79 keV). The long focal length required to make this possible is achieved
by the telescope being made up of two benches, one containing the focusing
optics (henceforth, the optics bench) and the other containing the detectors
at the focal plane (henceforth, the focal plane bench, with detectors FPMA and
FPMB), separated by a 10.15 m rigid mast which was deployed once the
spacecraft was in orbit.

While the carbon fiber mast is designed for minimal thermal flexing, there is
still a small amount of motion over time as the spacecraft orbits the Earth,
moves into and out of sunlight, and points at targets across the sky at a wide
range of angles relative to the Sun. If not accounted for, this motion would
cause images taken by NuSTAR to be blurred as the positions of the optics and
focal plane benches change relative to each other. In order to successfully
reconstruct an in-focus image, we need to know the relative positions and
orientations of the two benches at all times. This is done using a laser
metrology system[2, 3], made up of two laser-detector pairs that measure the
mast position several times a second over the course of an observation.

The laser metrology system for NuSTAR has worked well over the mission’s
lifetime. However, the intensity of both lasers as measured at their
respective position-sensitive detectors (PSDs) has declined since the
beginning of the mission (Fig. 1), with LASER0 operating at a little under one
half of its original intensity and LASER1 at approximately a quarter. While
steps have successfully been taken to slow this decline, such as altering the
duty cycle of the laser cycling, the cause is not fully understood. Although
the risk of a metrology laser failing before the end of the mission lifetime
is low, as a contingency for laser failure we want to be able to reconstruct a
source point spread function (PSF) using only the data from a single metrology
laser, in addition to any other knowledge of the telescope’s condition we have
at our disposal.

Figure 1:  The laser intensity over the duration of the mission for LASER0
(blue) and LASER1 (cyan) as measured by the respective position sensitive
detectors (see Section 2). Also indicated are times that the current to LASER1
was adjusted (dotted orange) and when the laser duty cycle was adjusted
(dashed grey). At the present time, the intensity continues to decline, albeit
slower than previously.  Figure 2:  An illustration of the NuSTAR telescope,
showing its two benches and mast. The X, Y and Z axes we use to refer to
motion between the benches are indicated with black dashed lines, and the
three degrees of freedom that are measured by the metrology system are show
with red arrows.

In this paper, we describe the metrology system in detail in Section 2. We
then present a method of reconstructing the NuSTAR PSF using a single
metrology laser, the solar aspect angle (SAA), and the mission elapsed time in
Section 3, and demonstrate its effectiveness in Section 4. We discuss the
consequences of which laser is used for reconstruction in Section 5. Finally,
we summarize and discuss potential future implementation in Section 6. For all
examples shown, we use an observation of GX 13+1, taken in 2017 (Obsid:
30301003002).

##  2 Metrology System

Between the optics bench and the focal plane bench of NuSTAR, there are six
degrees of freedom of motion that could conceivably be measured, but only
three that are significant enough to affect the image reconstruction:
translation in the X and Y axes (that is, the axes orthogonal to the mast) and
the rotation around the Z axis (that is, the degree to which the mast twists)
[2]. We illustrate the telescope, its axes, and the degrees of freedom we
measure in Fig. 2. The mast length and angle from parallel of the two benches
are sufficiently constant and deviations small that the effect on the aspect
reconstruction is negligible.

These degrees of freedom are measured using two lasers mounted on the optics
bench (henceforth LASER0 and LASER1), shining on two independent PSDs mounted
on the focal place bench (PSD0 and PSD1 respectively). The PSDs are 20mm
photodiodes that can measure the centroid of a light spot on their surface in
two dimensions [4]. The position of the centroid of the laser spot is measured
at a frequency of 4 Hz and recorded, after being corrected for the response of
the PSDs, in the Corrected PSD File (psdcorr) as part of the data reduction
pipeline using the NuSTAR Data Analysis Software (NuSTARDAS). Example laser
spot tracks are shown in black in Fig. 3.

Figure 3:  The tracks of the two laser spots on their PSDs for observation
30301003002. The original tracks are shown in black, and the track from PSD0
overlaid at the position of the PSD1 track is shown in red. The tracks have
been thinned to show 10% of the data points, for clarity.

These tracks are used to calculate the mast aspect solution that relates a
photon vector at the focal plane bench $\vec{v}_{\rm fb}$ to a photon vector
at the optical bench $\vec{v}_{\rm ob}$ [3, 5]. The mast solution is described
by a translation $T_{\rm mast}=[T_{x},T_{y},T_{z}]$ (with $T_{z}$ fixed to the
mast length), and a quaternion $Q_{\rm mast}=[q0,q1,q2,q3]$ where $q3$ is the
real component and thus the mast twist angle $\theta=2\cos^{-1}(q3)$, such
that:

$\vec{v}_{\rm fb}=Q_{\rm mast}\vec{v}_{\rm ob}+\vec{T}_{\rm mast}$ (1)

Aspect reconstruction is therefore the inverse of this process in order to
recreate the image made by photons entering the telescope at the optics bench
(in addition to the consideration of the sky-to-optics-bench and focal-plane-
bench-to-detector transforms, which we are not concerned with for the purposes
of this study). Aspect reconstruction is performed for each individual photon
based on the values of $Q_{\rm mast}$ and $T_{\rm mast}$ at the photon time of
arrival. The values of $Q_{\rm mast}$ and $T_{\rm mast}$ over time are stored
in the Mast Aspect Solution File (mast) for each observation.

The mast motion is due to thermal flexing caused by changes in the
illumination of the telescope by the Sun, both from occultation by the Earth
each orbit and from self-shadowing, which changes as a function of Solar
aspect angle (SAA). The SAA is the angle between the pointing of the
spacecraft and the Earth-Sun axis (where SAA = 0 means that the spacecraft is
pointing directly at the Sun, and SAA = 180 is directly anti-Sun) and remains
constant during an observation. The roll angle does not contribute to thermal
variations since the Sun is always kept on the same side of the spacecraft,
normal to the Solar arrays, such that the direction of the Sun around the
Z-axis does not change[1]. Therefore, the changes in $T_{x}$, $T_{y}$, and
$\theta$ over the course of a single observation are both periodic with the
orbital period and reasonably consistent in shape after settling due to the
SAA remaining constant (see Fig. 4, black). This means that the mast motion is
complex but to an extent predictable, since the orbit and SAA are known for
each observation. This predictability gives us the tools to compensate for the
potential loss of a laser, and to successfully reconstruct an image with just
a single metrology laser.

##  3 Single-laser Reconstruction Method

The method we use to perform aspect reconstruction using a single metrology
laser (e.g. LASER0) is as follows: we assume that LASER1 has failed and create
a synthetic PSD1 track by translating the measured PSD0 track to the would-be
position of the PSD1 track, as predicted from the date and SAA (see Section
3.1). We use this track to generate a new psdcorr file and use it as an input
to the NuSTARDAS module numetrology to generate a new mast file. We make
modifications to this new mast file based on the SAA (see Section 3.2) and use
this corrected mast file as input to nupipeline which proceeds to run as it
normally would to generate aspect corrected event lists. It is important to
note that these corrections do not affect the energy reconstruction but only
the photon distribution on the focal plane detector.

Figure 4:  The values of the X transform $T_{x}$, Y transform $T_{y}$, and
mast twist angle $\theta$ over time for observation 30301003002. The original
two-laser solution is plotted in black, the mast values derived from the
single-laser PSD tracks are plotted in red, the ideal corrected values are
plotted in blue, and the corrected values estimated using the SAA are plotted
in green.

###  3.1 PSD Track Translation

The first step in this process is to translate the PSD0 track to the position
of the PSD1 track. This is done by using the quaternions and vectors stored in
the appropriate CALDB alignment file for the observation date to transform the
track from PSD0 coordinates to focal plane bench coordinates, apply a vector
to move the PSD0 track to the position of the PSD1 track, and transform this
track back to PSD1 coordinates (see Fig. 3). The position of the PSDs and the
layout of the focal plane bench is shown in Fig. 5. In order to determine the
length of the baseline separating the tracks and the angle of rotation from
the X axis, we measured the median of both PSD tracks in focal bench
coordinates for all observations taken over the course of the NuSTAR mission.

Figure 5:  The layout and selected components of the focal plane module of
NuSTAR, showing the location of the FPM detectors and the PSDs, as well as the
origin of the telescope coordinate system. Viewed from the positive Z
direction (i.e. the direction of the optics bench).

We find that the baseline and angle are both functions of SAA, with the
baseline additionally a function of date (Fig. 6). The baseline can be
approximated with a third-order two-dimensional polynomial, which we fit to
the data for the full duration of the mission to date. The variation in
baseline is likely dominated by slight changes in the inclination of the
lasers, where a variation of $\sim 100$ microradians corresponds to $\sim 1$
mm distance at the focal plane bench, and would be affected by long-term
trends in the mast flexing, leading to the time dependence. The focal plane
bench itself is made from aluminum and subject to thermal expansion amounting
to $\sim$0.2 mm on an orbital timescale, so while it may contribute to the
scatter of the baseline relation, it is not a major factor in the baseline
variation.

The angle is a complex function of SAA, with a slight upward trend in the
angle after 2017 due to a known minor impact event in September of that year
[6]. Given the effect of a slight change in angle is minimal (see below), we
fit a spline curve to the data before 2017 and use that as our relation for
predicting the angle.

While there is a small amount of scatter, the effect of changes in the value
of baseline or angle at this scale is to move the derived sky positions of the
photons, and therefore the entire resulting image, by a consistent small
amount in sky coordinates (around 5 sky pixels per millimeter deviation).
However, this does not have an effect on mapping the image to celestial
coordinates, and therefore we are confident that these relations alone are
sufficient for obtaining the parameters for the PSD track translation.

Figure 6:  Left, the measured baseline between the median positions of the two
PSD tracks in focal bench coordinates, plotted as a function of Mission
Elapsed Time (MET) and SAA, and colored by time for clarity. The best-fitting
2D polynomial is shown in the form of a grid in black, and the residuals
between the data and this model are shown beneath in blue and red. Right, the
measured angle from the X axis between the two PSD tracks, plotted as a
function of SAA and colored by time. The data prior to 2017 is fitted with a
spline, plotted in blue.

In conclusion, it is possible to predict the position of the PSD1 track from
the SAA and time of an observation. Using this information, the translated
PSD0 track can replace the absent PSD1 track in the psdcorr file, and used in
the next steps of NuSTAR data reduction.

###  3.2 Mast Transform & Quaternion Correction

The next step in the NuSTAR data reduction process is to run the numetrology
routine using the new psdcorr file as an input, which produces the mast aspect
solution and stores it in a new mast file. This file is then used as an input
to nupipeline with runmetrology=no, which turns off the automated metrology
data reduction, in order to produce the event lists, which store the derived
photon positions. We performed this for our example observation of GX 13+1, a
bright source which allows us to clearly see the PSF shape. However, running
nupipeline on the new mast file as-is yields a distorted PSF, particularly for
FPMB (see Fig. 7).

Figure 7:  The NuSTAR PSF from observation 30301003002 after aspect
reconstruction for FPMA (left) and FPMB (right), in the following cases, from
top to bottom: original aspect reconstruction with two functioning metrology
lasers, aspect reconstruction with a single laser after only performing the
PSD track translation step, aspect reconstruction with a single laser after
performing the PSD track translation and applying ideal corrections to the
mast file, aspect reconstruction with a single laser after performing the PSD
track translation and applying corrections to the mast file estimated from
spline fits to mast function parameters. The FWHMs of Gaussian fits to the
PSFs are plotted with dashed black lines. The intensity is log-scaled to show
the wings of the PSFs.

To discover the cause of this distortion, we compared the original mast files
with those produced by the single-laser approximation by plotting $T_{x}$,
$T_{y}$, and $\theta$ against time in both cases (see Fig. 4). We find that
with a single laser track, most information regarding the two-dimensional
transform is retained; the mean value and the amplitude of the variation is
different, but the general shape of the curve over time is similar. However,
all information about the mast twist angle motion is lost and is measured as a
constant over the duration of the observation. In order to improve the single-
laser reconstruction of the PSF, we need to adjust the single-laser mast file
so that the variations in the transform and twist angle more closely resemble
those of the two-laser solution.

To begin, we created an ideally corrected single-laser mast file by adjusting
the mean and amplitude of the $T_{x}$ and $T_{y}$ curves over time to match
those of the original mast file, and fitting a simple sine function to the
mast angle curve with the period fixed to the NuSTAR orbital period. To verify
that this correction allows us to better reconstruct the PSF, we ran
nupipeline with the settings runmetrology=no and inmastaspectfile set to the
corrected mast file. This results in a PSF that is typically a little larger
than the two-laser solution (by around 10%), but no longer as distorted (see
Fig. 7). Taking this level of fidelity as sufficient, we repeated this process
for every observation in the NuSTAR archive with a bright point source
($>$50,000 counts) and recorded the following correction parameters: the
amplitude of the $T_{x}$ and $T_{y}$ tracks, and the mean, amplitude, and
phase of the best-fitting sine curve to the $\theta$ track. (The mean of the
$T_{x}$ and $T_{y}$ tracks is functionally equivalent to the position of the
translated PSD track – that is, any difference from the original mast file
value is due to acceptable scatter in the estimation relation, so we do not
consider these further.) We plotted these parameters against the SAA for each
observation in order to see whether they can be predicted from the SAA.

First, we examined the amplitude of the variations in $T_{x}$ and $T_{y}$ as a
function of SAA. Both the original amplitude and that of the single-laser
solution vary approximately as a function of SAA, although there is a large
amount of spread in the relation (Fig. 8). We found that a much tighter
relation could be obtained if we plotted the difference between the new and
old amplitudes (see Fig. 9). By fitting a spline to the data, we can use the
SAA of an observation to estimate the amount to add to the single-laser
$T_{x}$ and $T_{y}$ amplitudes to bring them to the value of the two-laser
amplitudes.

Figure 8:  The maximum amplitude of the variation in $T_{x}$ and $T_{y}$, for
the original two-laser solution (blue) and the single-laser solution (red),
plotted against SAA. Figure 9:  The difference in maximum amplitude of the
variation in $T_{x}$ (green) and $T_{y}$ (orange) between the single-laser
solution and the original two-laser solution, plotted against SAA and fitted
with a spline.

The mean and amplitude of the sine curve that best fits the mast twist angle
variation are both a relatively tight function of SAA, and can simply be
fitted with a spline to be able to estimate these parameters directly from the
SAA of an observation (Fig. 10). The phase of the sine curve does not vary
with a simple function of SAA (Fig. 11, top), so we compared this phase with
that of the orbit phase of the telescope as measured from the time it emerges
into sunlight from behind the Earth’s shadow (whether or not the telescope is
in sunlight is recorded in the DAY column of the Orbit file). We found that
the difference between these two phases is a function of SAA, albeit not
continuous (Fig. 11, middle). Since phase is a periodic quantity, we added to
selected data points intervals of 1 to bring them into a single continuous
function, which we fitted with a spline curve (Fig. 11, bottom). At this
point, we have successfully demonstrated that all information for a sine
approximation to the mast twist angle motion can be obtained from the
observation SAA and orbit data.

Figure 10:  Top, the amplitude of the best-fitting sine curve to the mast
twist angle motion over time, plotted against SAA and fitted with a spline.
Bottom, the mean of the best-fitting sine curve to the mast twist angle motion
over time, plotted against SAA and fitted with a spline. Figure 11:  Top, the
phase of the best-fitting sine curve to the mast twist angle motion over time,
plotted against SAA. Middle, the difference between the telescope phase as
measured from first emergence into daylight, and the phase of the mast angle
modulation, plotted against SAA. Points below the dashed line can be
incremented by 1 to create a continuous curve in phase. Bottom, as Middle,
with the indicated points incremented by 1 (and 2 in a few cases), and the
resulting curve fitted with a spline.

Therefore, with no knowledge of the contents of the two-laser mast file, it
can be approximated by taking the single-laser mast file and correcting it by
adjusting the transform variation amplitudes and replacing the constant mast
twist angle with a sine wave approximation, the parameters for which can all
be derived from the observation SAA and the Orbit file.

##  4 Results

In order to determine whether this method can successfully reconstruct the PSF
for all observations, and how well it does so, we performed both steps of the
process – estimating the PSD translation parameters from the SAA and
observation date, and the mast file adjustment parameters (i.e. the
differences to apply to the $T_{x}$ and $T_{y}$ transforms and the parameters
for the sine curve approximation to the $\theta$ variation) from the SAA and
orbit – on all observations of a single bright point source ($>$50,000
counts). We generated new psdcorr and mast files, corrected the mast files
using the derived adjustment parameters, then ran nupipeline using the
corrected mast file as an input for the metrology. We created intensity images
from the resulting event files and fitted them with a 2D Gaussian model which,
while the NuSTAR PSF is a more complex and peaked shape than a Gaussian, still
provides a good indication of whether the PSF is extended or elongated
compared with the PSF of the two-laser scenario.

###  4.1 General PSF fitting results

We plot the ratio of the FWHM semimajor to semiminor axis of the 2D Gaussian
fit to the PSF against the SAA for each observation, both for the single-laser
reconstructed images and the original images generated using the two-laser
solution, to see how well this single-laser approximation method reproduces
the two-laser results across different SAAs (Fig. 12, top). We only plot
observations for which a successful automated Gaussian fit could be made to
the data, which we define to have a semimajor axis less than 20 pixels (49.2
arcsec). Bad fits are often due to multiple bright sources in the field of
view or the presence of stray light. We also filter out fits for which the
original PSF is elongated with a major-to-minor axis ratio greater than 1.3,
usually due to the source being located at a high off-axis angle.

Figure 12:  Top, the ratio of the semimajor to semiminor axis of the 2D
Gaussian fit to the PSF. Bottom, the size of the semimajor axis of the 2D
Gaussian fit to the PSF in pixels, where each pixel covers an angular size of
2.46 arcsec. Fits to the original PSF reconstructed using the two-laser mast
solution are plotted in blue, and fits to the PSF reconstructed using the
single-laser approximation are plotted in red, with mean values (calculated
ignoring data with SAA = 74–78 and 114–116) are plotted in dark blue and dark
red respectively. Only observations for which a good fit was obtained are
plotted (see text).

We find the elongation of the PSF for the single-laser approximation to be
very similar to that of the two-laser solution except for two major spikes at
SAA $\approx 76$ and SAA $\approx 115$. If we ignore observations with SAA in
the ranges 74–78 and 114–116, we find the mean axis ratio for the original
two-laser solution to be $1.07\pm 0.04$, and the mean ratio for the single-
laser approximation to be $1.11\pm 0.09$. In terms of the elliptical
distortion of the PSF, the single-laser approximation performs almost as well
as the two-laser solution except for two small ranges of SAA.

If we plot the size in pixels of the semimajor axis of the Gaussian fit by SAA
(Fig. 12, bottom), there are two major peaks which correspond to those where
the PSF is elongated, at SAA $\approx 76$ and SAA $\approx 115$. There are
also additional, smaller peaks at SAA $\approx 105$ and SAA $\approx 125$, and
a general excess at SAAs between 80 and 95. Ignoring the same peaks as before,
we find that the mean semimajor axis of fit to the two-laser PSF is $6.6\pm
0.5$ pixels, or $16.2\pm 1.2$ arcsec (this is close to the actual PSF FWHM of
18 arcsec, which is smaller than the HPD due to the peaked shape of the PSF).
The mean semimajor axis for the single-laser PSF is $7.4\pm 1.0$ pixels, or
$18.1\pm 2.6$ arcsec.

Figure 13:  The X and Y transforms for observations 30002041004, of SS 433 at
SAA = 75.16 (left), and 60061239002, of NGC 4992 at SAA = 114.42 (right).
Colors as in Fig. 4. The estimated track (green) is not a good approximation
to the original track (black) for $T_{y}$ in the case of 30002041004, and for
$T_{x}$ in the case of 60061239002.

Therefore the reconstructed PSF using a single laser is systematically a
little larger than that of the two-laser solution, but only severely distorted
at two narrow ranges of SAA around SAA = 76 and SAA = 115.

We investigate observations in these SAA ranges to determine what is causing
the high level of distortion. We find that at these particular SAAs, the
periodic variation in the Y transform for SAA $\approx$ 76 and X transform for
SAA $\approx$ 115 cannot be replicated well by the mast solution derived from
the transposed PSD track, even when its amplitude is adjusted, due to a
double-peaked or otherwise complex shape to the variation that is not retained
in a single PSD track (Fig. 13).

Figure 14:  Top, the ratio of the semimajor to semiminor axis of the 2D
Gaussian fit to the PSF. Bottom, the size of the semimajor axis of the 2D
Gaussian fit to the PSF in pixels, where each pixel covers an angular size of
2.46 arcsec. Left, the results for the original two-laser mast aspect
reconstruction. Right, the results for the single-laser mast aspect
reconstruction. The FPMA results are plotted in green, and the FPMB results
are plotted in orange. Mean values (calculated ignoring data with SAA = 74–78
and 114–116) are plotted in dark green and dark orange respectively.

We also examine whether there is any significant difference between the
results for the FPMA and FPMB detectors (Fig. 14). We find that while there is
a slight systematic difference between FPMA and FPMB, this is fairly
consistent with the difference seen in the original two-laser solution, so
there is no additional discrepancy between the two detectors introduced by the
single-laser reconstruction method.

###  4.2 Her X-1

In order to more closely examine changes to the PSF between the two-laser and
one-laser mast aspect reconstruction methods, we performed a detailed study of
five observations of Her X-1 taken at different dates and SAAs, comparing the
data products generated using the two-laser solution to those generated using
the single-laser approximation. We present the observations used in Table 1.
Her X-1 has been used as a PSF calibrator for many X-ray observatories due to
its brightness and low $n_{H}$, which ensures that there is no dust scattering
halo extending the PSF.

Table 1: The NuSTAR observations of Her X-1 studied for this investigation. Observation ID | Date | SAA
---|---|---
30002006002 | 2012-09-19 | 79
30002006005 | 2012-09-22 | 77
30402009004 | 2019-06-23 | 119
30602003002 | 2020-08-12 | 98
90102002002 | 2015-08-03 | 103

For each observation, we ran through the entire process detailed in Section 3
to produce single-laser reconstructions of the images and studied the shape of
the PSF compared to that of the two-laser solution. Following a method based
on An et al. (2014) [7], we modeled the background using nuskybgd [8] and
subtracted it from the image of Her X-1. Then we found the centroid of the PSF
and measured the enclosed energy function (EEF) of the PSF within circular
apertures with radii at 1-pixel intervals up to a radius of $\sim$10
arcminutes (240 pixels), over the following energy bands: 3–4.5 keV, 4.5–6
keV, 6–8 keV, 8–12 keV, 12–20 keV, and 20–79 keV. We also plotted the
difference between the EEF in each band and the lowest-energy band (see Fig.
15).

Figure 15:  Enclosed energy functions for Her X-1, from observation
30602003002. Top panel, EEFs for the original two-laser solution. Bottom
panel, EEFs for the single-laser reconstruction. In each panel, the results
for FPMA are plotted on the left, and FPMB on the right. The upper plots show
the EEFs in each energy band, as a fraction of the total enclosed energy, and
the lower plots shows the differences between the EEFs in each band and the
EEF in the 3–4.5 keV energy band. Figure 16:  The fractional difference
between the two- and one-laser scenarios as a function of radius, for
observation 30602003002. Line colors and styles are the same as in Fig. 15.

We see that the PSF tends to be more peaked at higher energies, up to the
point that our imperfect background modeling becomes significant, and there is
very little difference in the PSF shape between the one- and two-laser cases.
We can investigate the difference between the two scenarios more closely by
plotting the fractional difference between the two-laser and one-laser EEFs at
each radius (Fig. 16). For observation 30602003002 we can see that, within the
first 50 arcseconds, the enclosed energy fraction for the two-laser solution
is higher than that for the single-laser approximation, meaning that the PSF
reconstructed with just one laser is a little more blurred out than the
original PSF, consistent with our results from Section 4.1 showing a slightly
larger PSF size for reconstruction using a single laser. The maximum
fractional difference is $\sim$10% at a radius of about 10 arcseconds, and
with a region of radius greater than $\sim$50 arcseconds, the difference in
enclosed energy is minimal, at $<$1%. The results for the other observations
are qualitatively similar, with 5–10% maximum difference between the one- and
two-laser scenarios being typical, and the difference becoming minimal by 50
arcseconds from the centroid.

Figure 17:  The spectrum of Her X-1 extracted from a 50 arcsecond radius and
fitted with a power-law with a high-energy cut-off, along with the residuals
from the model, for FPMA (left) and FPMB (right). The spectrum from the
original two-laser PSF is plotted in black, and the spectrum from the single-
laser reconstruction is plotted in red.

We can examine any differences in energy another way by looking at the
spectrum extracted from the original PSF and that reconstructed from a single
laser, using nuproducts to extract the spectrum and associated response files.
We plot the spectrum of Her X-1 extracted from a source region of 50
arcseconds (subtracting background extracted from a 100 arcsecond region as
far as possible from the source location), along with the residuals from
fitting it with a simple power-law model with a high-energy cutoff (Fig. 17).
We can see that the difference between the one- and two-laser scenarios is
negligible compared even with the difference between the FPMA and FPMB
detectors, and spectral fit parameters are the same to within $<1$%. This also
applies to the rest of the observations, at extraction radii between 50
arcseconds and 4 arcminutes, and with more sophisticated models to fit the
various features of the spectrum.

Finally, we also examine the ancillary response functions (ARFs) generated for
this source. For each observation, we plot the ratio between the two- and one-
laser ARFs as a function of energy, at a range of extraction radii between 50
arcseconds and 4 arcminutes (Fig. 18). We find that different extraction radii
do not, for the most part, affect the ratio between the ARFs. The ratio is
low, within $\lesssim 0.5$%, at low energies, and increases with energy, up to
a maximum of 2–3% at high energies. This is within the calibration errors at
the respective energies [9].

Figure 18:  The ratio between the ARFs generated from the single-laser
products and the two-laser products, for FPMA (left) and FPMB (right). Ratios
for different source extraction radii are plotted as different shades of the
color for each observation, though most of these lines are overlapping.

In conclusion, we do not expect there to be any significant difference in the
data products extracted from source regions greater than 50 arcseconds for
images reconstructed from our single-laser approximation method.

##  5 Changing the laser used for reconstruction

The previous analysis assumes the scenario that LASER1 is the one to fail and
LASER0 is operational, and thus uses the LASER0 track to reconstruct the mast
aspect solution. We used this assumption because, of the two lasers, the
intensity of LASER1 has been declining faster than that of LASER0, and we
expect that, should a laser fail in the future, it is more likely to be
LASER1. However, we still investigated the opposite scenario for completeness,
assuming that LASER0 fails and performing the previously-described process in
a similar manner using LASER1: we used the same estimated baseline and angle
(adding $\pi$ to reverse the translation direction) to make a copy of the PSD1
track at the position of the PSD0 track, and produced a new mast file. Since
the parameters of the sine wave approximation to the mast twist angle do not
depend on the PSD tracks at all, these estimators remain the same. The only
two parts that we need to regenerate for this case are the differences in the
amplitudes of $T_{x}$ and $T_{y}$, which we show in Fig. 19. The SAA relation
for the $T_{x}$ amplitude difference is very similar regardless of which laser
is used, and the relation for $T_{y}$ is close to the inverse of that for
using LASER0 for mast aspect reconstruction.

Figure 19:  The difference in maximum amplitude of the variation in $T_{x}$
and $T_{y}$ between the single-laser solution and the original two-laser
solution, using LASER1 as a basis rather than LASER0, plotted against SAA and
fitted with a spline. Colors are as in Fig. 9.

When we make these adjustments to the mast file, run the pipeline, and fit the
resulting images with a 2D Gaussian as in Section 4.1, we find that the
average elongation of the PSF (not counting the three largest peaks) is
$1.24\pm 0.2$, greater than for the scenario in which we use LASER0, with
three large peaks in PSF elongation at SAA $\approx$ 104, $\approx$115, and
$\approx$126, and several other ranges of SAA with greater PSF distortion than
the two-laser scenario (Fig. 20). In this case, increases in the semimajor
axis always drive increases in the distortion, with the semiminor axis not
increasing to keep the PSF roughly circular as it does in some cases for the
scenario in which we use LASER0. In summary, the failure of LASER0 would lead
to a generally worse scenario than the failure of LASER1.

This asymmetry between how well we can reconstruct the PSF using the PSD0
track versus the PSD1 track is unsurprising, as we would not expect NuSTAR’s
mast motion to be exactly symmetrical in terms of the extent to which it
affects each side of the spacecraft. This may also indicate that the location
of either the laser or the optic has a small error leading to the larger
reconstruction error. Fortunately, it happens to be the case that the laser
best suited for reconstructing the mast motion by itself is the one that at
this time appears less likely to fail.

Figure 20:  Results for the scenario of using LASER1 to reconstruction the
mast aspect solution. Top, the ratio of the semimajor to semiminor axis of the
2D Gaussian fit to the PSF. Bottom, the size of the semimajor axis of the 2D
Gaussian fit to the PSF in pixels. Lines and colors as in Fig 12, with mean
values calculated ignoring data with SAA = 101–107, 114–116, and 124–128.

##  6 Discussion

Through analysis of the mast behavior over time, as a function of SAA and of
orbital phase, we have developed a method for successful PSF reconstruction
should one of the metrology lasers on board NuSTAR fail. We do this by using
the remaining PSD track, the SAA, the date, and the orbital phase to generate
a synthetic mast file that can be used in the NuSTAR pipeline in order to
produce sufficiently accurate data products. We found that, except for two
narrow ranges of SAA, this method can produce reconstructed PSFs that are
slightly larger but otherwise undistorted and, more importantly, exhibit no
spectral changes. When compared with the PSF from the original two-laser
solution, the central emission is slightly smeared out, but with an extraction
region of 50 arcseconds or greater, differences in the data products between
the one- and two-laser scenarios are negligible. Therefore, we are satisfied
that acceptable PSF reconstruction can be performed with a single metrology
laser.

These results are a promising sign that the loss of a metrology laser on
NuSTAR would not be a mission-ending problem. We plan to implement the
generation of single-laser approximation mast files alongside the ordinary
running of the NuSTAR pipeline, in order to confirm that these results hold up
for future data and to easily enable a switch to using this method alone
should a metrology laser fail within the next few years.

The loss of a metrology laser would have implications for the scheduling of
observations in order to maintain high-quality PSF reconstruction. The SAA
ranges 74–78 and 114–116 would need to be avoided in order to prevent
significant PSF distortion. Additionally, because we are unable to calibrate
any of our SAA functions down to SAA = 0, this may have adverse effects on the
usefulness of Solar data. In the case of LASER0 failing instead of LASER1, the
SAA ranges to avoid would be larger, and the performance of PSF reconstruction
generally worse. However, it may be possible to address the issue of bad SAA
ranges by finding more sophisticated ways of modeling or adjusting the single-
laser mast file to more closely match the two-laser version, such as by
investigating ways by which the shape of the $T_{x}$ and $T_{y}$ variations
can be modeled rather than approximated by the measured laser track. We are
also investigating an alternative machine-learning approach to the problem
which may be able to successfully reconstruct the PSF even in SAA ranges that
are problematic using this method.

Looking forward, there are positive implications for future missions seeking
to use deployable masts in order to achieve a long focal length, since we have
demonstrated that, as long as sufficient records of SAA and orbital phase are
kept over the duration of the mission, it is possible to create a solid
contingency plan for if a metrology system fails.

### Acknowledgments

We thank our anonymous reviewers for helpful comments that improved this
paper. This work was supported under NASA Contract No. NNG08FD60C and made use
of data from the NuSTAR mission, a project led by the California Institute of
Technology, managed by the Jet Propulsion Laboratory, and funded by the
National Aeronautics and Space Administration. We thank the NuSTAR Operations,
Software, and Calibration teams for support with the execution and analysis of
these observations. This research has made use of the NuSTAR Data Analysis
Software (NuSTARDAS) jointly developed by the ASI Science Data Center (ASDC,
Italy) and the California Institute of Technology.

## References

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Hannah P. Earnshaw received their PhD in astronomy from Durham University, UK,
in 2017, and currently works as a data and calibration scientist for NuSTAR at
California Institute of Technology. Their research interests include
ultraluminous X-ray sources and transient extragalactic X-ray binaries.

Kristin K. Madsen received her PhD in astrophysics from the University of
Copenhagen in 2007. She has more than 10 years of experience in X-ray optics
and astrophysical research and is one of the principal instrument and mission
scientists for NuSTAR. She currently works as a staff scientist at Goddard
Space Flight Center. Her research interests include active galactic nuclei,
X-ray binaries, pulsar wind nebulae, and hard X-ray optical design.

Karl Forster is a graduate of the University of Leicester and the University
of Hertfordshire, UK, and received his PhD in astronomy from Columbia
University. His research interest includes high-energy emission from active
galactic nuclei, and he has developed and managed science operations for NASA
missions GALEX and NuSTAR.

Brian W. Grefenstette is a research scientist at California Institute of
Technology. He is one of the principal NuSTAR mission scientists, working on
calibration of the focal plane detectors. His science interests include
nuclear astrophysics, X-ray binaries, supernovae, and supernova remnants.

Murray Brightman received his PhD in astrophysics from Imperial College London
in 2010 and is now a Staff Scientist at the NuSTAR Science Operations Center
at Caltech. His research interests are broadly in the X-ray emission from
black holes and neutron stars.

Andreas Zoglauer received his PhD in 2005 from the Technische Universität
München, Germany, for the development of novel simulation and data analysis
tools for the MEGA telescope. He currently works at the University of
California at Berkeley, and his research interests include the development of
new simulation and analysis techniques, and the optimization and performance
prediction of future instruments.

Fiona A. Harrison received her PhD in physics from the University of
California, Berkeley, in 1993. She is the California Institute of Technology
(Caltech) Benjamin M. Rosen Professor of Physics and the Kent and Joyce Kresa
Leadership Chair of the Division of Physics, Mathematics, and Astronomy. Her
research is focused on the study of energetic phenomena ranging from gamma-ray
bursts and black holes on all mass scales to neutron stars and supernovae. She
is the principal investigator for NuSTAR.

###### List of Figures

  1. 1  The laser intensity over the duration of the mission for LASER0 (blue) and LASER1 (cyan) as measured by the respective position sensitive detectors (see Section 2). Also indicated are times that the current to LASER1 was adjusted (dotted orange) and when the laser duty cycle was adjusted (dashed grey). At the present time, the intensity continues to decline, albeit slower than previously.
  2. 2  An illustration of the NuSTAR telescope, showing its two benches and mast. The X, Y and Z axes we use to refer to motion between the benches are indicated with black dashed lines, and the three degrees of freedom that are measured by the metrology system are show with red arrows.
  3. 3  The tracks of the two laser spots on their PSDs for observation 30301003002. The original tracks are shown in black, and the track from PSD0 overlaid at the position of the PSD1 track is shown in red. The tracks have been thinned to show 10% of the data points, for clarity.
  4. 4  The values of the X transform $T_{x}$, Y transform $T_{y}$, and mast twist angle $\theta$ over time for observation 30301003002. The original two-laser solution is plotted in black, the mast values derived from the single-laser PSD tracks are plotted in red, the ideal corrected values are plotted in blue, and the corrected values estimated using the SAA are plotted in green.
  5. 5  The layout and selected components of the focal plane module of NuSTAR, showing the location of the FPM detectors and the PSDs, as well as the origin of the telescope coordinate system. Viewed from the positive Z direction (i.e. the direction of the optics bench).
  6. 6  Left, the measured baseline between the median positions of the two PSD tracks in focal bench coordinates, plotted as a function of Mission Elapsed Time (MET) and SAA, and colored by time for clarity. The best-fitting 2D polynomial is shown in the form of a grid in black, and the residuals between the data and this model are shown beneath in blue and red. Right, the measured angle from the X axis between the two PSD tracks, plotted as a function of SAA and colored by time. The data prior to 2017 is fitted with a spline, plotted in blue.
  7. 7  The NuSTAR PSF from observation 30301003002 after aspect reconstruction for FPMA (left) and FPMB (right), in the following cases, from top to bottom: original aspect reconstruction with two functioning metrology lasers, aspect reconstruction with a single laser after only performing the PSD track translation step, aspect reconstruction with a single laser after performing the PSD track translation and applying ideal corrections to the mast file, aspect reconstruction with a single laser after performing the PSD track translation and applying corrections to the mast file estimated from spline fits to mast function parameters. The FWHMs of Gaussian fits to the PSFs are plotted with dashed black lines. The intensity is log-scaled to show the wings of the PSFs.
  8. 8  The maximum amplitude of the variation in $T_{x}$ and $T_{y}$, for the original two-laser solution (blue) and the single-laser solution (red), plotted against SAA.
  9. 9  The difference in maximum amplitude of the variation in $T_{x}$ (green) and $T_{y}$ (orange) between the single-laser solution and the original two-laser solution, plotted against SAA and fitted with a spline.
  10. 10  Top, the amplitude of the best-fitting sine curve to the mast twist angle motion over time, plotted against SAA and fitted with a spline. Bottom, the mean of the best-fitting sine curve to the mast twist angle motion over time, plotted against SAA and fitted with a spline.
  11. 11  Top, the phase of the best-fitting sine curve to the mast twist angle motion over time, plotted against SAA. Middle, the difference between the telescope phase as measured from first emergence into daylight, and the phase of the mast angle modulation, plotted against SAA. Points below the dashed line can be incremented by 1 to create a continuous curve in phase. Bottom, as Middle, with the indicated points incremented by 1 (and 2 in a few cases), and the resulting curve fitted with a spline.
  12. 12  Top, the ratio of the semimajor to semiminor axis of the 2D Gaussian fit to the PSF. Bottom, the size of the semimajor axis of the 2D Gaussian fit to the PSF in pixels, where each pixel covers an angular size of 2.46 arcsec. Fits to the original PSF reconstructed using the two-laser mast solution are plotted in blue, and fits to the PSF reconstructed using the single-laser approximation are plotted in red, with mean values (calculated ignoring data with SAA = 74–78 and 114–116) are plotted in dark blue and dark red respectively. Only observations for which a good fit was obtained are plotted (see text).
  13. 13  The X and Y transforms for observations 30002041004, of SS 433 at SAA = 75.16 (left), and 60061239002, of NGC 4992 at SAA = 114.42 (right). Colors as in Fig. 4. The estimated track (green) is not a good approximation to the original track (black) for $T_{y}$ in the case of 30002041004, and for $T_{x}$ in the case of 60061239002.
  14. 14  Top, the ratio of the semimajor to semiminor axis of the 2D Gaussian fit to the PSF. Bottom, the size of the semimajor axis of the 2D Gaussian fit to the PSF in pixels, where each pixel covers an angular size of 2.46 arcsec. Left, the results for the original two-laser mast aspect reconstruction. Right, the results for the single-laser mast aspect reconstruction. The FPMA results are plotted in green, and the FPMB results are plotted in orange. Mean values (calculated ignoring data with SAA = 74–78 and 114–116) are plotted in dark green and dark orange respectively.
  15. 15  Enclosed energy functions for Her X-1, from observation 30602003002. Top panel, EEFs for the original two-laser solution. Bottom panel, EEFs for the single-laser reconstruction. In each panel, the results for FPMA are plotted on the left, and FPMB on the right. The upper plots show the EEFs in each energy band, as a fraction of the total enclosed energy, and the lower plots shows the differences between the EEFs in each band and the EEF in the 3–4.5 keV energy band.
  16. 16  The fractional difference between the two- and one-laser scenarios as a function of radius, for observation 30602003002. Line colors and styles are the same as in Fig. 15.
  17. 17  The spectrum of Her X-1 extracted from a 50 arcsecond radius and fitted with a power-law with a high-energy cut-off, along with the residuals from the model, for FPMA (left) and FPMB (right). The spectrum from the original two-laser PSF is plotted in black, and the spectrum from the single-laser reconstruction is plotted in red.
  18. 18  The ratio between the ARFs generated from the single-laser products and the two-laser products, for FPMA (left) and FPMB (right). Ratios for different source extraction radii are plotted as different shades of the color for each observation, though most of these lines are overlapping.
  19. 19  The difference in maximum amplitude of the variation in $T_{x}$ and $T_{y}$ between the single-laser solution and the original two-laser solution, using LASER1 as a basis rather than LASER0, plotted against SAA and fitted with a spline. Colors are as in Fig. 9.
  20. 20  Results for the scenario of using LASER1 to reconstruction the mast aspect solution. Top, the ratio of the semimajor to semiminor axis of the 2D Gaussian fit to the PSF. Bottom, the size of the semimajor axis of the 2D Gaussian fit to the PSF in pixels. Lines and colors as in Fig 12, with mean values calculated ignoring data with SAA = 101–107, 114–116, and 124–128.

###### List of Tables

  1. 1 The NuSTAR observations of Her X-1 studied for this investigation.