Diagnosing Space Telescope Misalignment and Jitter Using Stellar Images

The requirements for a weak-lensing survey can be most simply stated as limits on the tolerable error in the second moments of the PSF. We will measure the Gaussian-weighted moments defined as the zeroth moment

the higher moments

where n = 1, 2... labels the order of the moments. The two first moments and correspond to f₁(x,y) = x and y, respectively. The three second moments P_xx, P_xy, and P_yy correspond to , , and , respectively. In analogy, we can write down the third moments of the PSF, P_xxx, P_xxy, P_xyy, and P_yyy. These are calculated under a weight function which we take to be

The width of the weighting Gaussian is chosen to be 2 times the Airy radius,

where λ is the wavelength of the incident light, f is the focal length, and D is the telescope aperture.

We also compute quantities derived from the second moments. The ellipticities and stellar size are

These quantities appear in many approaches to weak-lensing shear measurement (Heymans et al. 2006 and references therein). The true PSF of an exposure will depend on the field position of the star (x_⋆,y_⋆) yielding e₁(x_⋆,y_⋆), e₂(x_⋆,y_⋆), . Our goal is to produce an accurate model estimate , etc. We will evaluate our success by calculating the rms residual errors in the ellipticity models,

and the fractional residual error in the PSF size,

If the physical state of the telescope can be described by a small number of time-variable parameters {p_i}, then the PSF is some function PSF(x,y|p_i,x_⋆,y_⋆) of focal-plane position, telescope state, and the field position of the star. We note that a great advantage of space-based observatories for WL work is that the stability of their environment allows us to engineer a telescope for which only a small number of degrees of freedom will vary significantly. For the SNAP telescope, the engineering specifications are that all optical systems are stable to well below the WL specification, except for the following:

• 1.  
  The alignment of the secondary mirror, which may vary due to imperfect performance of the feedback system that stabilizes the temperature of the mirror support structure;
• 2.  
  The telescope line of sight (LOS) may vary during an exposure and smear the PSF due to noise and the finite bandwidth of the attitude control system (ACS) or due to high-frequency reaction-wheel vibrations that transfer through the structure to optical elements, particularly the secondary mirror.

In § 3 we describe in detail the model we adopt for these disturbances.

Ground-based telescopes pose a more difficult challenge for PSF modeling, because they have a very large number of time-varying degrees of freedom. Indeed the atmospheric distortions have infinite degrees of freedom, formally. Our analysis thus cannot be considered valid for ground-based observatories. Jarvis & Jain (2004) propose instead that a principal-components analysis be performed on the ensemble of PSF patterns observed by the telescope, so that the coefficients of some finite number of principal components become the parameters for the PSF model. Jain et al. (2006) discuss how changes in Seidel aberrations would be manifested as PSF-change patterns, and might serve as a parameter set for PSF modeling. Jarvis et al. (2008) carry out the analysis and show that the physical model describes a substantial part of the PSF size and anisotropy over the field of view. The success of these methods will depend upon how well-behaved the telescope and atmosphere are, particularly whether the optically significant perturbations are described by a small number of variables. An alternative approach, applied by Wittman (2005) to PSF ellipticities induced by the atmosphere, is to determine by a priori analysis that the disturbance will be below the WL threshold.

We simulate the following strategy:

• 1.  
  Locate the stellar images in each exposure.
• 2.  
  Measure PSF quantities at each star location; in our case, the second and third Gaussian-weighted moments.
• 3.  
  Find the PSF parameters that best reproduce the stellar data.
• 4.  
  Use the and the PSF model to derive the desired , etc., at any location in the focal plane.

In the simulation we can then evaluate the rms residual errors of the PSF model.

In the simple case where the PSF does not depend on the field position of stars, we can consider this procedure to be, essentially, averaging the measured PSFs (and moments) of the observed stars. In this case, we expected the error on the PSF moments to be determined by the (quadrature) sum of the signal-to-noise (S/N) levels of all the available stars. In particular, if all the stars are dominated by Poisson noise from source photons, then , where N_γ is the total number of photons collected from all usable stars in the exposure. We can therefore expect that

where α is some coefficient of order unity ( for a Gaussian PSF).

More realistically, the PSF does depend on the field position of stars. In this case, we are using the PSF parametric model as a means of interpolating between the stellar positions. If the parameters are not too numerous, and cause readily distinguished PSF patterns on the focal plane, then we expect equation (9) to continue to hold. Our simulation will show that this is indeed the case for the SNAP design, and we will aim to estimate the coefficient α.

Paulin-Henriksson et al. (2007) instead consider the PSF to be locally constant, and ask how large a region will contain enough stars to adequately constrain this locally constant PSF. They then consider this region size to be smallest on which WL observations can be successful. In reality both the time-varying PSF contamination and the WL signal will have nontrivial angular power spectra and we have to compare the bandpowers of each in setting our specifications. Stabenau et al. (2007) investigate how PSF time variation will translate into multipole patterns for a SNAP-like sky-scan strategy. In this paper we will simply calculate the rms residual PSF errors and note that they have characteristic angular scales similar to the telescope field of view.

We adopt standard scalar diffraction theory to evaluate the optical contribution to the PSF. The wavefront on the focal plane {x,y} generated by a point source is

Here is the coordinate on the entrance pupil with components ξ and η, C is an uninteresting constant, P(ξ,η) is the entrance pupil function, k = 2π/λ where λ is the wavelength of the band-limited optical light used to form the image, and f is the effective focal length of the telescope optics. is the optical path difference caused by the lens/mirrors system, which we expand using Zernike polynomials (Noll 1976) as basis. The second exponential describes the phase differences caused by the off-axis incident light ray in the direction and the third exponential is the phase differences caused by the different distance light has to travel beyond the lens or mirrors and reach the focal plane. The optical point-spread function is

Figure 1 shows the pupil function of SNAP telescope.

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The OPD map of the perfectly aligned telescope and the derivatives with respect to misalignment parameters are calculated using ray tracing through the telescope's optical system. The OPD is projected onto a Zernike basis, with results shown in Table 1.

where Z_n is the n th Zernike polynomial, D_m is the m th misalignment parameter, is the OPD's Zernike coefficients for pristine telescope, and is the Zernike coefficients of .

In the SNAP telescope design (Lampton 2002; Sholl et al. 2004), the secondary-mirror position is expected to be the only optical dimension to vary significantly with time due to thermal drift (Sholl et al. 2005; Stabenau et al. 2007. The secondary mirror has 5 degrees of freedom (DOF) that include shifts D_x and D_y in transverse directions, the defocus D_z, plus tilts around the x axis (T_x) and y axis (T_y).

In practice we find that is a very weak function of field for these parameters, and hence for the current simulation we take to be constants. The pristine-telescope Zernike coefficients retain field dependence, but the axisymmetry of telescope design reduces the freedom to the radial direction.

The optical PSF must be convolved with the charge-diffusion pattern of the CCD detector. Charge diffusion is modeled as a Gaussian with fixed charge-diffusion length σ_d = 4 μm. If the charge-diffusion length were free to vary, it would be degenerate with an isotropic telescope jitter (see § 3.4). The PSF after charge diffusion is

where PSF₀ is the optical PSF. We execute the convolution with fast Fourier transforms (FFTs).

Guiding errors and mirror vibrations, also known as jitter, alter the effective PSF of a finite-length exposure. Each exposure hence has a unique PSF map, even if the optics are otherwise stable. If the observatory is free to rotate on all three axes, as for a space-borne observatory or an altitude-azimuth terrestrial telescope, then the effect of jitter varies across the field of view and is not a simple convolution of the image with a fixed kernel. Stellar images in the exposure can be used to infer the full field dependence of the jitter on the PSF. We demonstrate here that as few as two stars are sufficient to fully reconstruct the jittered PSFs, as long as the jitter amplitude is much less than the width of the PSF.

3.4.1. Effect of Jitter on the PSF

Assume that the modulation transfer function (MTF; the Fourier transform of the PSF) at x = (x,y) from the optic axis is known to be T₀(k) in the absence of telescope jitter. If the jitter has displaced the stellar image by some amount Δx = (Δx,Δy) —which varies in time—then the transfer function becomes

The PSF for stellar images in the integrated exposure is the time-averaged value

If k[Δx] ≪ 1, then the exponential can be approximated by a Taylor expansion:

The effect of the jitter on the PSF is then fully described by the mean displacement 〈Δx〉 and by the covariance matrix C_Δx = 〈ΔxΔx^T〉. Further detail of the jitter history is irrelevant. The linear term is simply a displacement of the entire PSF, and the quadratic term describes a convolution of the jitter-free PSF with a very narrow jitter kernel. With a telescope of diameter D, focal length f, and wavelength λ, physical optics forces MTF = 0 for k < 2D/(λf). Therefore the Taylor expansion is valid if

in other words the jitter must be much less than the size of the Airy disk.

3.4.2. Field Dependence of the Jitter MTF

If the observatory axis is displaced by angles θ  = (θ_x,θ_y,θ_z), i.e., pitch, yaw, and roll, then the displacement of the image of star i at (x_i,y_i) is

If the roll is nonzero, then the image displacement is field dependent. Now considering the observatory misalignment (jitter) to be a function of time,

In the small-jitter limit, therefore, we find that the effect of the jitter on the PSF at every point in the field of view is fully described by the six independent elements of the jitter covariance matrix C_θ = 〈θθ ^T〉. We can write

In practice, therefore, if we have an exposure for which the jitter-free PSF pattern is well determined, then we can completely determine the jittered PSF anywhere in the focal-plane by knowing the elements of C_θ. The jitter covariance matrix could be determined from perfect knowledge of the PSF of any two stars in the exposure. If there are a larger number of stars in the exposure, then the six jitter covariances are highly over determined, hence they are easily derived for every exposure, even if there are other degrees of freedom in T₀ that must be determined from these stars.

If the jitter amplitude is not small, then there can be a much larger number of moments of the jitter history that are important, and a finite number of stellar images may not in general recover full knowledge of the effect of jitter on the PSF over the field. We will defer consideration of this limit for another paper.

We analyze a fiducial case in which the secondary mirror is translated 1 μm and rotated 1 μrad from its correct position. This would be considered a very large error for the optomechanical system. We have verified that the choice of fiducial model does not influence the rms residuals to the fit.

When analyzing jitter, we assume an rms motion of 36 mas in pitch and yaw with 700 mas rms in roll (as expected from SNAP telescope design). The rms cross-correlation between axes is taken to be small by comparison. Again, these do not strongly affect the results.

We use the star counts from the COSMOS HST survey as representative of high galactic latitude fields. The star counts in the COSMOS field (Robin et al. 2006) are well fit by (see Fig. 2)

where m is stellar magnitude in the F814W band.

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If the star is too bright, it would saturate the SNAP CCDs. We hence conservatively assume that only stars with 19 < m < 23 will be used for morphometry. The bright limit roughly corresponds to 50,000 photons for a 300 s exposure. The faint limit corresponds to ≈1200 photons per star, which is comparable to the sky background in the exposure. Fainter stars will contribute little to the PSF knowledge.

The instrumented SNAP focal-plane area is ≈0.7 deg². This places approximately 2100 measurable stars on the focal plane, with a total photon count of N_γ ≈ 2.3 × 10⁷ per 300 s exposure. This suggests that an ideal morphometry process would yield e^rms, , well below the required weak-lensing specification as discussed in § 1.

We generate a sample of 2100 stars with random positions across the focal plane. The magnitude of the stars is randomly generated according to the magnitude distribution of equation (24). The PSF, including optical distortions and charge diffusion, is computed and used to determine the mean number of photons detected in the CCD pixels. The number of detected photons per pixel is drawn from a Poisson distribution, and the quantum efficiency and gain of the SNAP pixels are used to compute pixel values. Random dark noise of 5 photo-electrons (as expected in the SNAP CCDs) is added to each pixel. The PSF moments defined in equations (1) and (2) and equation (5) are then computed from the pixel values.

For each available star in an exposure, we calculate the second and third moments of a PSF image to which shot noise, sky noise, and read noise have been added. With the resulting PSF moments as data, we search for the best fit misalignment parameters by minimizing χ²

where N_⋆ is the total number of stars, N_mom is the number of independent PSF moments per star (N_mom = 3 if only second moments are used), M is star moment (with noise), is star moment calculated from the model (no noise), and σ_moment is the rms of star moments. In general, σ_moment depends on star magnitude, filter band, and the position of the star on the focal plane. We assume a fixed filter band and neglect the (< 10%) dependence on position. We produce a lookup table of σ_moment versus source magnitude by Monte-Carlo methods before doing the fit.

The PSF moments depend nonlinearly on misalignment parameters. As an example, Figure 3 shows the dependence of P_xx on the defocus parameter D_z. The model-fitting procedure is hence nonlinear, so slower than a linear χ² fit.

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5.1.1. Using Second Moments of PSFs

We first simulate a single exposure of SNAP, using only the PSF second moments to constrain the secondary-mirror misalignment. The fiducial misalignment parameters, their fitted values, and the 1 σ uncertainties are listed in Table 2. D_z is precise to < 10 nm which is well below the achievable mechanical stability. Hence the morphometric information can greatly improve knowledge of defocus. The other parameters are as much as 20 times less precise: accuracies of ≈0.2 μm in D_x and D_y seem quite disappointing, for example.

To better understand this wide range of precisions, we examine the eigenvectors and eigenvalues of the parameter covariance matrix, shown in Table 3. The best-constrained eigenvector is along the D_z direction, i.e., defocus. Two eigenvectors are ≈80 times more poorly constrained, but this is not hard to understand. If the secondary mirror were spherical with radius R, it would have only three degrees of freedom, as misalignments with D_x = -RT_y and D_y = -RT_x leave the sphere invariant. A nonspherical secondary mirror breaks these degeneracies and results in finite (but poor) constraints on these eigenvectors.

The "Residual" row in the table gives the projection of the fitting error onto the eigenvectors. It shows that the fitted parameters deviate from the fiducial mostly in the directions that have weak constraints. All the deviations are at or below 1.6 σ level, which is a sign that the fitter is working properly.

The quantity that matters to weak lensing is the precision of the PSF knowledge. The apparent poor fit to some of the misalignments reflects the insensitivity of the PSF moments to certain combinations. This insensitivity also means, however, that poor knowledge of these eigenvectors has little adverse effect on our PSF model. We compare the noiseless PSF ellipticities and size generated using the fitted misalignment parameters with these generated using fiducial misalignment parameters. Figure 4 (left panel) shows the distributions of the residual PSF moments for a realization of a single exposure. We see that the PSF errors are well below the 10^-3 target for all three quantities. The spread in PSF errors reflects the variation across the focal plane.

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5.1.2. Including Higher Moments of PSFs

Table 4 shows the results of fitting secondary misalignment and jitter parameters jointly. Adding the 6 jitter degrees of freedom to the model roughly doubles the uncertainties on the 5 misalignment degrees of freedom. The parameter D_z is still very precise (at 20 nm level). The jitter parameters are determined without significant degeneracies among each other nor with the misalignments; this is because these parameters influence the PSF moments with rather distinct dependences on field angle.

The residual PSF ellipticities and sizes for one realization of joint misalignment/jitter fitting are shown in Figure 5.

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As mentioned before, the spreads of the residual PSF e₁, e₂, and σ_⋆ distributions in this single realization are due to field dependence. Different realizations of the data (star locations and random noise change) produce different mean and spread of the residual moments distribution. Figure 6 (left panel) shows the distributions of e^rms and from 50 realizations. We produce a single measure of the efficacy of the morphometry procedure by averaging the rms PSF residuals over all focal-plane positions in many realizations, as per equations (6)–(8). The mean values (in quadrature) of the distributions in Figure 6 (left panel) are exactly those. They are labeled by the arrows in the plot and listed in Table 5 as well. Calculated using equation (9), the α values are also tabulated in Table 5. Taking the average value, we find α ≈ 1.8. As a result, we have

The ensemble average residuals are consistent with zero; i.e., the PSF models are unbiased, to the 2.5 × 10^-5 accuracy of our 50 realizations.

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To test our hypothesis that the PSF errors will scale as , we repeat the simulation using 100 stars instead of the fiducial 2100 stars. The distributions of e^rms and are shown in the right panel of Figure 6. Again, the quadrature means of these distributions are labeled by the arrows in the plot and listed in Table 5. We find α ≈ 1.7: so the rms residuals do indeed scale as expected.

From Figure 5 it is clear that part of the residual errors in the PSF ellipticities are from a shift in the mean across the field of view, while the rest is from errors that vary across the field of view. The two different types of PSF modeling errors will propagate into different angular scales in the WL power spectrum. For the SNAP simulation, we find that roughly half of the e modeling variance is in the mean across the field of view.

Because essentially all the residual variance arises from shot noise in the stars, it will be uncorrelated from exposure to exposure.