TPF-I SWG Report - Exoplanet Exploration Program - NASA

D ESIGN AND A R C H I T E C T U R E T RADE S TUDIES
4.9.2 Point-Spread Function
The point-spread function (PSF) described here is not the instantaneous response of the array on the sky.
Rather, it is correlation map of the sky obtained from a point source after synthesis of all the observations
(see Section 4.1). It is the analog of the PSF for an optical telescope, or the dirty beam of interferometric
synthesis imaging. The PSF for a phased array is more complex than that of a standard imaging
interferometer (which makes measurements from only pair-wise combinations of collectors). A full
analysis that derives the structure of the PSF can be found in Lay (2005).
The basic features of the PSF for the X-array (2:1 aspect ratio) are depicted in Figure 4-23. This is the
broad-band version of Figure 4-4, obtained by co-adding the PSFs for several spectral channels. The main
peak is centered on the cross that denotes the actual location of the point source. This main peak is
flanked by four satellite peaks, each of which has a set of sidelobes. Note that the sidelobes are
suppressed in the broad-band PSF relative to the monochromatic case, but the satellite peaks are not. The
phase chopping described in Section 4.1 results in an inverted anti-Sun image. The number, amplitude,
and location of the satellite peaks relative to the main peak are determined by the shape of the nulling
configuration. We define the angular resolution of the array to be the FWHM of the main peak, which can
be predicted from the average length of the imaging baselines in the configuration:
λ
θ ≈ 0.48 av , (3)
res
B img
where the average wavelength was taken to be 12 μm. This relationship was used to determine the size of
the array needed to observe a given star (see next section). In addition, the rms level of the satellite and
sidelobe structure in the PSF was used as a metric to assess ease of deconvolution.
4.9.3 Array Size Criterion
Combining Eqs. (2) and (3) we obtain a requirement on the average length of the imaging baselines:
λav
Bimg
≥ 1.6 . (4)
θMHZ
The criterion for inner working angle is
λav
Bnull
≥ 0.9 . (5)
θMHZ
For most configurations, it is Eq. (4) that drives the size of the array needed to observe a given target. The
big advantage of the X-array over other configurations is that the length of the imaging baselines is
decoupled from the length of the nulling baselines; the long dimension of the array is stretched to meet or
exceed the angular resolution requirement of Eq. (4), while the short dimension is held to just meet the
inner working angle requirement of Eq. (5). For the other configurations in Fig. 4-8, the array size needed
to meet Eq. (4) leads to null baseline lengths that greatly exceed the minimum requirement in Eq. (5) and
incur a large penalty from increased stellar leakage.
4.9.4 Deconvolution Approaches
The ability to extract the information about the planet from the reflected light from a planet in orbit in the
habitable zone of a star up to 30 pc away is crucial to the design and architecture of the TPF
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