3.1.

Effective Area Corrections

As demonstrated above, the GR component appears as early as 2′ off-axis but only becomes significant above 3′. In most cases, planned science targets are located within 2′ of the OA, but that is not always the case. In these instances, a source may contaminate itself with its own GR. This leads to an increase in the source spectrum, and, because of the nonuniform nature of the GR, it is not practical to treat it as an additional background. Instead, we have modified the effective area to properly account for the GR inclusions.

While the GR increases the area, the aperture stop, responsible for reducing the diffuse background by limiting the FoV, also decreases the area with increasing off-axis angles due to the clipping of the edges of the optical path between detector and optic. This is primarily a geometrical effect, but, because each reflecting shell has a different energy response, the selective rejection of different shells introduces a spectral dependency for both the aperture and the GR component, which is also subjected to the aperture correction.

To complicate matters further, thermal gradients along the mast cause it to flex on an orbital time scale,⁴ and the motion smears out the PSF on the detector plane, resulting in a time-dependent clipping of the area. However, the problem is completely determined by knowing the relative location of the aperture stop, stationary with respect to the detectors, and the focal plane bench, with respect to the OA location, stationary in the optical bench frame. The two benches relative motions are tracked and reconstructed by a laser metrology system.³ Figure 7 lays out the geometry and time-dependent terms. The coordinate frame is set in the optical bench frame since this is where the OA is stationary. The circle represents the aperture stop, and the blue track is the motion of its center to the frame. The source, shown as the purple track, will follow its own motion based on the combined spacecraft jitter and mast motion. Because the aperture stop is not centered on the source, it is necessary to take the azimuth into account as well.

Fig. 7

GR and aperture stop correction schematic. The optics and their OA are stationary in the optical bench coordinate system. The circle represents the aperture stop and as a function of time we keep track of the center of the aperture with respect to the optics. The source moves along a different path caused by the spacecraft jitter. To cover all motions, we raytrace 20 azimuth angles for every 18 deg, 14 off-axis angles binned every arcminute, and $10\times 10$ aperture stop positions at 1-mm resolution.

To generate the corrections, we ran raytraces that covered a phase space of 20 azimuth angles for every 18 deg, 14 off-axis angles binned every arcminute, and $10\times 10$ aperture stop positions at 1-mm resolution. We assumed a fixed PSF size for all energies with scattering parameters inside the raytrace adjusted to reproduce the observed polychromatic Hercules X-1 PSF. Thus, we did not take into account the secondary term of the PSF varying in size as a function of energy.¹¹

The above terms all go into calculating the aperture stop correction, but, in addition to these, the GR correction also requires the specification of an extraction region. This necessity is illustrated in Fig. 8 where we have shown an off-axis observation of the bright accreting black hole x-ray binary Cygnus X-1 (a), the raytraced observation (b), and the same raytrace where we have separated the focused photons from the GR component (c). The amount that the GR component contributes is dependent on the size of the extraction region, and, due to the obvious complexity of the GR pattern, the corrections were only derived for circular regions, which we include in steps of 20″ in radius.

Fig. 8

(a) Off-axis observation of Cygnus X-1 (NuSTAR observation ID: 10014001001). The sharp edges at the bottom are the detector edges. (b) Raytrace of the Cygnus X-1 observation. Detector extent not included. (c) Contours of the raytraced image are shown decomposed into the focused source component (properly double bounced photons) and the GR component to illustrate how different extraction regions centered on the source sample different amounts of the GR.

An example of the magnitude of the aperture and GR corrections is shown in Fig. 9. The aperture correction is less than unity because it is removing photons from the effective area, and it is largest for low-energy photons since the majority of these come from the outer shells of the optic and thus are more prone to being blocked by the aperture than light focused by the inner shells. The correction is only important for off-axis angles $>3^{\prime }$. For the GR corrections, the net effect is an increased effective area. The suppression of the low energies is because of the aperture stop correction to the GR. Both corrections are of a linear nature as a function of the off-axis angle, and, in practice, we interpolate the correction tables when generating the observation specific instrument response.

Fig. 9

(a) Correction to the effective area of proper double bounced photons due to the aperture stop. (b) Correction to the effective area due to the inclusion of GR photons for different extraction region sizes. The GR-correction also contains aperture corrections of the GR photons.

3.2.

The “Streak”

The “streak” is an artifact that is rare because it requires a fairly exact alignment of the source to the optics and detectors. It is caused by the absence of glass between mirror segments as shown in Fig. 2(b), which allows the photons to pass right through the optics without reflecting off any surface and propagate down to the focal plane. This gap occurs at 60-deg intervals, and, to reach the focal plane, the source must be aligned azimuthally with one of the gaps and be within the correct off-axis angle. Since there are no gaps between shells with a radius smaller than the intermediate mandrel, the smallest angle at which a photon can make it through to the focal plane is given by the radius of the intermediate mandrel at an $R\sim 108.7\mathrm{mm}$ (shell 69), ${\theta }_{\min }\sim {37}^{\prime }$. The largest angle is determined by the size of the optics and is ${\theta }_{\max }\sim {65}^{\prime }$.

These streaks are rare, but they have been observed at several locations in the Galactic Center. They were caused by the same source, and, once the full mosaic was compiled, as shown in Fig. 10, they were identified as originating from the binary 1A 1743-294 in an outburst during the observations.¹²

Fig. 10

During a Galactic Plane survey,¹² the streak appeared at two different locations, and, by generating the sky mosaic and tracing back the streaks, it was discovered that they came from 1A 1742-294, an x-ray binary that was in outburst.

3.3.

Back Reflections

In BR, the photons strike the backside of the upper mirror of the adjacent shell first then reflect again off the front side of the mirror shell. Because the graze angles of adjacent shells are only slightly different, the photons exit at almost the same angle at which they came from the source.

Like for the GR, the BR is limited to certain off-axis angles, and at any particular off-axis angle only a few shells contribute at a time. As shown in Fig. 3, the condition for a photon to exit the optics is constrained by the opening angle of the lower shell. For the innermost shell, the allowed angles are $\sim 14\pm 9^{\prime }$, and for the outermost shell it is $\sim 24\pm {24}^{\prime }$. The aperture stop further limits these angles, and the geometrical area obtained without including the reflection is shown in Fig. 3.

Reflection from the backside requires very high fluxes to be detected and cannot be seen from typical astrophysical sources. This component was, however, observed during the solar observations on 11 October 2014 when a solar X-class flare went off a few hours before the scheduled observations. We show in Fig. 11 an example of how the component looks with the accompanying raytrace simulation (a), the full mosaic of the actual observation (b), and its raytrace (c). The slew was away from the solar north pole, and the bright streak is the flare entering in through the mirror gap. In the simulations, the mirror gap is larger than in reality; we simulated the as-designed width of the double street, spacer to spacer (see Fig. 2), which assumes that the mirror overhang between the spacers is nonfocusing due to the surface not being constrained to a conical shape anymore. From imaging data, there is some indication that at least part of the mirror overhang is properly focusing, but, taken together with the jagged edges of the glass, we have no good way of estimating how much that is. The sharp circular edges are caused by the aperture stop. In the bottom of the mosaic, the GR component is just visible as a brighter area before transitioning to the BR. We ran the simulation with a longer exposure time than the actual observation to get better fidelity and more detail in the transition area between GR and BR and to reduce the errors on the effective area calculation.

Fig. 11

(a) Two examples of the GR and BR distribution on the detector from actual NuSTAR data and simulation. The fidelity of the simulation is greater than for the actual since the detectors were operating at 99% dead-time with practical exposure times on the order of $\sim 20$ s. (b) Solar observation, north pole slew. The additional counts apparent at $y$-axis 50 to 60″ are from the diffuse cosmic x-ray background. (c) Raytrace of the same observation. The narrow stripes show where the spacers are blocking the x-rays. The broad stripe has no mirror, and we see photons in the actual observation at that location because it happens to be centered such that the flux from the solar x-flare is allowed to pass through. In the bottom of (c), the first 10′ are GR.

To estimate the true effective area of this component, we require knowledge of the reflectivity coefficients off the backside of the mirror. We calculated the reflectivity from the inverted multilayer stack with an added 0.21-mm ${\mathrm{SiO}}_2$ substrate at the top and found that very few photons make it into the stack through the substrate at the angles in question, making the glass substrate the primary contributor. Because of the inefficiency of ${\mathrm{SiO}}_2$ as a reflective surface, only soft photons that can undergo total external reflection have a nonnegligible contribution. This causes the BR component to cut-off sharply between 3 and 5 keV as shown in Fig. 12(a). The mean effective area between 3 and 5 keV [black curve in Fig. 12(b)] is obtained from the raytrace using the mean ${\mathrm{SiO}}_2$ reflectivities between 3 and 5 keV for the first reflection and the NuSTAR multilayer recipes⁹ for the second reflection. We did not include the optics thermal cover and detector window Be absorption, and the area has been scaled to the photons falling on the detector area only.

Fig. 12

(a) The reflectivity curves of a photon reflecting off a 0.21 mm ${\mathrm{SiO}}_2$ substrate at three different grazing incidence angles. (b) The average effective area between 3 and 5 keV for the detector from the raytrace assuming averaged reflection coefficients and effective area obtained from the actual NuSTAR observation of the sun.

To obtain an independent verification of the effective area, we studied the full mosaic of the solar observation. Thanks to the streak, we can extract a flux for the solar flare for the tiles where it was present. The solar flux is extracted from the detector by laying down an area around the streak and using that as the effective area, including all absorption effects in the photon path.¹³ We then extracted the photon count from the remaining detector, limiting the energy range to be between 3 and 5 keV. The effective area is obtained by dividing the detector photon rate with the expected flux and as shown in Fig. 12(b) there is good agreement between the two estimates.

The errors on the effective area from the observation mainly come from evaluating the area of the streak, which is almost certainly narrower than the extraction region laid down on the detector. The jagged edges of the mirror also vary the gap width and cause additional scattering; thus, we estimate a 50% error on the area. On the raytrace side, we assumed the mean reflection values between 3 and 5 keV for the backside of the mirrors without weighting it by an input spectrum. This will underestimate the area at higher off-axis angles due to there being more low-energy than high-energy photons in the averaged energy interval. In comparison, we show the raytrace effective area at 3 keV, which as expected shows better agreement with the measured area.

In comparison, the on-axis effective area in the same energy band is $\sim 500{\mathrm{cm}}^2$ focused on a small area instead of the entire detector. If we take the typical extraction area to be the PSF half power diameter, which is 1′ or 2.9 mm, then the area of the extraction region is $6.6{\mathrm{mm}}^2$ and the effective area per detector area is $\sim 500/6.6=75{\mathrm{cm}}^2{\mathrm{mm}}^{-2}$. The BR flux covers on average about 70% of the detector area, which is $1120{\mathrm{mm}}^2$, so the effective area per detector area for $1{\mathrm{cm}}^2$ of BR is $\sim 1\times {10}^{-3}{\mathrm{cm}}^2{\mathrm{mm}}^{-2}$. The source count rate is thus reduced by a factor of $\sim 7.5\times {10}^4$ in a comparable typical extraction region and thus is completely negligible if caused by typical astrophysical sources.

4.1.

Primary Stray Light

SL is the term given to photons that arrive directly at the focal plane without having undergone any reflection or transmission through the optics.

The geometry of the SL is determined by the aperture stops, Fig. 1(b), and the silhouette of the optical bench. A schematic of the optical bench is shown in Fig. 13 and outlines the angular extent of the bench as seen from the detectors. The circle marks the FoV of the sky as seen from the center of one detector up through the aperture stop. Within this FoV, there are slices of sky that are not blocked by either the aperture or the bench, and if a source should be located there it may directly reach the focal plane. If a source does happen to occupy that space, it can be blocked by choosing a different position angle (PA) of the observatory, which rotates the outline of the bench on the sky. Different areas of the detector see different areas of the sky, and SL may occur from sources within 1 deg to 4 deg of the observed target.

Fig. 13

(a) The OB as seen from the sky. Hexagonal plates mark the location of the two optics. (b) Projected outline of the OB in degrees on the sky as seen from the detectors. Red circle is the projected opening of the limiting aperture stop for the center of one of the modules. This circle is displaced depending where on the detector one is looking up from, with a max displacement of 2 deg.

This component places by far the tightest constraints on the planning of observations. The SL can, in most cases, be completely avoided by picking a suitable PA, but this in turn limits the times a year a target can be observed. Figure 14(a) shows an example of careful planning that enables a source to be observed despite multiple SL sources. We also show an example in Fig. 14(b) where both SL and GR were present to illustrate the visual difference between the two components. The characteristic circular shape of the SL comes from the aperture stop, and, because of the simple geometry, it is straightforward to predict the location of the SL.

Fig. 14

(a) SL pattern in NuSTAR observation obsID: 90201027002. The source can be seen in the top right quadrant. The truncation of one crescent is the optical bench blocking the photons. The PA was chosen to allow the source to fall in a SL-free region. The image has been smoothed with a 3 pixel Gaussian. (b) Example of NuSTAR observation obdID: 90201020002 that has both GR and SL.

In the rare instances that a science target cannot be scheduled to avoid SL, the SL needs to be treated as an additional background. Fortunately, there are a couple of mitigating factors that make analysis straightforward.

With careful analysis, most SL regions can, therefore, be dealt with even if they overlap the actual target of the observation.

4.2.

Absorbed Stray Light

Similar to the primary SL, absorbed stray light (ASL) also arrives at the focal plane directly from the source, but the angles are larger, going all the way out to 10 deg, and they do so by transmission through the material of the aperture stops.

The aperture stops were designed to be 1.88 mm, layered into 0.75 mm Al, 0.13 mm Cu, and 1.0 mm Sn. However, as we will show below, they appear to have been built without the 1.0 mm Sn. This allows strong hard spectral sources to transmit through the apertures above $\sim 20\mathrm{keV}$. Unlike the primary unabsorbed SL, this component is very weak and only a handful of the brightest astrophysical sources (e.g., the Crab, Cygnus X-1, and GX 9+9) have been capable of producing a significant detection. Over five years of operation, less than ten observations have been impacted.

Figure 15 shows a schematic of the geometry. The top is the limiting aperture, and it leaves a circular SL as illustrated in shade. For strong sources, the high-energy flux is capable of transmission through the aperture stop and thus photons that have progressed through the first aperture stop and managed to slip through the opening of the second, leave another circle, or crescent, of once absorbed photons. Photons that have transmitted through the first and the second aperture stop material leave a third circle of twice absorbed photons. The top of the detector module is a square opening (“the can”) that only allows photons to pass through the opening; there is no transmission from photons that hit the side. Any photons arriving at angles that are larger than the can’s FoV are rejected. Due to their complexity [Fig. 1(b)], we do not model the fixtures that hold the apertures in place, and observations have shown that we can ignore its extent since the “can” excludes photons that arrive at angles where absorption from the fixture would have been important. Note, however, that a photon may transmit through AP2 or AP3 without having encountered AP1 or AP2.

Fig. 15

Aperture stop schematic. There are three aperture stops, surrounded by 0.13 mm Cu and 0.75 mm Al, and a square opening in the detector module acting as the fourth. Rays that enter through the top aperture and hit the detector without transmitting through any of the other apertures is the primary SL (blue circle). Rays that transmit through the first aperture are single absorbed (orange circle), and rays that transmit through the first and second aperture stop are double absorbed (green circle).

Figure 16 shows the predicted ASL from Cygnus X-1 at an off-axis angle of 3.98 deg (a) and the actual detector image of the observation (b). It is easy to see that there is additional flux on the detectors, but, without the predictive plots, distinguishing the boundaries is not straightforward. The flux is, therefore, not even across the ASL region the way it is for the primary SL, and treating it as an additional background is difficult without precise knowledge of both the actual source spectrum, the layout of the absorbing elements, and their relative occupation underneath the target source.

Fig. 16

(a) Predicted ASL. The four aperture plots individually show the projection of each aperture (AP1-3) and the “can,” and the two plots labeled “FPMA” and “FPMB” are the summed aperture images on the two modules (for certain angles, the two can be different). In these two latter plots, the color shading is a visual aid to distinguish the different regions and should not be taken literally. In the four aperture plots, the smallest (lightest shade, green) crescent shows where SL passes through unabsorbed, and the second largest circle (light blue) where it is absorbed by a single layer of Cu + Al. The darkest region (dark blue) represents the fixture, which we do not model due to its complexity, and observations show that we can ignore its extent since the “can” excludes photons that arrive at angles where the fixture would have been important, like in this example. (b) Actual ASL. Without the predicted ASL pattern it would be difficult to see where it is once and twice absorbed.

The ASL spectrum itself can be easily identified because of its characteristic low-energy absorption and peak flux at 20 to 40 keV. Figure 17(b) shows the Crab spectrum as extracted from the single absorbed and double absorbed regions (a). We also extracted the Crab spectrum from the SL region and applied to it the absorption from single and twice absorbed 0.75 mm Al and 0.13 mm Cu. This unfortunately shows the absence of the Sn, which would have completely suppressed the spectrum had it been present.

Fig. 17

(a) Actual and predicted ASL pattern for Crab observation (NuSTAR observation ID: 10110005001) positioned 1.9-deg off-axis. The white regions are of single absorbed and black regions are of double absorbed. (b) The spectra of the single absorbed and double absorbed regions together with the Crab SL spectrum from the top of the detector with the as-built absorption of Cu and Al applied.

Because of the complicated patterns, albeit predictable, the ASL currently requires specialized, nonstandard treatment.