Accurate Photometry of Saturated Stars Using the Point-spread-function Wing Technique with Spitzer

Our technique requires accurate centering of the images of the reference star (Sirius) and the star of interest. The PSF of IRAC is roughly point and axisymmetric; that is, its center position can be determined by examining the non-saturated flux distribution around the star (i.e., radial profile). Determining the accurate position of the centroid of a heavily saturated star image is a two-step process. We first determine a "best-guess" center using the PSF diffraction spikes, and then the centroid is fine-tuned using the non-saturated radial profile distribution of the star. The "best-guess" center is determined by the intercepts of the diffraction spikes by examination of the image. In image display software (such as ds9), an initial guess of the centroid can be determined by drawing a symmetric line for each pair of diffraction spikes diagonally and horizontally (see the top left panel in Figure 1 for an example). These lines intercept within a few resampled pixels if they are truly symmetric for a pair of the spikes, and a rough center can be determined as the center of the intercepted area. Using this rough center, a radial profile is then constructed using the median value within a concentric annulus that has a width of 5 resampled pixels (12). Multiple radial profiles were obtained by shifting the center by ± 3 resampled pixels (i.e., a search area of 17 × 17, roughly the full width at half maximum of the PSF) around the best-guess center in both x and y directions, resulting in a total of 7 × 7 = 49 profiles. Comparing all these radial profiles, we found that, outside 30'', all the profiles agree well, within a few percent, while the interior profiles between 10''–20'' depart from each other but still agree within 10%. A fiducial radial profile is computed by averaging all the profiles. Figure 2 shows an example of all the profiles normalized to the fiducial one. The best centroid was determined from the profile that has the minimum deviation in the χ² sense from the average profile between 30'' and 180''. The profile using this best centroid is also shown as the thick, black line in Figure 2. At both bands, the shifts between the initial guess and the best centroid are small, within ±1 resampled pixel. We note that the exact centroid position has no impact on the radial profile outside the saturated core (r > 10''), and our technique (as described below) is not sensitive to this "best-guess" center as long as we can perfectly align two saturated images.

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Instead of determining an accurate absolute centroid for the star of interest, we only care about the centroid position relative to that of Sirius (i.e., the goal is to align the two images perfectly). Achieving this alignment is also a two-step process. First, a best-guess center of the stellar image is determined using the symmetry of the diffraction spikes, similarly to the Sirius data as presented in the previous section. Second, the best alignment is determined by minimizing the residuals in difference images between Sirius and the target of interest. Because of the flux difference between the star and Sirius, the images are first scaled to match in brightness. The scale factor is determined by matching the surface brightness level in the radial profiles at a radius of 40''. The scaled target image is roughly centered on the centroid of the Sirius image, and then 11 × 11 shifted images are generated over a range of ± 5 resampled pixels around the "best-guess" centroid. Difference images are generated by subtracting the scaled and shifted images from the Sirius image. The best shift position is then determined visually by examining the difference images (see Figure 3 for an illustration). When the two images are misaligned, the PSF structures (spikes and diffraction rings) in the difference image are amplified as echos in a positive and negative pattern, but these artifacts mostly disappear when the two images are perfectly aligned. For each of the stars, the best match (shift values in x and y directions) was determined independently for the data from both channels 1 and 2.

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All 11 stars were measured in both bands relative to Sirius using this procedure. Once a star was aligned with Sirius, its final radial profile was constructed using the best centroid position, as shown in Figure 4. To construct the final radial profiles, we used the median value within the concentric annulus as the radial flux value, and determined its associated uncertainty as the standard deviation in the annulus divided by the square root of the number of resampled pixels within the annulus. Using the median value ensures that the profile is not biased by bright field stars. As shown in Figure 1, there is a bright (saturated) field star in the field of HD 126660 roughly at a radius of 70'', resulting in larger uncertainties near that radius. The profiles between HD 126660 and HD 215648 are very similar within 40'', but depart slightly due to the saturated field star in the HD 126660 image and the different values of the background offset. Nevertheless, all radial profiles are self similar in each band up to a radius of 120'' from the star. Outside that radius, the radial profiles depart from that of Sirius, and more severely for fainter stars where any small background offset affects the profile strongly. We note that the background value is not the true background value in the field, rather it is a relatively small background mis-match arising when combining the mosaics; therefore, it can be either positive or negative. Overall, the final derived radial profiles have good S/N over a dynamical range of 10⁴ for bright stars like Sirius, β Gem and Procyon, ∼10² for the faintest star, HD 165459, and 10³ for the rest of the targets.

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We determine the relative brightness between the star of interest and Sirius by comparing their radial profiles. The brightness difference between a star and Sirius should be the flux ratio of the two profiles radius by radius, which is expected to be constant in an ideal situation (e.g., PSF is stable and the median radial profile is a true representative of the PSF structure, with no offset from the background). In reality, there are a number of challenges in comparing the radial profiles. Bright background sources within the extended PSFs and their associated instrumental artifacts could bias the results; we masked obvious examples. However, subtle changes in the underlying background are harder to detect. In fact, by necessity, the background level is determined far from the stellar positions, while the radial profile comparison is dominated by closer-in positions, so offsets are definitely possible. Other sources of uncertainty can arise, resulting in inconsistencies in the profiles indicated by elevated noise in their values at some radii.

Figure 5 shows the radial distribution of the flux ratio between Sirius and each target star. For bright stars like β Gem and Procyon, the flux ratio profile is relatively flat within the uncertainties (computed by the error propagation of the two profiles) within radii of 120''. For fainter stars as shown in Figures 5(b) and (c), various non-flat structures are seen, which are not necessarily the same for both bands. We adopted two different methods to compute the average values of the flux ratios.

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3.3.1. Method 1: Rejecting Points Using the Accumulative Average Profile

As mentioned earlier, in an ideal case, we expect a flat radial distribution in the flux ratio. This is certainly the case at small radii as the brightness of the star drops radially. Therefore, the flux ratio at small radii should carry more weight in determining the true flux ratio. The situation is similar to determining the optimal aperture size in photometry: we need to find a size that is large enough to contain most of the signal but small enough so that artifacts from the background placement do not dominate the signal. For each of the profiles, we therefore computed the accumulative average profile (in flux ratio) such that at each radial point the value is the weighted average of all the data points prior to that radius (similar to the enclosed circle energy distribution). Because the accumulative average is more weighted by the values in the previous radii, the average is less affected by adding a new deviant point and remains relatively flat for bright stars (see Figure 5(a)). We then used the accumulative average profile as the guide and computed the average flux ratio by retaining the points that are within ± 1σ of the accumulative average profile (blue dots in Figure 5). To reserve only the high S/N points, a threshold was also imposed when computing the accumulative average profile. The higher the threshold value, the smaller the number of points that are qualified to derive the average value. We have tried several different thresholds as given in Table 2. For bright sources, this threshold does not make any difference; furthermore, the average and median flux ratios are also indistinguishable within 1σ uncertainty.

Table 2. Derive Flux Ratio Relative to Sirius at Three S/N Thresholds Using Method 1

	S/N = 20				S/N = 30				S/N = 40
Name	N	$\bar{f}$	$\tilde{f}$	σf	N	$\bar{f}$	$\tilde{f}$	σf	N	$\bar{f}$	$\tilde{f}$	σf
Channel 1
β Gem	87	1.2040	1.2044	0.0058	87	1.2040	1.2044	0.0058	87	1.2040	1.2044	0.0058
Procyon	73	1.8563	1.8576	0.0120	73	1.8563	1.8576	0.0120	73	1.8563	1.8576	0.0120
HD 30652	71	23.737	23.738	0.188	68	23.745	23.739	0.186	67	23.749	23.739	0.185
HD 102870	38	28.760	28.775	0.159	38	28.760	28.775	0.159	38	28.760	28.775	0.159
HD 142860	42	39.504	39.542	0.306	42	39.504	39.542	0.306	42	39.504	39.542	0.306
HD 19373	38	39.686	39.680	0.451	37	39.706	39.680	0.440	36	39.704	39.680	0.446
HD 126660	27	46.939	47.257	0.807	26	47.023	47.258	0.691	25	47.103	47.258	0.572
HD 215648	27	49.541	49.549	0.476	27	49.541	49.549	0.476	27	49.541	49.549	0.476
HD 173667	37	57.637	57.861	0.920	31	57.867	57.886	0.694	29	57.939	57.938	0.658
HD 10476	26	72.061	72.187	0.774	26	72.061	72.187	0.774	25	72.014	72.175	0.751
HD 165459	19	1552.60	1544.38	18.11	18	1554.71	1552.71	16.06	17	1555.66	1552.71	16.02
Channel 2
βGem	62	1.2720	1.2721	0.0142	62	1.2720	1.2721	0.0142	61	1.2724	1.2721	0.0141
Procyon	77	1.8646	1.8661	0.0162	77	1.8646	1.8661	0.0162	75	1.8645	1.8661	0.0154
HD 30652	43	24.107	24.090	1.409	42	24.101	24.090	1.376	38	24.08	24.09	1.43
HD 102870	29	28.796	28.851	0.435	29	28.796	28.851	0.435	27	28.74	28.77	0.49
HD 142860	34	39.850	39.964	0.507	34	39.850	39.964	0.507	29	39.89	39.97	0.49
HD 19373	51	41.085	41.096	0.366	48	41.119	41.130	0.357	45	41.12	41.11	0.42
HD 126660	23	46.736	46.666	2.838	23	46.736	46.666	2.625	21	46.68	46.64	2.11
HD 215648	36	50.506	50.612	0.585	36	50.506	50.612	0.585	34	50.50	50.61	0.78
HD 173667	21	58.033	58.417	1.895	19	58.256	58.616	1.643	12	58.56	58.88	1.40
HD 10476	28	74.355	74.543	0.933	26	74.356	74.543	0.922	24	74.39	74.56	0.90
HD 165459	18	1556.25	1565.47	47.90	17	1566.77	1566.19	35.19	15	1565.34	1565.47	33.49

Note. N is the number of points used to derive the average ($\bar{f}$), median ($\tilde{f}$) and standard deviation (σ_f ) of the flux ratio.

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3.3.2. Method 2: Extending the Fitting Range by Adjusting the Small Background Offset

We now test the approach to gain confidence that we can use our 3.6 and 4.5 μm photometry as one of the steps to establish absolute flux calibration between Sirius and fainter stellar standards. We have used several approaches to check the accuracy of our photometry. The first one is to use the color information for stars with very similar spectral types. Our measurements in the two IRAC bands are independent: they are made with two separate optical trains and detectors and, of course, reduced separately. Thus, the uncertainty in the color difference between these bands reflects the photometric errors well. There are five stars that have the same spectral type (F5/6 V) as shown in Table 3. Of them, HD 215648 has an M dwarf companion 111 away (Raghavan et al. 2010) within the saturated PSF core, making the measured color more negative relative to Sirius than what it should be. We have assigned an approximate spectral type to this companion of M3V based on its K_S mag and estimated that its effect on the [3.6]−[4.5] color is 0.003 mag. Making this adjustment, the average [3.6]−[4.5] color for the five stars is −0.0080 ± 0.0034 mag, i.e., the rms scatter is 0.0067 mag. Another estimate of the expected color can be derived using synthetic photometry from the theoretical stellar models (BOSZ database of a F5V star and the corresponding CALSPEC spectrum of Sirius as in Bohlin et al. 2017; Mészáros et al. 2012). The synthetic color is −0.0010, consistent with our measurements. We can compare this with the tabulation of the similar WISE [W1]−[W2] color by Pecaut & Mamajek (2013), where we force zero color for A0V stars by subtracting the value for them. Averaging the value for F5V and F6V, we find a color of −0.0095. Pecaut & Mamajek (2013) estimate the errors in the WISE colors to be ∼1%. These results suggest that our derived photometry is probably accurate to within ∼0.006 mag per band, consistent with the uncertainty derived in the last section.

The second method of checking our result makes use of the V − K_S versus K_S − [3.6] color–color relation. We adopt the relation from Paper I where it is derived using high-quality V, K_S, and Spitzer 3.6 μm photometry as:

$\begin{eqnarray}\begin{array}{rcl}{K}_{{\rm{S}}}-[3.6] & = & -0.001228\ {x}^{4}+0.014382\ {x}^{3}\\ & & -0.042158{x}^{2}+0.057587x,\end{array}\end{eqnarray} \tag{ 1 }$

where x = V − K_S. This equation is a fit to 950 stars, each with photometric errors of only ∼2%, so nominally it has small errors in its midrange where we take the colors for this discussion. For the F-type (and the G0V) stars, the nominal errors are <0.002 mag. The dominant error may be due to the uncertainty in reddening, estimated in Paper I as only about 0.002 mag. In any case, the errors are small compared with 1%.

Using this relationship, we can then predict a star's [3.6] magnitude relative to Sirius. Note that this color relationship only applies to main-sequence stars and adopts a calibration system that references to Sirius (A0V star with V = K_S = −1.395 mag; Paper I). Since these bright stars are heavily saturated in 2MASS, the K_S magnitudes listed in Table 3 are derived and transferred from heritage infrared photometry by Paper I. ⁴ As shown in the last column of Table 3, the rms scatter in the residual is 0.010 mag among the 11 stars (and the average residual is zero). This value reflects the combination of the errors in the K_S value as well as that in the Spitzer photometry, so it suggests that the uncertainty in the latter ([3.6] measurements relative to Sirius) for individual stars is <1%. Given the excellent color agreement in our first test, a similar uncertainty is expected in the [4.5] measurements.

Yet another comparison uses the DIRBE 2.2 and 3.5 μm photometry from Smith et al. (2004) for Sirius, β Gem, and Procyon (the other stars we measured with Spitzer are too faint for accurate DIRBE photometry). We have made this comparison in two ways: (a) comparing the stars with DIRBE at 2.2 μm and correcting as in Equation (1) to IRAC [3.6] and (b) comparing the stars with DIRBE at 3.5 μm with no correction to IRAC [3.6]. For method (a) we have transformed the DIRBE measurement to 2MASS K_S. This was achieved by comparing transformed K_S and DIRBE photometry for more than 400 stars (see Paper I for the transformations and tests of their accuracy). The difference between the two K-band measurements is constant with little color dependence over the relevant range for Procyon and β Gem, so all that was required was to adjust the zero-point assumed for the DIRBE photometry. The predicted differences relative to Sirius are 0.2095 and 0.2110 mag respectively for β Gem and 0.664 and 0.670 mag respectively for Procyon, to be compared with the measured values of 0.202 and 0.671. If we average the two determinations for each star from DIRBE, the residual of observed minus prediction is −0.008 mag for β Gem and +0.004 mag for Procyon.

All of these tests using other sources of photometry indicate that our Spitzer PSF wing photometry is accurate to 1% or better.

Accurate photometry on the Rayleigh–Jeans portion of stellar spectral energy distributions (SEDs) is relatively insensitive to details of the physics of the stellar emission. This behavior is the foundation of the Infrared Flux Method (IRFM) for estimating stellar diameters and deriving an absolute calibration from stellar sources (Blackwell et al. 1979). Alternatively, for stars with very accurate diameter measurements through interferometry, infrared measurements become a sensitive test of stellar models.

It is beyond the scope of this paper to compare our results with detailed stellar models. However we can make a simple test by seeing if the fluxes scale in proportion to the diameter of a star squared (e.g., the solid angle (θ_LD) it subtends) times its temperature (T_eff) under the Rayleigh–Jeans assumption, as the very simplest relation physics would predict. Results for this very simplistic model are shown in Table 4 excluding β Gem (a K giant), HD 10476 (its later spectral type undermines this simple assumption) and HD 165459 (too faint to have direct diameter measurements). It works remarkably well, with flux ratios at IRAC [3.6] for all of the stars relative to Sirius within ∼2% of the predictions of the model, and most of them consistent with the model within the errors from propagation of the uncertainties in stellar temperatures and the diameters. Bohlin et al. (2017) have found that modern models of the spectrum of Sirius are consistent with the diameter determined by Davis et al. (2011; and used in our comparison) within the errors, i.e., to within ∼1%. Combined with the results in Table 4, the simplistic Rayleigh–Jeans model appears to be surprisingly accurate.