Determination of the Photometric Calibration and Large-Scale Flatfield of the STEREO Heliospheric Imagers: II. HI-2

Abstract

In this article we describe the methods used to determine the photometric calibration parameters for the outer Heliospheric Imagers (HI-2) onboard the Solar Terrestrial Relations Observatory (STEREO) spacecraft from measurements of background stars, and we present those values that represent small corrections to the values predicted from pre-launch calibrations. Conversion factors to physical units are also derived. We determine the degradation of these instruments over the course of the mission to date; this is found to be around an order of magnitude slower than for white-light instruments on other spacecraft. We compute a correction to the large-scale flatfield for HI-2A, allowing for vignetting in the outer parts of the images. In addition, we consider the effects of pixel saturation and the implications for the use of the HI-2 instruments for stellar photometry. We also discuss the limitations of the currently employed geometrical projection assumptions.

Appendices

Appendix A: Magnitude Scales

In this appendix we describe the steps required to compute the predicted count rates for stars, given their spectral types and V-magnitudes.

1.1 A.1 Calibrating the Spectra

The spectra given by Pickles (1998) are all normalised to a value of 1.0 at 555.6 nm. Gray (2005) gives the following equation to relate the physical flux at 555.6 nm to V-band magnitude [$m_{v}$]:

$$ \log F_{555.6} = -0.4 m_{v} - 10.449~\mbox{W}\,\mbox{m}^{-2}\,\mbox{nm}^{-1}. $$

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However, this is exact only for A0V types, and it holds only approximately for other spectral types (the relation is accurate to about 10 % apart from late-M class stars). To obtain more accurate conversions, we use a numerical integration of the spectrum multiplied by the V-band profile to determine physical units. The improved conversion factors are illustrated in Figure 11.

Figure 11

The flux density of different spectral types of star at $\lambda=555.6~\mbox{nm}$ for $m_{v}=0.0$. Solid = main sequence, dashed = supergiants, and dash–dot = giants. The horizontal dotted line is the value from Equation (12).

1.2 A.2 HI-2 Magnitudes

As we explained in Section 2.1, the HI-2 responses are much broader-band than those of HI-1 (Figure 1b and Article 1) as the optics do not have a filter coating. The resulting passband is also much broader than the V-band used for most catalogue magnitudes (the region of sensitivity spans the B, V, R, and I bands). Because of this, stars of different spectral types and the same V magnitude may have very different intensities in the HI-2 passbands (Figure 12). We therefore define a HI-2 magnitude such that

$$ m_{\mathrm{hi2}} = - 2.5 \log_{10} \biggl( \frac{\int_{0}^{\infty} F(\lambda) T_{\mathrm{hi2}}(\lambda) \,\mathrm{d}\lambda}{\int_{0}^{\infty} F_{0}(\lambda) T_{\mathrm {hi2}}(\lambda) \,\mathrm{d}\lambda} \biggr), $$

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where $F(\lambda)$ is the spectral flux density of the star and $F_{0}(\lambda)$ is the spectral flux density of a star of type A0V, and $m_{v}=0$; $T_{\mathrm{hi2}}$ is the passband of the HI-2 instrument under consideration. For any spectral type, a HI-2 magnitude correction may be determined by evaluating Equation (13) for an $m_{v}=0$ spectrum for that type.

Figure 12

The spectra of various main-sequence types compared with the V and HI-2A passbands. The vertical dashed line indicates a wavelength of 555.6 nm. N.B. the scaling of the spectra is arbitrary and designed to make them all visible.

Because the HI-2 instruments (as for any CCD-based detectors) are photon counters and not bolometric detectors, the HI-2 magnitude will not give a measure of the apparent brightness of a star in the HI-2 field of view; a redder star will appear brighter as more photons are needed to produce the same flux at longer wavelengths. It is, therefore, also useful to define a photonic magnitude that indicates the V-magnitude of an A0V star that would appear to have the same brightness in the HI-2 detectors. This is defined as

$$ \mu_{\mathrm{hi2}} = - 2.5 \log_{10} \biggl( \frac{\int_{0}^{\infty} F(\lambda) T_{\mathrm{hi2}}(\lambda) \frac{\lambda}{hc} \,\mathrm{d}\lambda}{\int_{0}^{\infty} F_{0}(\lambda) T_{\mathrm{hi2}}(\lambda) \frac{\lambda}{hc}\,\mathrm{d}\lambda} \biggr), $$

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with a correction being defined in the same manner as for the HI-2 magnitude. This photonic magnitude is used in defining the magnitude limits of the calibration sample.

The relation between the HI magnitudes and photonic magnitudes and the visual magnitude scale for HI-2A is summarised in Figure 13. A more detailed summary is provided in the supplemental file M0_summary.csv (the contents of the columns are described in the file READ.ME), which tabulates the properties of $m_{v}=0$ stars as observed by both of the HI-2 instruments.

Figure 13

Relation between HI-2A magnitudes and the visual magnitude scale for luminosity classes I (super giants), III (giants), and V (main sequence). (a) Regular magnitude and (b) photonic magnitude. The curves for HI-2B are very similar.

Appendix B: Saturation and Non-linearity

In the HI detectors, saturation is a non-trivial phenomenon because the images are binned and summed onboard the spacecraft before being transmitted to Earth. As a result of this, there is expected to be a gradual onset of saturation, which is not easy to distinguish from non-linearity in the detector system. Fortunately, this can be investigated because approximately once a day, a single-exposure unbinned image is transmitted. These images are in units of DN with the only correction being the subtraction of the bias level.

To investigate any possible non-linearity, we took all of the stars from our sample passing within 300 CCD pixels of the image centre and plotted the brightness of the highest pixel in the stars as a function of their HI-2A photonic magnitudes (Appendix A). The results are shown in Figure 14. It is clear that there is a cutoff for HI-2A at about magnitude 4.3, at which point the peak counts reach a maximum. Below this level, the response does not deviate significantly from linearity. For HI-2B, where the peak count rate is lower for a given brightness due to the large PSF, the behaviour is similar, although the cutoff is not as well defined and appears to be at about magnitude 3. The maximum counts observed are about 15650 DN for HI-2A and 15710 DN for HI-2B, which is equal to the maximum value that the 14-bit A-to-D converter can record ($2^{14}-1$ DN) minus the DC bias levels of about 735 DN for HI-2A and 675 DN for HI-2B. This implies that the limit of the A-to-D converter is below the full-well level of the CCD.

Figure 14

The peak counts of stars as a function of brightness. (a) All of the measurements as a function of photonic magnitude, HI-2A is the upper group of points, shown in red, HI-2B is the lower group of points, shown in blue. (b) The medians for each star as a function of brightness, along with fits to the points below the apparent cut-off levels. In (b) points below the cutoff are marked by asterisks, those above by plus signs.

Since the peak level due to the $\mbox{F}+\mbox{K}$ coronal brightness is only about 2000 DN in a single exposure in the region of the HI-2 fields closest to the Sun, there is no need to apply any non-linear corrections to the instrumental conversion factors or to be concerned about saturation for heliophysics investigations. However, the effects of partial saturation must be taken into account when using the HI-2 instruments for stellar work such as variable star or bright nova studies. A detailed study of the way in which measurements of bright stars are affected by saturation is beyond the scope of this analysis.

Appendix C: Derivation of the Diffuse Correction

In this appendix we present a derivation of the geometrical correction to the flux-conversion factors for extended sources, the results of which were quoted in Section 4.2.2.

As we stated there, the projection of the HI instruments is well approximated by the AZP projection (Equation (9)):

$$R = \frac{F_{p}(\mu+1)\sin\alpha}{\mu+ \cos\alpha} . $$

This projection is represented by the geometrical construction in Figure 15a (unlike the figures in Calabretta and Greisen (2002) and Brown, Bewsher, and Eyles (2009), we have shown a value of $\mu$ between 0 and 1, which is the situation for the HI projections).

Figure 15

(a) The main quantities of the AZP projection and the angles used in the derivation of the diffuse geometrical correction. This figure differs from those in Brown, Bewsher, and Eyles (2009) and Calabretta and Greisen (2002) in that they both show a value of $\mu> 1$, while here we show the situation found in the HI cameras where $0 < \mu< 1$. (b) The image plane showing the pixel and its azimuthal extent.

To compute the diffuse correction, we need to determine the area of sky (or globe) that will be projected onto a defined region of the image plane (for convenience we refer to the region as a pixel). Since the AZP projection is circularly symmetrical, we do not need to consider the azimuthal position of the pixel on the CCD, only its distance from the CCD centre and its size. Let that pixel be at the point $P$ (Figure 15) a distance $R$ from the field centre and be square with a linear size $\delta R$. This pixel observes a region of the sky at an angle $\alpha$ from the optical axis. Let $P^{\prime}$ also be the location of the projection of the area of sky seen by the pixel onto the image plane from the centre of the sphere [$O$], i.e. where that area of sky would be projected in the tan projection (Brown, Bewsher, and Eyles, 2009; Thompson and Wei, 2010). The angle subtended on the sky by that pixel (which is assumed to be small) in the $\alpha$-direction is given by

$$ \delta\alpha= \delta R \frac{{\mathrm{d}} \alpha}{{\mathrm{d}} R} = \delta R \Big/ \frac{{\mathrm{d}} R}{{\mathrm{d}} \alpha} = \frac{\delta R}{F_{p}(\mu + 1)} \frac{(\mu+ \cos\alpha)^{2}}{\mu\cos\alpha+ 1}. $$

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In the image plane (Figure 15b), the pixel subtends an angle $\delta\phi= \delta R / R$ at the intersection of the optical axis and the image plane [$C$]. Its angular size on the sky [$\delta\phi^{\prime}$] is the angular size of a line segment at $P^{\prime}$ subtending $\delta\phi$ at $C$, as seen from $O$. From basic trigonometry $\frac{CP^{\prime}}{OP^{\prime}} = \sin \alpha$, so that (in the small-angle approximation) the angular size of the pixel perpendicular to the plane of Figure 15 is given by

$$ \delta\phi^{\prime}= \delta\phi\sin\alpha= \frac{\delta R \sin \alpha}{R} = \delta R \frac{\mu+ \cos\alpha}{F_{p}(\mu+ 1)}. $$

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Since both $\delta\alpha$ and $\delta\phi^{\prime}$ are small angles, we may determine the solid angle subtended by the pixel on the sky by using $\delta\Omega= \delta\phi^{\prime}\delta\alpha$, i.e.

$$ \delta\Omega= \biggl(\frac{\delta R}{F_{p}(\mu+ 1)} \biggr)^{2} \frac{(\mu+ \cos\alpha)^{3}}{\mu\cos\alpha+ 1}. $$

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We may thus define a correction factor [$\rho$] by which the raw image must be divided to correct for pixel projected area as

$$ \rho= \frac{\delta\Omega(\alpha)}{\delta\Omega(0)} = \frac{(\mu+ \cos\alpha)^{3}}{(\mu+ 1)^{2}(\mu\cos\alpha+ 1)}. $$

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Since the quantity that we measure directly is the distance from the CCD centre rather than the sky angle, it is then useful to invert the AZP projection of Equation (9), which can be relatively readily done as follows: let $\gamma= F_{P}(\mu+1)/R$, and rearrange to give

$$\mu+ \cos\alpha= \gamma\sin\alpha, $$

square both sides and recall that $\sin^{2} \alpha= 1 - \cos^{2} \alpha$:

$$\mu^{2} + 2 \mu\cos\alpha+ \cos^{2} \alpha= \gamma^{2} \bigl(1 - \cos^{2} \alpha\bigr), $$

or

$$\bigl(1+\gamma^{2}\bigr) \cos^{2} \alpha+ 2 \mu\cos\alpha+ \bigl(\mu^{2} - \gamma^{2}\bigr) = 0. $$

This can be solved to give

$$ \cos\alpha= \frac{-\mu\pm\gamma\sqrt{1-\mu^{2}+\gamma^{2}}}{1 + \gamma^{2}}. $$

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The positive value for the discriminant is the one appropriate to obtain values of $\alpha< 90^{\circ}$. This is a more convenient formulation for our purposes than that given by Calabretta and Greisen (2002) since we need $\cos\alpha$ rather than $\alpha$.

Appendix D: Deviations from AZP

Throughout the analyses presented in the main body of this article, we have followed Brown, Bewsher, and Eyles (2009) and assumed that the AZP projection (Equation (9)) is an accurate representation of the actual projection of the HI-2 instruments. This is a purely empirical association, however; the HI optics were not designed so as to yield an AZP projection. Therefore in this appendix we describe some checks on the accuracy of that assumption as applied to HI-2.

To determine this, we measured the location of each star brighter than $m_{v}=5.0$ in the Yale catalogue (Hoffleit and Warren, 1995) in the first image of the first, sixth, etc. day of each month throughout the mission using an interpolated maximum pixel location, i.e.

$$\begin{aligned} x =& \frac{\sum_{i=-1}^{1} ( (m+i) I_{m+i,n}^{2} )}{\sum_{i=-1}^{1} I_{m+i,n}^{2}}; \\ y =& \frac{\sum_{j=-1}^{1} ( (n+j) I_{m,n+j}^{2} )}{\sum_{j=-1}^{1} I_{m,n+j}^{2}}. \end{aligned}$$

Here $(m,n)$ is the location of the local maximum and $I$ is the counting rate in the image. We found that the dominant effect is a radial distortion, which is shown in Figure 16.

Figure 16

The radial deviation of the actual HI-2 projections from the AZP projection that has been assumed. (a) HI-2A and (b) HI-2B. In both cases the greyscale plot shows a 2D histogram of the displacements of stars brighter than magnitude 5.0, and the overlaid trace shows the median displacement. The vertical dashed line indicates the edge of the circular field of view.

The general trends are very similar in the two instruments, and in each case the median displacement at any radial distance within the 512-bin radius field of view of the HI-2 cameras is less than 0.5 image bins (1 CCD pixel). Beyond that, in the corners of the CCD the deviation becomes considerably larger, exceeding 4.0 bins by the extreme edge of detection. The shape of the variation shows that the true projection must be at least a three-parameter relation, and so the distortion cannot be corrected by simply adjusting the AZP parameters.

It is possible to achieve an improved match by fitting the AZP parameters in only the inner part of the field of view and then adding a power-law correction in the outer parts to change Equation (9) into

$$R = \frac{F(\mu+ 1) \sin\alpha}{\mu+ \cos\alpha} + D \max(\alpha- \alpha_{D},0)^{4}, $$

where $D$ is a coefficient quantifying the deviation from AZP and $\alpha_{D}$ is the angle beyond which deviation from AZP starts. However, even this fails in the corners of the field of view (beyond 512 bins from the centre of the CCD). Therefore, since the errors incurred in position determination by using the AZP projection are small compared with the size of the HI-2 PSFs and the accuracy with which a CME can be located, we consider that the improved accuracy does not justify the additional algebraic complexity. It is also noted that the implied corrections to the off-axis diffuse correction (Section 4.2.2) are smaller than 1 %. For stellar analyses, the deviations are significant, but they are not large enough to cause mis-identification of stars.

The double-peaked nature of the HI-2B histogram (Figure 16b) has its origin in a slight distortion that has the shape of a banana, where objects well North and South of the CCD centre are displaced slightly East of the AZP predicted position. It seems probable that this originates from the same manufacturing issue that affected the focusing of HI-2B (Eyles et al., 2009).